Physics and Chance: Philosophical Issues in the Foundations of Statistical Mechanics

PHILOSOPHICAL ISSUES IN THE FOUNDATIONS OF 5 J\.TISTICAL MECHANICS

Lawrence Sklar

Philosophical issues in the foundations of statistica mec anics

LAWRENCE SKLAR

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Lawrence. Physics and chance : philosophical issues in the foundations of statistIcal mecnamcs / Lawrence ~K1ar. p. cm. Includes bibliographical references and index. ISBN 0-521-44055-6 1. Statistical mechanics. 2. Phvsics - Philosoohv. 1. Title. QC174.8.S55 92-46215 1993 CIP 530.1'3 - dc20 ~K1ar,

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Contents

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Preface

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Philosophy and the foundations of physics The structure ot this book l. Probability 2. Statistical explanation 2 0/'

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Gibbs' statistical mechanics

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Statistical explanation I. Philosophers on explanation 1

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3. The KAM Theorem and the limits of ergodieity Ergodieity and t e rationalization of equilibrium statistical mechanics 1. Ensemble probabilities, time probabilities, and

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measure zero pro em

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the non-equilibrium theory

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References Index

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.

The aim of this work is to continue the ex loration into the foundational questions on the physical theory that underpins our general theory of siderations into the fundamentals of our physical description of the world. a p YSlca eory lS s a lS lca mec nlCS. The history of this foundational quest is a long one. It begins with an intense examination of the remises of the theo at the hands of ames Clerk Maxwell, Ludwig Boltzmann, and their brilliant critics. It continues to the philosophical community. And the quest has persisted as a set o cu concep a a enges, ln a s y rna e ever ric er y e development of ever more sophisticated technical resources with which to treat the roblems. I ho e that this book will encoura e others in the philosophical community to join with those in physics who continue to

has often guided me to the crucial questions to be addressed. James Jo ce and Robert Batterman, as students and as collea ues, have been enormously helpful to me in my thinking about these issues. I have Frank Artzenius, and Abner Shimony. manuscript of this book. I am also grateful to Terence Moore of Cambrid e Universi Press, to the two referees for the Press for their help in bringing the book to publication, and to Ronald Cohen for his editorial The research contained in this book has been supported by a number fully acknowledged here.

1

I. Phlloso h

and the foundations of hies

There are four fundamental theories that constitute, at present, the foundational illars of our h sical theo of the world: eneral relativi , quantum mechanics, the theory of elementary particles, and statistical of these fundamental theories presents its own budget of scientific and would examine the so-called foundational issues in these areas vary in a marked and interesting way from t eory to t eory. General relativity - at present the most plausible theory of the structaken to include the theory of gravitation - is in many ways the most remain: Should we accept general relativity, or some alternative to it like the Dicke-Brans scalar-tensor t eory? Are t ere genera izations 0 t e theory that might encompass other forms of interaction over and above ?

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tions of causal "niceness," for example? But we are, at least, clear about of the theory is its totally classical nature, and all expect that some day we wi ve a new quant1ze t eory to ta e its pace. ut at east we know what theory we are talking about when we begin to explore the Philosophically, too, the situation is particularly clean in the case of

logy, and so on have been discussed long enough and hard enough by

2

Physics and chance

the answers, ar least there is a general consensus about the right questions to ask. uantum mechanics is a ain a theo whose scienti c claims are at least fairly clear. The basic structure of the fundamental theory is avail-

what the theory should be. Just as general relativity runs into the fundamental problem that it is not a quantized theory, leaving us to speculate about 'ust what our ultimate uantized s ace-time theo will be so quantum mechanics runs into difficulties when one tries to extend it to

fields, whether in the vein of Wightmannian axiomatic theories, theories of scattering of t e LSZ type, S-matrix t eories, or somet . g else. Philoso hicall uantum mechanics like eneral relativi resents us with a now well-defined set of foundational questions to be answered.

in the orthodox theory. This understanding will require the resolution of suc su si iary pro ems as wave-partic e ua ity, e non-causa interde endence of se arated s stems the nature of uncertain relations the existence or non-existence of hidden variables, and so on.

what the questions are that need to be answered. In the latter case we at east ow qUlte e nlt1ve y w at some POSSl e answers are, even we aren't sure which are ri ht and which are wron . In the uantum case I think many will agree that most (perhaps all) of the answers themselves

understand much more clearly what they are saying. e eory 0 e ementary partlc es agam as ltS own speCla avor as an area for foundational research. Here we are concerned with the a eold question of the fundamental constitution of matter. This theory presents

Introduction

3

particles by, perhaps, gluon forces, But just why there are the structures and forces there are, and just what fundamental laws govern the comosition of macro-ob'ects out of their micro-constituents remains something of a mystery to us, despite the recent progress of theories such as

theory also remain less well characterized and far less well explored than those concerne with space-time theories an quantum mechanics, Althou h some attention has been aid to such uestions as wh ex lanation so frequently consists in reducing macro-objects to their microscopic

theories devoted solely to accounting for the thermal behavior of macroSCOPIC matter, t ese ranc es 0 p YSICS now stan eSl e e 0 er three as fundamental com onents of our scientific world-view, Independent of the other theories, they are an indispensable supplement

to move from the fundamental theory governing the behavior of the mlcroscoplC constituents 0 matter an t e t eory 0 e constitutlon 0 macroscopic ob'ects out of their microscopic parts to a full explanation of the macroscopic behavior in terms of the microscopic constitution

distribution of radiation and the laws of particle scattering are among the e wor w ose escnp lon an exp ana lon requITes vanous aspects 0 the application of thermodynamics and statistical mechanics, Originally formulated to provide the general laws governing the thermal

4

Physics and chance

world ranging from the distribution of elementary particles upon interaction to the distribution of stars interacting by gravity in massive clusters can all be sub'ected to the thermod namic view oint. Thermodynamics has a peculiar place when viewed in the context of

all physical phenomena are to be described. This framework is modified and generalized by general relativity in the curious revolutionary shift due to Einstein which makes the s atio-tem oral arena of henomena a dYnamic and causal participant in events and assimilates the gravitational

quantum terms. In addition to these two fundamental theories (theories not yet proper y reconCI e to eac 0 er we nee t e e -t eoretic rinci les that describe the s ecific nature of material existents in their most primitive and elementary form - that is, the field theory of the

discipline such as thermodYnamics. If we consider the dYnamic evolue c aractenze y t e t eones note , tlon 0 t e wor as It wou there seems little lace for additional fundamental lawlike structure. We can describe any system at a time by its quantum state, and project its

by the general quantum theory, and the specific rules of dYnamical intere e ementary partlc es. at e se action prOVI e y t e e t eory 0 is there to do? That there is another, fundamental, level of descri tion that is vital and fruitful - a level involving such concepts as equilibrium,

apparently radically different fundamental natures, is the surprising truth o ermo ynamlcs. Many no longer think of thermodYnamics as an autonomous science, however. Since the middle of the nineteenth century there has been a

Introduction

5

of internal motion of microscopic parts of matter, progressing through the development of the kinetic theory of gases in the nineteenth century, and evolvin into the com lex and multi-faceted disci line of statistical mechanics, there has been a continuous program designed to show us

Part of the motivation of at least some of the discoverers and developers of statistica mechanics has een the hope that the eeper un erstandin of thermod namic henomena rovided b this new a roach would eliminate the need to invoke thermodynamic-like principles of the

are reducible to, and intelligible in terms of, the features posited by the ot er n amenta t eories 0 t e wor an e aw i e nature 0 t ose features as these other fundamental theories describe them. This ho e for the elimination of thermodynamic principles as autonomous elements

theses that remain underivable from the fundamental laws of kinematics or ynamlCS. ese are now ta en to e 0 a pro a 1IStlC or statlstlca nature. Just what these seemingly autonomous principles of statistical mech-

themselves be derived from the fundamental laws of general kinematical an ynamlca t eones IS arge y e su Ject matter 0 IS 00. Unlike the relatively closed disciplines of general relativity and uantum mechanics, statistical mechanics presents us not with a single, well-

aspects of the theory are well understood and universally accepted, such crucla areas as e correc approac 0 e In ro uc 10n In 0 e eory of irreversibility and the approach to equilibrium and the proper statistical mechanical definition of entropy are the subject of intense and

we find the same fundamental issues at the foundational level arousing con roversy among con emporary eoris s as arouse con oversy m e early years of the theory. Not that no progress has been made nor that the issues are as unclear now as they were then. But it is still true that h r r n h

6

Physics and chance

concerned with an application of the conceptually clarifying techniques

sophers than either our theory of space-time or quantum mechanics. Not that foundational issues in statistical mechanics are ignored, of course, for there exists alar e rou of h sicists who devote substantial effort to the task of clarifying this theory at its most fundamental level. But

attack on the so-called problem of the direction of time, philosophers have generally kept their distance from the conceptual perplexities of statistical mechanics. I believe that some of this reticence can be accounted for by the very

But in the case of

task. 1S context es to a ope to brin to ether in one lace a sufficient! e1ementa ,com rehensive, and organized survey of the fundamental physical and philosophical

would like to understand what the crucial debates ope a 1S wor W1 e su C1en y compre ciently accessible, that it will encourage some to detailed, technical works of the physicists that

are all about. Further, enS1ve, an yet su proceed to the more are more difficult of

Introduction

7

upon it. I doubt if I shall here resolve the major conceptual questions to everyone's satisfaction. On some uestions I will offer m own a raisal of the answers and my own conclusions about the right direction in which to

other. I will consider my ambitions amply fulfilled if this work can stimulate the same application of energy an ta ent in attac . g the ndamenta roblems I will surve and ex ound as is resent! devoted to the fundamental philosophical questions in space-time theories and quantum

think of and that might bear fruit under a kind of extensive investigation that is not possi e or t em ere. D. The structure of this book

which we will be concerned. As is not uncommon in philosophy, these orm a rat er ense networ 0 Issues, so t at reso vlng one questIon seems to resu ose fior resolution of all the others. To some extent then, the sorting out of issues and the order in which they will be treated

exploration of the substantive issues will follow a preliminary outline of e IStOry 0 ermo ynamlcs an statlstlca mec anlcs. IS survey 0 the hi h oints in the develo ment of these fields is essential to introduce to the reader the main problems and their overall context. 1. Probability

8

Physics and chance

a mea ure 0 su jec ive gr e 0 e ie. om e concep a issues concerning the role of probability in statistical mechanics, we shall see, hinge on a prior understanding of probability from a general philosophical h r wh in the particular theories we are examining will vary with our conception

role of probability in the particular context of statistical mechanics that are rather more specific to the very special function played by probabilities in this articular realm of h sics. In other words even havin ado ted some general philosophical stance, there will be additional conceptual

I do not expect to resolve to general philosophical satisfaction the issue of the correct interpretation of probabilistic assertions. I will lay out briefl what some of the ma'or alternatives are and what some difficulties with them are taken to be. I will opt for one interpretation as most

sophical questions I certainly don't have available. Attention will be paid, owever, to t e pro ems more spec' c to t e ro e 0 pro a i ity in statistical mechanics because these have been rather ne lected in the literature in comparison with the broader, and admittedly more funda-

Scientific theories are su osed to ex lain what ha ens in the world. The features of the world that statistical mechanics sets out to explain

existence of a small set of macroscopic parameters sufficient to charactenze equll num an a somew at arger set su Clent to c aractenze some a roaches to e uilibrium, the lawlike inter-relationshi s amon these macroscopic parameters - these are the sorts of things to be ac-

statistical and probabilistic assertions postulated by it. What is the nature o an exp anatlon a res s upon a s a lS ca or pro a llS lC assump 10n. Again, we have available to us a rather wide variety of accounts in methodological philosophy of science as to just what statistical explana-

Introduction

9

once again it will be seen that statistical explanation as it functions in statistical mechanics has its own idiosyncratic features that deserve special attention in their own ri ht. We shall see that man of the erennial debates in the foundations of statistical mechanics rest upon assumptions

debates even if not resolving them.

calculate certain quantities. Associating these quantities with macroscopica y measure parameters we 0 er an exp anation 0 the equili rium ro erties of the s stem. But what justified our choice of fundamental statistical assumptions?

the appropriate one to choose to associate with the macroscopic quantity? ere 1S an attempt to 0 er at east partia answers to some 0 ese uestions that relies heavil on certain features characteristic of the laws governing the micro-constituents of the macroscopic system. This is so-

laws governing its micro-constituents and other features that are the result, ra er,o e vast num er 0 m1cro-constituents at rna e up a macros stem. I shall t ,b disentan lin a number of distinct uestions that tend to get muddled together in the literature, to make clearer just how

line will be that there are a number of quite distinct questions one can as an at some resu ts are answers to some questions ut not to others. Putting this simply, the response is likely to be, "of course that is true," but, as we shall see, failure to make the fine distinctions here has

The study of non-equilibrium in statistical mechanics presents quite a different set of scientific problems than those that arise in the equilibrium

10

Physics and chance

cally, in this case from the equilibrium case. In particular, the relationship of the generalizations to the underlying dynamical laws governing the micro-constituents is uite different in the two cases. The roles of initial conditions and laws in the explanations, the modes

5. Cosmology and statistical mechanics Since Boltzmann, it has been frequently alleged that a full understanding

Recent work on observational and theoretical cosmology has made the interre ationship 0 cosmo ogica an statistica mechanica features of the world a vital area of scientific ex loration. The aim of this section will be to concisely present the present state

philosophically. Once again, what I hope to show is that there are a num er 0 quite istinct questions to e answere ,questions at can easil be confused with one another. When these are disentan led from one another, the earlier exposition of statistical explanation will allow us

concerning the origin of irreversibility.

for in the theory of thermodYnamics. The alleged relationship of thermoYnam1cs to statlstlca mec an1CS, ten, 1S t at 0 one t eory t at as been reduced to another. We will examine a number of standard hilosophical accounts of the reductive relationship and seek to place the

Introduction

11

outstanding general methodological issues. Indeed, in some cases there will have to be a simple dogmatic adoption of one plausible position amon others. Rather I want to focus on the roblems that are s ecial to this particular reduction, maintaining, as I will, that once again the

has very special features vis-a.-vis other physical theories. Its descriptive concepts have peculiarities not share in general y theoretical concepts of h sics. And certainl this will be true of the conce ts of statistical mechanics as well. Naturally, then, the interrelation of these two theories

7. The direction of time

Most of the attention that has been paid by philosophers to the founda-

The issue has been explored rather as a means of providing resources use m lscussmg t e so-ca e pro em 0 t e rrectlon 0 tIme. ere, the resources of statistical mechanics have been invoked in order to offer a reductionistic account of the very notion of past-future asymmetry it-

dYnamics and, allegedly, explained by statistical mechanics, that grounds our very notIon 0 e lstmctlon etween past an ture. e I ea ere is present in Boltzmann's work, and it has received its most detailed exegesis by Reichenbach.

ible to the entropic aSYffiffietry of the world in time. Failure to clearly un ers n JUs w a tn 0 re uc Ion one oug 0 ave In mm ere nas led, I believe, to some confusion in the literature. I will then explore, to some degree, the plausibility of the reductionist program. It is to be

difficult problem than has previously been gained.

12

Physics and chance

ence to, and exposition of, a substantial amount of physics, the history of science, and philosophy. In order to keep the work within reasonable bounds man fascinatin conce tual roblems in the foundations of statistical mechanics will simply have to be ignored altogether or, at best,

of phase changes. The scientifically knowledgeable rea er will be especially struck by the overall focus in this book on classical statistical mechanics and the apparent neglect of quantum statistical mechanics. Because we know

what is the point in directing our philosophical attention to a version of the theory we now to e incorrect an outmo e ? The reason is clear. It is that the articular conce tual roblems on which we focus - the origin and rationale of probability distribution assumptions over initial

the theory is essential. For example, quantum ensembles of finite systerns cannot escape recurrence resu ts as can c aSSlca ensem es 0 nlte s stems. And, some aIle e, onl b reference to uantum limitations can the limits on initial ensembles be correctly delineated. Where one

varieties of classical statistical mechanics as the objects for conceptual exp oration. is way 0 olng t figs is not 1 losYncratic, ut common in the physics literature devoted to foundational issues. Whether things can be carried off profitably in this way can only be determined by

in the problems treated but who lack an extensive back-

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mathematical physics. Naturally this approach will frequently make the treatment of both ohvsics and ohilosoohv skeletal. Some of the difficultv incurred bv this approach will be remedied, I hope, by the bibliographical information .....: ........:! ...................... .:.I .:.I ........ 1-...... 'rJ... .... ................... .......... ...."" . . "" .................J... -r ... .lVC. .... . 1 ua VC • • .Vll;;; VI" ..l

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works that I have found most useful in laying out the essential philosophical and physical issues in a clear way. The bibliographies are annotated in the hope that they will provide a road map into the literature for those who would like to pursue the material covered in this work . 'c• ,1 ..J...L:> 1-..... 11 ...... ............. .. , "".... , ~

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ing out in greater depth the exposition of philosophical positions expounded and criticized here. Given the lack of coherence in the physics, noted earlier. this exPlicit 2uide to the literature is. I believe. a particularly essential part of this work. ....

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physical theory is marked with an asterisk (*), Material demanding comprehension ot more advanced mathematics and phySICS IS marked with two asterisks (**).

2

selective historical survey of the development of thermodynamics, kinetic prehensive. Nor is the material presented in a manner that would suit historians of science. Neither issues of chronolo and attribution nor the far more important questions of placing the specific scientific results in goal is merely to present some important developments in the history of will facilitate our later conceptual exploration. Many of the scient' c resu ts note in this chapter wi 0 y e mentioned here and not referred to again. They are being noted simply to pects of the historical development - those dealing with the conceptual detail, especially in Chapters 5 through 7. The reader ought not to be iscourage ,t en, . certain conceptua an oun ationa issues seem to be too sketchily treated in this chapter to be grasped with full clarity, nor xt

abeyance. These conceptual and foundational questions and issues will

sions into novel areas. I then take up the development of the kinetic approac to matter, t e eve opment 0 lnetlc t eory at t e an s 0 Maxwell and Boltzmann toward a genuinely statistical or probabilistic theo the develo ment of formal statistical mechanics and some as ects of the evolution of foundational questions through the probing critiques developments are noted. Once again these developments are treated

15

Historical sketch foundational issues they will be taken up in detail in Chapter 7. cs

erties and laws The discovery of the right concepts with which to describe nature and the discovery of the lawlike relations among these features are processes that cannot be disentan led from one another. The a ro riate conce ts are the ones that allow the formulation of lawlike regularities among the

of development of a science, a progression that requires great philosop ica su t ety in 011 er to e y characterize , ut whose supe cia nature is clear. In the be innin we characterize the world in terms of concepts directly connected to our sensory awareness of things. Gradually

qualitative and comparative to fully metric concepts. Finally, as the concepts are uti 1ze to construct aws escn 1ng t e regu arities we iscover in nature we move to a sta e where the ve meanin of the relevant conceptual terms is more closely associated with the role of the terms in

One point of origin of the full thermodynamic theory is the discovery 0 erst so-ca e equatton 0 state or gases. e re evant aw is formulated usin conce ts whose scientific role is alread wellestablished in kinematics (the description of motion) and dynamics (the

the pressure imposed upon it, and that it exerts on the walls of its conta1ner, 1S constant. But this law neglects the additional relevant parameter of temperature, and it is in the introduction of the concept of temperature, and the

with felt hotness and coldness. The introduction of thermometers detaches

16

Physics and chance

equa 10n 0 s a e k is a constant. Even the consideration of simple experiments on the resulting temmixed suggests the necessity of an extensive quantity - heat - to parallel s me temperature but twice as much heat content. Moving to the results obtained on the mixing of different substances, there develops a slow awareness of the need for such notions as heat ca ad - how much heat needs to be added to a given substance to raise its temperature to

deal with static properties of matter, it is soon realized that a theory of the laws of change of the features introduced is needed as well. Temerature difference is invoked as the drivin force for heat ow and invoking the notion of the distribution of temperature over matter, soon

formulated governing the combined flow of matter and heat, although the erivation 0 these equations requires the concomitant progress in the develo ment of the full abstract thermod amic theo . Here ro ress is obtained by imposing stringent assumptions upon the flow of matter

to move to more complex equations for which those first derived prove o y approxlmations. Eu er's equatIons, or examp e, escri e t emotion of an incom ressible non-viscous fluid. Retainin incom ressibili but allowing viscosity (momentum transfer by diffusion rather than by transfer

ity, and so on leads to the ability to characterize ever more complex sltuatl0ns In w lC matter an eat are 0 In a c anglng state nve y their interactive forces.

2. Conseroation an irreversi ility Conservation. Concurrent with the gradual development of ever more comp ex an gene constltutlve equatlons - e vanous equatl0ns re tmg the lawlike interrelations amon the descriptive arameters of particular kinds of matter such as ideal gases in their macroscopically static states,

Historical sketch

17

general principles that are taken to hold for all particular material systerns, and that together with the more particularized constitutive equations rovide the resources needed for the full theo . There are two parallel paths in this progress, and insight into both strands develops

.

the overarching principle of the conservation of energy. The other is the growing realization of the fundamental role p ayed by irreversibility in the theo assimilated after the conce tual innovations of absolute temperature and entropy into the famous Second Law of Thermodynamics.

frictional motion to raise the temperature of an object suggested the specu ation that heat, the gain 0 which accounte or a rise in temperature mi ht be nothin more than some kind of motion of internal constituents of matter. This internal motion was not directly evident because,

scopically determined is

metric experiments on mixing can be accounted for by the idea that heat is some t tng t at can in a it matter an e trans erre rom one piece of it to another. Somethin like the densi of this so-called caloric substance in the matter is taken to account for temperature, with subtle

18

Physics and chance

as a quasi-substance that can carty energy across matterless space. Next, iii engines 0 vi a y impo n 0 the Industrial Revolution. Most important of all, though, is the growing ability to carefully quantify amounts of heat and amounts of mechanical motion and the abili to show that the disa earance of a iven uanti of overt motion always results in a corresponding increase in a specific

observable motion. When this is combined with the growing prominence of conservation principles in the foundation of physics, the suggestion is unavoida Ie that there is one kind of uali of thin s - ener - that is never created or destroyed. The apparent disappearance of quantity of

the cost of a corresponding disappearance from the world of heat, is then rea y nothing more than the conversion of motion from the realm of the overtl observable to that of the invisible microsco ic and vice versa. By the midpoint of the nineteenth century, Clausius is writing "

Irreversibility. The facts about the interconvertibility of mechanical energy an eat ou ine in east section e , y means 0 t e princip es of the conservation of ener and the identification of heat as non-overt internal energy, to a degree of assimilation of the physics of thermal

Historical sketch

19

motion of the working fluid, and so on. Far more important, though, is the realization that even an ideally efficient en ine is limited in its abili to convert in ut heat into out ut usable mechanical motion. The crucial observation is that in order for

the working fluid is discharged at the end of the mechanical cycle. For examp e, input steam rom the oi er must e at a higher temperature than the tern erature of the condenser that turns the out ut steam into water to eliminate back pressure in the steam engine. Heat can only do "

can be extracted from the heat than is allowed by the temperature gap etween input an output reservoirs. eru e y, eat at ig temperature is valuable because work can be extracted from it b usin an extraction reservoir, at the end of the cycle, of lower temperature. Heat available at

work.

ity to get useful mechanical work out of heat energy is limited not simply Y t e act t at eat is energy an t at t e conservation aw lscusse earlier prohibits our extracting mechanical energy greater than the heat energy consumed. It is limited also by the necessity to operate our engines

The results are soon generalized into a principle of the irreversibility o processes in e wor . nergy mus e aval a e in a 19 qua 1 state to do work. Performing the work degrades the energy into "low quality." If the conversion of heat into work in the process is ideal, only

the processes that reduce them from ideality are all those that allow the qua 1 0 energy 0 egra e Wi ou pro uClng wor in e process. e net result is that there is a steady degradation of the quality of energy throughout the world. Very soon, speculation arises about the ultimate 11

20

Physics and chance

becomes a cliche of popular science.

3. Formal thermostatics are converted into a formal theory of surpassing elegance, usually called thermodynamics. Throughout, the trick is to borrow as much conceptualization as one can from the familiar mathematical resources of d namics adding such intrinsically thermal concepts as are necessary, and modifying

the new context must be taken into account. One egins with the notion 0 a system. The concepts 0 inematics and d namics are a ro riated as allowin for descri tions of s stems and of their changes over time. Thus we can talk about the volume of

of the world (adiabatic systems) and those that can exchange energy eit er wit an In etermlnate environment or wi one anot er. e tota ener flow into or out of a s stem is divided into those arts of the flow that are the result of overtly observable mechanical work, and the

overtly unchanging state called the equilibrium state. Two systems, each In IVl ua y In equl 1 num w en energe lca y IS0 ate ,mayor may no leave their respective equilibrium states when they are allowed to exchange energy with one another. When the condition of the systems is

Historical sketch

21

equivalence class of systems in mutual equilibrium. At this point, the work of Camot on heat engines is applied. The tern erature of the reservoir into which heat is dischar ed in an ideal engine able to convert all of its input heat into overt work is taken as the

heat in differing ideal engines. The net result is the definition of the full absolute temperature sca e. One then examines the routes b which a s stem in one e uilibrium state can be converted into a system in a, distinct equilibrium state. One

entropy value to each equilibrium state of a system. Entropy is so defined at a system w ose energy 1S ess egra e an so more ava1 a e or transformation into mechanical work) has a lower entro than its more degraded counterpart. The crucial fact needed to justify the introduction

ing in a cycle, will produce no effect other than the transference of heat rom a coo er 0 a warmer 0 y. n e e V1n- anc vers10n, 1 1S impossible to construct a device which, operating in a cycle, will produce no effect other than the extraction of heat from a reservoir and the "

undergoing a transition is never less than its initial state." e ne an e egan vers10ns 0 e orma eory 0 ermo ynaffilcs are available. Perhaps the high point is the 1909 formulation of the Second Law by C. Caratheodory. His basic postulate, "In the neighborhood of

The extension of thermodynamics concepts beyond their original range

chemical potentials in a manner parallel to pressure. Another important

that which is transmuted into internal energy of microscopic constituents

Statistical equilibrium thermodynamics. In the orthodox thermo-

unchanging values of the macroscopic parameters. An isolated system se es to an equi I flum state 0 e pressure, vo ume, an so on. system in perfect energetic contact with an "infinite" reservoir at a given tern erature settles to an e uilibrium state in which its tern rature and energy content are fixed. As we shall see in Sections 11,6; III; and IV this \

micro-mechanics are also introduced.

single macroscopic state but by a probability distribution over a class of

probabilistic theory, we would not be surprised to find ourselves under orIn

terms, and on a postulational basis, a novel thermodynamics is derived

24

Physics and chance

cally connected component systems. Instead of the static uniformity throughout all these pieces that one would expect in the older account, in this new account a distribution of numbers of com onents in a diversity of macroscopic states is now predicted by the theory.

and the application of the Zeroth Law of Thermodynamics, suffice to generate the specific form of the fluctuational nature of equilibrium identical to the forms familiar from the robabilistic theo built on the underlying micro-structure and micro-mechanics. •

In the orthodox presentation of thermodynamics, we attribute thermoynamic features temperature, entropy on y to systems that are in e uilibrium. In the formal version of the theo the ve meanin lness of such attributions is restricted to equilibrium states by the way the

Of course this does not mean that the theory tells us nothing about change. e very pro ems t e t eory was origina y esigne to so ve are t ose that arise when we ask for exam Ie what states can be obtained from others by processes that leave the system energetically isolated from its

and ends up with it in another. But we have no resources within this t eory to a equate y escn e systems not 1n equ11 num, nor to ea thermod micall with such details of the transition from one e uilibrium to another as the rates of flow of quantities.

equilibrium thermodynamics, one dealt with temperature distributions over a system an W1 t e ows 0 eat nven y tempera re erences. Other such laws, connecting fluxes or flows with their driving forces, exist as well, and _provide the starting point from which a fuller

Historical sketch

25

introduce such notions as the density of mass at a point, a field of velocities of mass components, a field of energy density, and so on. The trick is to extend this notion of a field of uantities to such urel thermodynamic quantities as temperature and entropy. But how can it be

entropy are only defined, within orthodox thermostatics, for systems in equilibrium? The solution is to invoke the notion of local equilibrium. If thermod namic features of thin s chan e slowl enou h from oint to point and from moment to moment, we can hope to think of a non-

with appropriate temperatures and entropies. The full understanding of just what constitutes "s ow an sma enough c ange" wou ta e us into an anal sis that oes be ond the henomenolo ical level to which we are now restricted, but the theory can be developed on a postulational

Once the appropriate field-like concepts have been introduced, one invo es a ance equations, agaIn amllar rom continuum mec anics. These are the local versions of the various rinci les of the conservation of mass, the conservation of energy and momentum, and so on. The

is, of course, not generally conserved. In a small region it can even ecrease y oWIng out t roug e oun anes 0 e regIon. at IS crucial is that within a small re ion the entro enerated is never smaller than the entropy that "flows out," so that we have a local version of the

Clausius-Duhem inequality). n non-equll num t eory e notlon 0 a s ea y-s ate process ta es a role analogous to that taken by the equilibrium state in thermostatics. Even in a system that is prevented from reaching equilibrium by inter-

26

Physics and chance

intricately interact with one another. In the case of a system that is globally ",,,,-g r n rui' i po u a e a recipro i relation among flows, so that the effect of the driving force of one flow on the flow of another quantity is matched by a reciprocal effect. The ve meanin fulness of the formal statement of the rinci Ie first of all requires that the forces and flows be characterizable in a linear way, '-'.L ..

mechanically when the system is near its "neutral" position. The reciprocity relations can then be derived from certain very general postulates of invariance. For exam Ie a rinci Ie of "detailed balance" amon states leads to these reciprocity relations.

it is possible to show that the equilibrium state is stable against small perturbations. In this exten e theory it is possible to show t at the stead -state rocesses also ossess a stabili ro e . An small deviation of the system from its steady-state flow will die out, restoring the

maximize the entropy of the system in question, in near equilibrium nonequi i rium eory one can s ow assuming e inearity an reciprocity conditions) that the stable stead -state flows will be 'ust those rocesses compatible with the restraints that minimize the production of entropy.

dum. When the system with which we deal is in a state that is far from equll num, many 0 e tracta e eatures 0 systems near-equi 1 num no Ion er a I. We cannot ho e that in eneric situations, forces and fluxes will be related in a simple linear way; that there will be simple

steady-state systems will hold. Nor can we expect an elegant principle of mintma entropy pro uctl0n to 0 genera y. If a s stem is deviant enou h from e uilibrium, so that even locall we cannot view it as at least momentarily in something like an equilibrium

Historical sketch

27

subregions, then we can, once again, apply the field-like generalizations of the thermodynamic notion. As before we introduce field-like mechanical features such as veloci and momentum fields and field-like thermodynamic features such as a field of temperatures and of entropy density.

momentum, and energy. Again, as before, a field-like surrogate for the Second Law is intro uce . Although entropy in a region may ecrease due to flow of entro across the re ion's boundaries the net result of entropy flow and entropy production in the region is always positive.

boundary conditions, it may maintain an unchanging state of flow. Sometimes these stea y-state situations wi e sta e an easi y repro uci e, so that a far-from-e uilibrium s stem no matter how often re ared will always fall into the same steady-state situation. In other cases, though,

will apparently spontaneously change into different steady-state flows. One situatIon 0 partlcu ar Interest IS were we manlpu ate a ar- rome uilibrium stem b slowl va in an extemall controllable arameter. In some cases this results in the generation of a unique, always repro-

of available options for steady-state flow. Under other appropriate circumstances or particu ar va ues 0 parameters, t e system WI s ow a fascinatin oscillatory behavior, switchin back and forth with clock-like regularity between a number of distinguishable steady-state flow modes.

stable steady-state behavior of near equilibrium systems, but instead ecoffilng amp e y vanous In s 0 pOSItIve ee ac mec anlsms. As a parameter changes, the system may, as a matter of small fluctuation, pick a particular steady-state mode, but once having moved into that

Some elegant experimental instances of steady-state behavior of sysems ar rom equll num ave een cons ruc e. one ea s a Ul between plates of differing temperature, one gets steady, stable diffusion of heat from the hotter to the cooler plate when the temperature dif-

28

Physics and chance

varying chemical concentrations, temperatures, and so on, one can generate cases of steady-state flow, of oscillatory behavior with repetitive order both in s ace and in time or of bifurcation in which the s stem jumps into one or another of a variety of possible self-sustaining flows,

"self-organizing" phenomena as those described may play an important role in biological phenomena (biological clocks as generated by oscillato flows s atial or anization of an initiall s atiall homo eneous mass by random change into a then self-stabilizing spatially inhomogeneous

caloric theory dominated the scientific consensus, so throughout the caloric perio t ere appeare numerous specu ations a out just w at in 0 internal motion constituted that ener that took the fonn of heat. Here, the particular theory of heat offered was plainly dependent upon one's

might think of heat as a kind of oscillation or vibration of the matter. ven an a vocate 0 lscreteness - 0 t e constltution 0 matter out 0 discrete atoms - would have a wide varie of choices, es eciall because the defenders of atomism were frequently enamored of complex

As early as 1738, D. Bernoulli, in his Hydrodynamics, proposed the mo e 0 a gas as cons 1 e 0 mlcroscoplC pa lC es ln rapl mo lon. Assuming their unifonn velocity, he was able to derive the inverse relationship of pressure and volume at constant temperature. Furthennore,

Historical sketch

29

common identical velocity, would be proportional to temperature. Yet the caloric theory remained dominant throughout the eighteenth century. The unfortunate indifference of the scientific communi to Bernoulli's work was compounded by the dismaying tragi-comedy of Herepath and

model of independently moving particles of a fixed velocity. He identie heat wi interna motion, but apparent y too temperature as proortional to article veloci instead of article ener . He was able to offer qualitative accounts of numerous familiar phenomena by means of

though it appeared elsewhere, it had little influence. (J. Joule later read Herepa 's wor an in act pu is e a piece exp aining an e en ing it in 1848 a iece that did succeed to a de ree in stimulatin interest in Herepath's work.)

the Royal Society in 1845. The paper was judged "nothing but nonsense" y one re eree, ut It was rea to t e oClety In a t oug not y Waterston, who was a civil servant in India), and an abstract was ublished in that year. Waterston gets the proportionality of temperature

the same, and even (although with mistakes) calculates on the model e ratlo 0 spec c eat at constant pressure to t at at constant vo ume. The work was once again ignored b the scientific communi . Finally, in 1856, A. Kr6nig's paper stimulated interest in the kinetic

Kr6nig's paper may have been the stimulus for the important papers of auslus In an aUSlUS genera lze rom onlg, w 0 a idealized the motion of particles as all being along the axes of a box, by allowing any direction of motion for a particle. He also allowed, as Kr6nig

of translational motion.

Physics and chance

30

in manreepa the average distance a molecule could be expected to travel between one collision and another. The r win rece tiveness f the scientific c mmu . theory was founded in large part, of course, on both the convincing

body of evidence for the atomic constitution of matter coming from other areas of science (chemistry, electro-chemistry, and so on). 2. Maxwell

In this paper we find the first language of a sort that could be interpreted in a probabilistic or statistical vein. Here for the first time the nature of ossible collisions between molecules is studied and the notion of the probabilities of outcomes treated. (What such reference to probabilities

uniformity with regard to the velocities of molecules, Maxwell for the rst time ta es up t e question of just w at in of istribution of ve ocities of the molecules we ou ht to ex ect at e uilibrium and answers it by invoking assumptions of probabilistic nature.

claim that the components in the yand z directions are "probabilistically In epen ent 0 e component In e x !fechon. rom t ese assumptions he is able to show that "after a reat number of collisions amon a great number of identical particles," the "average number of particles

v = 2

Av ex

Historical sketch

31

aware that his second assumption, needed to derive the law, is, as he puts it in an 18 7 paper, "precarious," and that a more convincing derivation of the e uilibrium veloci distribution would be welcome. But the derivation of the equilibrium velocity distribution law is not

from place to place, we will have transport of mass. But even if density stays constant, we can have transfer of energy from place to place by molecular collision which is heat conduction or transfer of momentum from place to place, which is viscosity. Making a number of "randomness"

, An improved theory of transport was presented by Maxwell in an 1866 paper. Here he offere a genera eory of transport, a theory that once a ain relied u on "randomness" assum tions re ardin the initial conditions of the interaction of molecules on one another. And he provided

molecular interaction, and on the relative velocities of the molecules, w ic , given non-equi i rium, ave an un nown istri ution. But or a articular choice of that otential - the so-called Maxwell otential which is of inverse fifth power in the molecular separation - the relative velocities

- that is, unchanging with time, and that this is so independently of the eta sot e orce aw among t e mo ecu es. symmetry postu ate on des of transfers of molecules from one veloci ran e to another allows him to argue that this distribution is the unique such stationary distri-

The paper then applies the fundamental results just obtained to a vanety 0 transport pro ems: eat con uctl0n, VISCOSIty, slon 0 one as into another, and so on. The new theory allows one to calculate from basic micro-quantities the values of the "transport coefficients," numbers

comparison of the results of the new theory with observational data, a oug e 1 cu les encoun ere In ca cu a Ing exac va ues In t e theory, both mathematical and due to the need to make dubious assumptions about micro-features, and the difficulties in exact experimen-

32

Physics and chance

3. Boltzmann In 1868, 1. Boltzmann published the first of his seminal contributions to 'n i h en 1'z for velocity found by Maxwell to the case where the gas is subjected to p p

,

In the second section of this paper he presents an alternative derivation of the equilibrium distribution, which, ignoring collisions and kinetics resorts to a method reminiscent of Maxwell's first derivation. B assuming that the "probability" that a molecule is to be found in a region

In a crucially important paper of 1872, Boltzmann takes up the problem of non-equilibrium, the approach to equilibrium, and the "explanation" of the irreversible behavior described b the thermod namic Second Law. The core of Boltzmann's approach lies in the notion of the distri-

some specified value of the energy x and x + dx. He seeks a differential equation that wi speci ,given the structure of this function at any time, its rate of chan e at that time. The distribution function will change because the molecules collide

this, some assumptions are made that essentially restrict the equation to a particu ar constitution 0 t e gas an situations 0 it. For examp e, the ori inal e uation deals with a as that is initiall s atiall homo eneous. One can generalize out of this situation by letting f be a function

by a "streaming" term that takes account of the fact that even without co isions t e gas wi ave its lStri ution in space c ange y emotion of the molecules unim eded aside from reflection at the container walls, The original Boltzmann equation also assumes that the gas is sufficiently

not be taken into account. In Section 111,6,1 I will note attempts at genera lzlng eyon t is constraint. In order to know how the ener distribution will chan e with time, we need to know how many molecules of one velocity will meet how

Historical sketch

33

of Collisions. One assumes the absence of any "correlation" among molecules of given velocities, or, in other words, that collisions will be "totall random." At an time then the number of collisions of molecules of velocity VI and V 2 that meet will depend only on the proportion of

This - along with an additional postulate that any collision is matched by a time-reverse collision in which the output molecules of the first collision would if their directions were reversed meet and enerate as output molecules that have the same speed and reverse direction of the

his famous kinetic equation:

of energy, as it was expressed in Boltzmann's paper. What this equation escri es is t e raction 0 mo ecu es wit ve ocity VI' I changing over time. A molecule of veloci V mi ht meet a molecule of veloci V and be knocked into some new velocity. On the other hand, molecules of

,

another).

,

Physics and chance

34

1S

justifying the claim that the

.

.

equation finally

rmo yn i : i qu qui" s a wi be ceaselessly and monotonically approached from any non-equilibrium state. It is to justifying the claim that the Maxwell-Boltzmann distribution is the uni ue 'stationa solution of the kinetic e uation that Boltzmann turns. The definition of H is arrived at by writing lex, t) as a function of velocity: H=

4. Ob 'ections to kinetic theory The atomistic-mechanistic account of thermal phenomena posited by the 'netic t eory receive a osti e reception rom a segment o. e scientific communi whose two most rominent members were E, Mach and P, Duhem, Their objection to the theory was the result of two program-

One theme was a general phenomenalistic-instrumentalistic approach to SC1ence. rom t 1S p01nt 0 V1ew, e purpose 0 SC1ence 1S e production of sim Ie, com act, lawlike eneralizations that summarize the fundamental regularities among items of observable experience, This view "

Historical sketch

35

phenomenological laws of thermodynamics. The other theme was a rejection of the demand, common especially amon En lish Newtonians that all henomena ultimatel receive their explanation within the framework of the mechanical picture of the world.

scientific treatment for only a portion of the world's phenomena. From this point of view, inetic eory was a misgui e attempt to assimilate the distinctive theo of heat to a universal mechanical model. There was certainly confusion in the view that a phenomenalistic-

irreversibility of thermal phenomena, seems to have been initially noted y axwe lmse ln correspon ence an y omson m pu lcation in 1874. The roblem came to Boltzmann's attention throu h a oint made by J. Loschmidt in 1876-77 both in publication and in discussion

guarantee to us that a gas whose micro-state consists of one just like the equll flum gas - excep a e lrec lon 0 mo lon 0 eac cons 1 en molecule is reversed - will trace a path through micro-states that are each the "reverse" of those traced by the first gas in its motion toward

36

Physics and chance

S(b') > S(a')

Figure 2·1. Loschmidt's reversibility argument. Let a system be started in

.

.

.

invariance of the underlying dynamical laws that govern the evolution of the th r must be a micro-state b' that evolves to a micro-state a' and such that the entropy of b', 5(b'), equals that of b and the entropy of a' equals that of a 5 a' as Boltzmann defines statistical entro ). 50 for each "thermodynamic" evolution in which entropy increases, there must be a corresponding "anti-thermod amic" evolution ossible in which entro decreases.

means that the second gas will evolve, monotonically, away from its equi i rium state. ere ore, Bo tzmann's H- eorem is incompati e wi the laws of the under! in micro-mechanics. (See Pi ure 2-1.) A second fundamental objection to Boltzmann's alleged demonstration

Reversibility Objection. In 1889, H. Poincare proved a fundamental t eorem on testa 1 lty 0 motIon at IS governe y e aws 0 Newtonian mechanics. The theorem onl a lied to a s stem whose energy is constant and the motion of whose constituents is spatially

at a given time in a particular mechanical state.

In 1896, E. Zermelo applied the theorem to generate the te eli e retnwan ,or ecurrence Jec lon, 0 0 zmann s mec anlcally derived H-Theorem. The H-Theorem seems to say that a system started in non-equilibrium state must monotonically approach equilib-

Historical sketch

37

Figure 2-2. poincare recurrence. We work in phase-space where a sin e oint re resents the exact microsco ic state of a system at a given time - say the position and velocity of \ I eve molecule in a as. Poincare shows for certain s stems, such as a gas confined in a box and energetically isolated from the outside world, that if the system starts in a certain microscopic state 0, then, except for a "vanishingly small" number of such initial states, when the system's evolution is followed out along a curve p, the system will be found, for any small region E of micro-states around the original microus, "a most a " suc state 0 to return to a mIcro-state In t at sma regIon E. systems started in a given state will eventually return to a microscopic state "very c ose to at Imtla state.

one likes. But such a state would have a value of H as close to the initial value as one li es as we . Hence Bo tzmann's emonstration 0 necess monotonic a roach to e uilibrium is incom atible with the fundamental mechanical laws of molecular motion. 5. The probabilistic interpretation of the theory

tion has any definitive answer. Suffice it to say that the discovery of the everS1 11ty an ecurrence Jectlons prompte t e 1scoverers 0 t e theo to resent their results in an enli htenin wa that revealed more clearly what was going on than did the original presentation of the theory.

meant. The language here becomes fraught with ambiguity and concepe tua 0 scunty. ut 1t 1S no my purpose ere e1 er 0 ay ou a possible things they might have meant, or to decide just which of the many understandings of their words we ought to attribute to them. Again,

38

Physics and chance

Maxwell's probabilism. In a train of thought beginning around 1867, vo 0 y e on flow of heat from hot to cold is only the mixing of molecules faster on the average with those slower on the average. Consider a Demon capable of seein molecules individuall a roachin a hole in a artition and capable of opening and closing the hole with a door, his choice

left, thereby sorting a gas originally at a common temperature on both sides into a compartment of hot gas and a compartment of cold gas. And doin this would not re uire overt mechanical work or at least the amount of this demanded by the usual Second Law considerations. From

Whether a Maxwell Demon could really exist, even in principle, iscussion. 1. Bri ouin an ecame in ater years a su ject 0 muc 1. Szilard offered ar uments desi ned to show that the Demon would generate more entropy in identifying the correct particles to pass through

Later, arguments were offered to show that Demon-like constructions cou aVOl at m 0 entroplc mcrease as e resu t o e emon s rocess of knowled e accrual. More recently, another attack had been launched on the very possibility

that the Demon, in order to carry out its sorting act, must first register in a memory t e act t at It IS one sort 0 partlc e or e 0 er Wl w lC it is dealin . After dealin with this article, the Demon must "erase" its memory in order to have a blank memory space available to record the

by the Demon and fed into its environment. It is this entropy generation, t ey argue, a more an compensa es or e en ropy re uc lon accomplished by the single act of sorting. In his later work, Maxwell frequently claims that the irreversibility

Historical sketch o

0

39

0

only due to limitations on our knowledge of the exact trajectories of the "in principle" perfectly deterministic, molecular motions. Later popular writin s however do s eak if va uel in terms of some kind of underlying "objective" indeterminism. •

with Loschmidt, Boltzmann began a process of rethinking of his and Maxwell's results on the nature of equilibrium and of his views on the nature of the rocess that drives s stems to the e uilibrium state. Various probabilistic and statistical notions were introduced without it being always

world") toward equilibrium emerged in Boltzmann's writings. One paper 0 1877 rep ied speci ca y to Losc idt's version 0 the Reversibilit Ob'ection. How can the H- Theorem be understood in light of the clear truth of the time reversibility of the underlying micro-

by taking the statistical viewpoint. It is certainly true that every individual mlcro-state as e same pro a llty. ut ere are vast y more mlcrostates corres ondin to the macrosco ic conditions of the s stem bein in (or very near) equilibrium than there are numbers of micro-states

will be many more of the randomly chosen initial states that lead to a un orm, equll flum, mlcro-state at t eater hme t an ere Wl e initial states that lead to a non-equilibrium state at the later time. It is worth noting that arguments in a similar vein had already appeared in a

40

Physics and chance

ways in which molecules can be placed in the momentum boxes, always onsi er a a e e ne y i ri ion, a pec a'on 0 e num er of molecules in each momentum box. For a large number of particles and boxes, one such distribution will be obtained by a vastly larger umber of assi nments of molecules to boxes than will an other such distribution. Call the probability of a distribution the number of ways it

of boxes go to infinity and the size of the boxes go to zero and one discovers that the energy distribution among the molecules correspondin to this overwhelmin I robable distribution is the familiar MaxwellBoltzmann equilibrium distribution. (See Figure 2-3.)

away from the approach that takes equilibrium to be specified as the unique stationary solution of the kinetic equation. As such it shares " recariousness" with Maxwell's ori inal ar ument. But more has been learned by this time. It is clear to Boltzmann, for example, that one must

awareness of this stems from considerations of collisions and dynamics at te us t at it is on y t e ormer met od t at wi ead to stationary distributions and not the latter. And as we shall see in the next section Boltzmann is also aware of other considerations that associate probabil-

Combining the definition of H introduced in the paper on the kinetic equation, t e ca cu ate monotonic ecrease 0 H imp ie y t at e uation the role of entro S in thermod namics (su estin that S in some sense is to be associated with -H), and the new notion of prob-

state is determined simply by the number of ways in which the macroy arrangements 0 e constituent mo ecu es 0 state can e 0 talne the s stem. As it stands, much needs to be done, however, to make this "definition" of entropy fully coherent.

41

Historical sketch A

•

•

• •

• •

•

•

·

t

•

0

• •

•

•

•

•

• •

• •

•

•

•

•

~

• •

• •

•

• • • • • • • • • • • •

•

x8 •

•

t

•••••• •••••• •••••• •••••• ••••••

,.,-

•

xFigure 2-3. Boltzmann entropy. Box A (and Box B) represent all possible c V

aIU'C~

,

,

•

,"c·

'...1

A

~

VI "1. ~

allU

.

"

~

1~

...I'

p

...I ~___ u p UlloV '...1

_~

a

VI

.11

5a~

.L

•

1. llav'C.

.L " Ul.... 1011.... Vl

T

~l11all ~UU-

"1.

~

~11

".,i." ~IU~

I11V-

.L

y,

."l

Ul'C~'C

~uu-

boxes are of small size relative to the size of the entire molecular phase space 1-.. + In_~~ ....,.... ....... 6 ....

1. "0'

~

n~ +hn+ U~

~

..... ~

1

~T:ll

1

............ ]

"

.....

.11~T 1-~ :~ ~n ~h 1-~~T ~~_ n ~~T L]

~

...., .................... ~ ..

....,~"""

.. ~ ........]

reasonable micro-state of the gas. An arrangement of molecules into boxes like th",t r.f'R;n

A ....",n h"",

'

.

...I

-0'

hu "'

-,

.

r

r.f..J··

,I"",., ;ntr.

"1

boxes. But an arrangement like that of Fig. B can be obtained only by a much 'r.f

"1'''''1']11"",r

~r. th"", .......

. ~

th~n th~t for

mllrh ' . ,
'r,

R Thp

fnr

A ;.,

l";'m rH ...trihn-

Pnl •• ,


..

tion (the Maxwell-Boltzmann distribution) corresponds to the arrangement that , i...

"

'

hv thp

to thp

of ..

that total number of molecules and total energy of molecules remain constant.

1

thp

• .

..

..,.

1

11~11'1o th-:lt tpNYl ......

11'1

A. ~ p-:lrhr -:I~ 1 ~~1

hp

l'

"/

.

.-

1. .1' ..1. ... 1. 1. ~1:.. " 'hI. " ., during which the system possesses this condition on the average," and as he takes Maxwell to mean 1t as "the ratio or the number or L1nnumerable similarly constituted systems] which are in that condition to the total number of systems." This is an issue to which we will return again and .1 1 u .... ~

• . 1"

......

......

~T~_ f-t...~

,y ..........1.1

1

•

« 0 .1.1....

~

...... l..L.1«1o loU' u....

.1lo,

l..L.1 .... loll.1.1....

• :l~"'T ........... ,,'-' "] evolve, partly inspired by the need to respond to critical discussion of his and Maxwell's ideas, espec1ally m hngland, and partly due to h1S own ruminations on the subject. To what extent can one stand by the new, statistical reading of the H-Theorem, now taken to be read as describing '-'

..... ~

thp " "1~ ...... ~ .....vv r

w.~

.....

1

11

_~u,

1

1

.1.

f-~

,,'-'

"

"1..1

"

......'-'....... r

f-~

'1

'-'~

V

• 'OhT ...... , .... "

~&

' ...

n1

]

"

'!'oi(-"

1

"1

of 1

"

1

.

...

of ~ r'\~~~

from

J

~~~~~

......

,

~T~

~_~

yy.....

" '.......

42

Physics and chance

ponding to a condition closer to equilibrium as more "probable" than transitions of the reverse kind. But if, as Boltzmann would have us believe all micro-states have e ual robabili this seems im ossible. For given any pair of micro-states, 5 h 52' such that 51 evolves to 52 after a

,

,

2'

1

while keeping speeds and positions constant - such that 5; is closer to equilibrium than 51' and yet 52 evolves to 51 over the same time interval. So "anti-kinetic" transitions should be as robable as "kinetic" transitions. Some of the discussants went on to examine how irreversibility was

tion" in the system, either in form of energy of motion of the molecules interc anging wi t e ae er or wi t e extema environment. In a letter of 1895 Boltzmann ave his view of the matter. This required, once again, a reinterpretation of the meaning of his kinetic equa-

unexpected kind, and, perhaps, unique in their nature in the history of science. In this new icture Boltzmann ives u the idea that the value of H will ever monotonically decrease from an initial micro-state. But it is still

himself points out, commenting on Boltzmann's letter, the trick is to rea lze at we examine t e system over a vast amount 0 time, e s stem will near! alwa s be in a close-to-e uilibrium state. Althou h it is true that there will be as many excursions from a close-to-equilibrium

equilibrium as the starting point, at a later time we ought to expect the system to e In a micro-state c oser to equi Inurn. ee 19ure - . But the theo of evolution under collision is now time-s mmetric! For it will also be true that given a micro-state far from equilibrium at one

Historical sketch

43

Figure 2-4. Time-symmetric Boltzmann picture. In this picture of the world, . ". . . "

in the equilibrium state. There are random fluctuations of the system away from . i ri m. The r at r the fluctuation of a s stem from e uilibrium the less frequently it occurs. The picture is symmetrical in time. If we find a system far from e uilibrium we ou ht to ex ect that in the future it will be closer to equilibrium. But we ought also to infer that in the past it was also closer to e uilibrium.

that it tells us that equilibrium is overwhelmingly probable. Isn't this a curious conc usion to come to in a wor at we n to e gross y distant from e uilibrium? Boltzmann takes up the latter paradox in his 1895 letter. He attributes "

intervals of time will, improbably, be found in a state far from equilibrium. Per aps t e reg10n 0 e cosmos 0 servat10na y ava1 a e to us 1S 'ust such a rare fluctuation from the overwhelmin I robable e uilibrium state that pervades the universe as a whole.

irreversibility and Boltzmann responds to two short papers of Zermelo's W1 two s ort p1eces 0 1S own. 0 tzmann s paper pomts out at the ieture ado ted in the 1895 letter of a s stem "almost alwa s" near equilibrium but fluctuating arbitrarily far from it, each kind of fluctuation

The 1897 paper repeats the picture of the 1896 paper, but adds the cosmo og1ca ypo eS1S 0 r. c ue z to 1t. n 1S paper, 0 tzmann makes two other important suggestions. If the universe is mostly in equilibrium, why do we find ourselves in a rare far-from-equilibrium ?

" " of a sentient creature. Therefore a sentient creature could not find itself eXls mg m an equ11 num reg10n, pro a e as regions no sentience can exist. Even more important is Boltzmann's answer to the following obvious

44

Physics and chance

owar equi i rium rna c e y a piece s oping away rom equll flum to non-equilibrium, then why do we find ourselves in a portion of the universe in which systems approach equilibrium from past to future? equilibrium support sentience equally well? Here the response is one e nee direction of time in which our local region of the world is headed toward equilibrium. There could very well be regions of the universe in which entro ic increase was counter-directed so that one re ion had its entro increase in the direction of time in which the other region was moving

increasing the "future" direction of time! The combination of cosmological speculation, transcendental deduction, and definitional dissolution in these short remarks has been credited b man as one of the most in enious proposals in the history of science, and disparaged by others as the last,

not settle on which conclusion is correct.

e ongtns

0

eory

the entire surface" of the phase space to which the particle is confined

display this same behavior: The reat irre ulari of thermal motion and the multi lici of forces that act on the body from the outside, make it probable that the atoms themselves, by virtue consistent with the equation of kinetic energy. (See Figure 2-5.)

Given the truth of this claim, one can then derive such equilibrium

Historical sketch

45

each phase point, weighting regions of phase points according to a measure at can eas y e enve. e Wl see t e etal s apter. Here we see Boltzmann's attem t to eliminate the seeming arbitrariness of the probabilistic hypotheses used earlier to derive equilibrium features.

general derivation of an important result earlier obtained by Boltzmann or specla cases. e me 0 s 0 0 mann sear ler paper a owe one to show, in the case of molecules that interact only upon collision, that in equilibrium the equipartition property holds. This means that the total

46

Physics and chance

o s h with the macroscopic constraints imposed on a given system, but having

region in the phase-space of points, each point corresponding to a given possible micro-state. If we place a probability distribution over those oints at a time we can then s eak of "the robabili that a member of the collection of systems has its micro-state in a specified region at time

value (such as the kinetic energy for a specified degree of freedom). The dynamic equations will result in each member of this collection or ensemble havin its micro-state evolve corres ondin to a ath amon the points in phase-space. In general, this dynamic evolution will result in

There is one distribution for a given time, however, that is such that e pro a i ity assigne to a region 0 p ase points at a time wi not va as time oes on because the initial robabili assi ned at the initial time to any collection of points that eventually evolves into the given

freedom is the same for each degree of freedom, so that our identificahon resu ts 1n envat10n 0 t e equ1part1t10n eorem or equ 1 num at is de endent onI u on the fundamental d namical laws, our choice of probability distribution, and our identification of equilibrium values with

47

Historical sketch

possible total micro-states of the system, each represented by a point of the

. . . .

.

.

systems whose points were originally in T -1 (A) have evolved in such a way that . , . . " " of the phase space have a constant probability assigned to it as the systems evolve.

t roug every p ase conslstent Wl t e energy. will see in detail in Cha ter 5.) Furthermore Maxwell asserts the encounters of the s stem of articles with the walls of the box will "introduce a disturbance into the motion

" He continues: It is difficult in a case of such extreme complexity to arrive at a thoroughly

sooner or later, after a sufficient number of encounters pass through every phase

Here we have introduced the "ensemble" a roach to statistical mechanics, considering infinite collections of systems all compatible with

48

Physics and chance

those that, like the hypothesized behavior of the swarm of molecules in i i giv n energy." In 1887 he utilizes the Maxwellian concept of an ensemble of macroscopically similar systems whose micro-states at a time take on eve realizable ossibili and the Maxwellian notion of a "stationa " probability distribution over such a micro-state.

distributions over such collections, of stationary such probability distributions, of (3) the identification of equilibrium values with averages of uantities that are functions of the micro-state accordin to such robability measures, and of (4) the postulates that rationalize such a model

, a set of issues that continue to plague the foundations of the theory.

m. Gibbs' statistical mechanics

J. Gi s, in 1901, presente ,in a sing e 00 0 extraof inary compactness and ele ance an a roach to the roblems we have been discussing that although taking off from the ensemble ideas of B~ltzmann and

Gibbs emphasizes the value of the methods of calculation of equilibflum quant1t1es rom stat10nary pro a 11ty 1Stfl utlons rev1ewe 1n e last section. He looks favorabl on the abili of this a roach to derive the thermodynamic relations from the fundamental dynamical laws with-

detailed constitution of gases out of molecules to rely on hypotheses a out 1S const1tut10n, an 1S s ept1c1sm is increase y resu ts, we known at the time, that seem in fact to refute either the kinetic theo or the standard molecular models. In particular, the equipartition theorem

Historical sketch

49

energy that actually goes on. This is a problem not resolved until the underlying classical dynamics of the molecules is replaced by quantum mechanics. In Gibbs' method we consider an infinite number of systems all having

6n such coordinates, 3 position and 3 momentum coordinates each for

each point molecule. In a space whose imensionality is the number of these sitions and momenta taken to ether a oint re resents a sin Ie possible total micro-state of one of the systems. Given such a micro-state

of the ensemble of systems can be viewed as a flow of points from those representing e systems at one time to t ose ynamica y etermine to re resent the s stems at a later time. Suppose we assign a fraction of all the systems at a given time to a

that region at the specified time. In general, the probability assigned to a regl0n Wl c ange as time goes on, as systems ave elr p ase oints enter and leave the re ion in different numbers. But some assignments of probability will leave the probability assigned a region of

Suppose we measure the size of a region of phase points in the most natura way POSSl e, as e pro uct 0 lts posltlon an momentum sizes, technically the integral over the region of the roduct of the differentials. Consider a region at time to of a certain size, measured in this

new lS new regl0n

the

What must PCq,p) be like that the probability assigned to region A is ?

50

Physics and chance

point. If we deal with systems of constant energy, then any function P ric . Gibbs suggests one particular such P function as being particularly noteworthy: P = exp('I' - £/9)

the ensemble of systems is so distributed in probability, Gibbs calls it canonically distributed. When first presented, this distribution is contrasted with others that are formall unsatisfacto the "sum" of the probabilities diverges), but is otherwise presented as a posit. The justi-

possess any specific energy, a collection all of whose members share a common energy. The phase point or each 0 these systems at any time is confined to a sub-s ace of the ori inal hase s ace of one dimension less than that of the full phase space. Call this sub-space, by analogy with

evolution into another region is assigned the same probability as that atter region at e initla tlme. uc a pro a llty assignment a a rea y been noted b Boltzmann and Maxwell. It amounts to assi nin e ual probabilities to the regions of equal volume between the energy surface

between nearby surfaces they bound. Gibbs calls such a probability distn utl0n on an energy su ace e micro-canoruca ensem e. There is a third Gibbsian ensemble - the rand canonical ensemble appropriate to the treatment of systems whose numbers of molecules of

51

Historical sketch 2. The thermodynamic analogies

From the features of the canonical and micro-canonical ensembles we will derive various e uilibrium relations. This will re uire some association of quantities calculable from the ensemble with the familiar thermo-

is quite cautious in offering any kind of physical rationale for the associations he makes. He talks of "thermo ynamic ana ogies" throughout his work maintainin onl that there is a clear formal analo between the quantities he is able to derive from ensembles and the thermodynamic

system out of its micro-parts, and avoids as well their attempt to someow rationa ize or exp ain w y t e ensem e quantities serve to ca culate the thermod namic uantities as well as the do. He begins with the canonical ensemble. Let two ensembles, each

represented by the original ensembles. The resulting distribution will be stat10nary on y e s 0 t e two ong1na ensem es are equa , g1v1ng an analo of 9 to tern erature, because s stems when connected energetically stay in their initial equilibrium only if their temperatures are

which also functions in the specification of the canonical distribution, to e eterm1ne y an a Justa e extema parameter. we 1mag1ne every s stem in the ensemble to have the same value of such an energy fixing parameter, and ask how the canonical distribution changes for a small

n

, i

, 1

are given by A = -de/da i . A bar over a quantity indicates that we are 1ng 1tS average va ue over t e ensem e. we compare 1S W1t t e thermodynamic,

we get the analogy of 9 as analogous to temperature, and -1'\ as

52

Physics and chance

u i ri u in their energies if they are represented by a canonical ensemble. The main conclusion is that if we are dealing with systems with a very large m er of de rees of freedom such as a s stem of as molecules the we will find the greatest probability of a system having a given energy ,

i

ensemble, and eventually to a picture of the physical situations that the two ensembles can plausibly be taken to represent. The conclusion Gibbs comes to is that a canonical ensemble with its constant e analo ous to temperature, best represents a collection of identically structured systems

(if they have many degrees of freedom) about a central value of their energy, but with systems existing at all possible energies istribute aroun this central value in the manner described b the canonical distribution. The micro-canonical ensemble, on the other hand, seems to be the

equilibrium values, but that are, with overwhelming probability, near equll flum an at equll flum on e average over tlme. exaffilnation of the fluctuation amon com onents in an ensemble whose members are micro-canonically distributed shows that these fluctuations will be

One can find thermodynamic analogies for the micro-canonical ensem e as one can or e canomca ensem e. e quantity ana ogous to entropy turns out to be log V; where V is the size of the phase space region to which the points representing possible micro-states of the sys-

are difficulties in this analogy when one comes to treating the joining oge er 0 sys ems 1m la y IS0 a e rom one ano er. Gibbs is also careful to point out that there are frequently a number of distinct quantities that converge to the same value when we let the

Historical sketch

53

quantity in one context, whereas another one of them might, in its functional behavior, be more analogous to that thermodynamic feature in some other context. Gibbs also points out that the values calculated for a quantity using the

will also coincide in the limit of vast numbers of degrees of freedom. Because it is usua y easier to ana vtica y eva uate quantities in the canonical ensemble framework this su ests this framework as the appropriate one in which to actually carry out calculations, even though

canonical ensemble of systems." In is iscussion 0 e ermo rnamic ana ogies or e microcanonica ensemble Gibbs also oints out clead the distinction between avera e values calculated by means of an ensemble and most probable values

3.

non-equi i rium ensem es

The Gibbsian program supplies us with a method for calculating equilibnum va ues 0 any ermo ynamlc quantity. lven t e constraints lmsed on a system, set up the appropriate canonical or micro-canonical ensemble of systems subject to those constraints. Calculate from this

equilibrium. e lneVlu ow are we, Wi in is ensem e View, 0 un ers an able approach to equilibrium of systems initially not in equilibrium? In Chapter XIII of his book, Gibbs shows, for a number of standard cases

the appropriate ensembles, the features of equilibrium so described will rna c e ermo rnamlC pre lC ion. or examp e, in e case Cl e , after the interaction the component of the new combined system that was originally at the higher temperature will have transferred heat to

But what right do we have to assume that the modified system will e appropnate y escn e y a canonlca or mlcro-canonlca ensem e. The problem here is this: We have already described by means of an ensemble the initial state of the s stem when it be ins its transition to a new equilibrium by, say, having the constraints to which it originally was

of the variation of the ensemble with varying external constraints, 11 or - og , ynamlc entropy. n apter 1 theorems about this ensemble analo e to entro . The ma'or conclusions are that (1) if an ensemble of systems is canonically distributed in distribution of the ensemble having the same average energy; and (2) a given lmlts 0 p ase gives a sma er average in ex 0 pro a llty 0 phase than any other distribution. Together, these two results give us, in a certain sense, the result that the canonical and micro-canonical distrienergy, the ensembles that maximize ensemble entropy. es. uppose, or examp e, we ave a mlcro-canonlca ensem e e ermined by some fixed values of constraints. We change the constraints. Can we expect the micro-canonical ensemble to evolve, in a manner accordance with the dynamical laws, into the micro-canonical ensemble

Now the ensembles we want to be the endpoint of evolution are the unIque ensem es at are suc t at two constraInts are met: t e number of systems between any values of functions of the phases that are constants of motion are constant· and 2 the value of - the mean of the index of probability, is minimized. Can we show, then, that the evolve with time toward the minimum value? That would show that the ensem e IS evo vlng 0 e unIque, s lonary ensem e consls ent WIt the constants of motion that are the standard equilibrium ensemble. That is can we use the "excess" of the - value of the ensemble at a time over the 11 value of the equilibrium ensemble as an indicator of how far the

o rst will alwa s

when the mixing began. But take any small, fixed, spatial region of the

portion of original fluid that was A. (See Figure 2-7.) en e constraInts are p ase space In 0 sma u non-zero regIons. changed, the ensemble initially fills a box entirely or the box is entirely unoccupied. As dynamic evolution goes on, the original compact ensem" " entire region of now, newly accessible phase space. Eventually we ex-

so spread, and not just spread uniformly in this new coarse-grained"

56

Physics and chance

c and 10% with a black ink as in A, the water and ink being insoluble in one

.

.

.

.

.

.

,

as in B. But a magnified look at a small region of the fluid in B, as shown in C, •

•

•

I

.

,

portion of any small volume filled by black ink is, however, 10%, if the ink is

in even this coarse-grained sense. He doesn't offer any proof, however, t at suc exceptlona ensem es Wl rea y e rare In t e wor. or is there an proof that the ap roach to e uilibrium, in this new sense, will be monotonic, nor that the time scale will be the one we want to

, ensemble that represents approach to equilibrium.

A

B

, appropriate time scale to represent observed approach to the new equilibrium.

58

Physics and chance

i oes poin ou e prova e ac a e lnl la coarse-gralne entropy can, at least, not be greater than the entropy of future states. But it is also the case that the fact that one can expect in the future ensembles statements about the past. The "increase of coarse-grained entropy top

y e .

That is, if we trace the dynamic evolution of the systems that constitute a non-equilibrium ensemble at a given time back into the past, we will enerate ensembles that are more uniform in the coarse- rained sense than the ensemble at the specified time.

symmetry is faced by later investigators: But while the distinction of prior and subsequent events may be immaterial with respect to mathematical fictions, it is quite otherwise with respect to events in the real world. It should not be forgotten, when our ensembles are chosen to illustrate the robabilities of events in the real world that while the robabilities of subsequent events may be often determined from probabilities of prior events, it i ra I the a e that obabilitie of rio event an be r those of subsequent events, for we are rarely justified in excluding from consid-

propriate ensemble description of various non-equilibrium processes. Here, Gi s ocuses not on t e evo ution 0 an ensem e representing an isolated s stem but on the course of chan e to be ex ected as external constraints are varied. First, he considers constraints being abruptly

now non-equilibrium in the newly accessible phase-space. In this case, one expects t e ensem e to c ange In suc a way at It na y, In t e "coarse- rained" sense, a roximates the e uilibrium ensemble for the newly specified parameters. And one expects, therefore, the coarse-grained

Historical sketch

59

nal constraints. For example, heat having flowed from the originally higher temperature body to the one originally at lower temperature, the two com onents will be at e ual tern eratures as re resented b the a ropriate "modulus" constant of the ensemble. .

IV. The critical exposition of the theory of P. an T. E nfest

In 1912, P. And T. Ehrenfest published a critical review of the then

cyclopedia of Mathematical Sciences, the article provided a brilliantly concise an i uminating, . sometimes controversia, overview 0 t e status of the theo at that time. This iece can be considered the culmination of the early, innovative days of kinetic theory and statistical

expansion and development. The piece is directed first to an exposition o t e onglna lnetlc t eory an Its c anges In response to t e ear y criticism, alon with a discussion of remainin unsolved roblems in the "statistical-kinetic" approach. It then moves to a critical exposition and

theory from Kronig and Clausius to Maxwell and Boltzmann culminated In 0 tzmann s lnetlc equatIon an IS proo 0 t e eorem. e Theorem led to the criticisms of the theory summed up in the Reversibility and Recurrence Objections of Loschmidt and Zermelo. These led

with the underlying dynamical theory. Despite this there are those who remalne unconVlnce t a o tzmann a aVOl e e n amenta problems of consistency posed by the Reversibility and Recurrence Objections. In order to show that he had, one must resolve many am-

ability theory step by step into hypothetical statements about relative requenCles In c ear y e ne s a IS lca ensem es. n e 0 er an , the revised Boltzmannian theory of non-equilibrium does require for its justification postulates whose plausibility and even consistency remain in "

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Physics and chance

1. The Ehrenfests on the Boltzmannian theory

brief survey of the evolution of assumptions about "equal frequencies of occurrence" in the early development of kinetic theory. Kronig asserted that the motions of the molecules bein ve irre ular one could "in accordance with the results of probability theory" replace the actual motion

frequencies of molecular conditions, some explicit and some only tacit. He assumed, for example, that the spatial density of the molecules was uniform that the fre uencies of molecular seeds did not va from place to place, and that all directions of molecular motion occured with

about frequency postulates where the frequencies are not all equal - the non-G eic erec tigtcases? C ausius a a rea yassume at at equi i rium there was a definite distribution of molecular velocities even if he didn't know what it was. Maxwell posited his famous distribution law,

potentials and to the cases of polyatomic gases from Maxwell's law, W lC 0 S or monatomic gases. But how can one derive the e uilibrium veloci distribution law? Maxwell and Boltzmann, treating dYnamic evolution of the system, showed

the theory.

definition of an ergodic system as one whose phase point "traverses

time. Therefore, every system will have the same time average - defined t

value of the system,

surface. Moreover, no example is known where the single G-path actually

Boltzmann's claim that in the gas model, all motions with the same total

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Physics and chance

to derive results in gas theory, despite the fact that they were originally er 0 1 1 i no usu y v n m n i n y o w 0 ave the use of these distributions in the foundations of their work. At this point, the Ehrenfests offer a plausible reconstruction of what "rna have been" the startin oint of Boltzmann's investi ation - that is of his suggested plan to use the micro-canonical ensemble average as the

behavior of a gas over an infinite time ought to be identical to its behavior at equilibrium. Can we calculate "time averages of functions of the phase over infinite time " in the limitin sense of course? In articular can we show that the infinite time average of a gas corresponds to the Maxwell-

(1) Ensem e average of 2 (3)

= time average of the ensem Ie average of f = ensemble avera e of the time avera e of = time average of f

bution over the ensemble. Equality (2) follows from the legitimacy of interc anging averaging processes. Equa ity 3, owever, is crucia. To derive it we must invoke the ostulate of er odici for it is that ostulate that tells us that the time average of f is the same for each system

Historical sketch

63

At this point, the discussion moves on to an attempt to make fully unambiguous the statistical reading of the kinetic equation and the HTheorem. In the rocess the Ehrenfests offer their demonstration of the consistency of these important non-equilibrium results with the under-

number of misunderstandings resulting from Boltzmann's loose language. Consider the phase space appropriate for a single molecule - that is, the s ace such that oints in it re resent the osition and momentum in all degrees of freedom, of one of the molecules of the system. Coarse-grain

of the gas at a given time. This results in a certain number of molecules ving eir representative points in t is sma p ase space, ca e Jl-space, in each i-th box. Call the number in box i a and a iven s ecmcation of the a /s a state-distribution, Z To each Z corresponds a region of points

with a given corresponding state-distribution, Z First, note t at t e region 0 -space correspon ing to e Z t at is t e Maxwell-Boltzmann distribution is overwhelmin liar e and dominates the allowed region of r-space open to the gas. If we assume ergodicity,

p,q

number of discrete time points separated by an interval d t from each o er. With this apparatus we can now present a tidied-up version of Boltzmann's claims about the behavior of H over time. We can say that

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Physics and chance

well above H o, when we look at neighboring time moments we will

Hb

Ha

He

Much less often, but with equal frequency, we will find:

Hb

He

or

Hb Ha

Only very rarely will we observe the following pattern to be the case:

Hb So from a state with an H much above its minimum value we will almost always find that the immediately succeeding state is one with a lower

the concentration curve of the bundle of H-curves. We then assert that t is curve monotonica y ecreases rom its initia ig Ha Za va ue, conver es to the minimum value H. and never de arts from it. At any time tn, the overwhelming majority of curves of Hwill be at a value

similar to the concentration curve. Next, we claim that the curve of the eorem, enve rom t e Inetlc equatton t at presupposes t e Stosszahlansatz, is identical to this concentration curve. Note that neither of the claims made (that the concentration curve will show this mono-

65

Historical sketch

--------===:::+:=--- Smax

Figure 2-10. The concentration curve of a collection of systems. A col-

.

..

.

.

The systems evolve according to their particular micro-state starting at the initial time. At later times 2 4 6 ... the collection is reexamined. At each time the overwhelming majority of systems have entropies at or near values 52' 53' 54, 5 5 ... which are lotted on the "concentration curve." This curve can monotonicall a roach the e uilibrium value 5 even if almost all the s stems, individually, approach and recede from the equilibrium condition in the manner de icted in Fi ure 2-4.

a consistent reading of Boltzmann's claims has now been produced. An

particular Z state overwhelmingly dominate the class of Z states compati e wi t at 0 servationa constraint. T is is nee e to connect e observational thermod namic initial state with the osited definite initial Za in the earlier treatment of the H- Theorem. (See Figure 2-10.)

Objection from Recurrence. A further variant of the Reversibility Objection - t at to eac state t ere IS t e reverse state, an t at 1 t e ormer leads to an increase of H in a time interval the latter must lead to a decrease of H in the same time interval of equal size - is noted to be

versibility: (1) The Stosszahlansatz gives, for each time interval, only the most pro a e num er 0 co ISlons, an t e eorem on y e most probable value in the change in H; (2) the actual number of collisions (and the actual change in H) fluctuates around this most probable value.

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Physics and chance

Consider a gas in state Za at a given time. How many collisions of a e y -s a. 0 a e as ime, n i un erpoints in r-space corresponding to the Z-state. We can formulate the statistical Stosszahlansatz as the claim that the sub-region of the region of oints in s ace corres ondin to the Z-state in which the num er of collisions is given by the Stosszahlansatz is the overwhelmingly largest

Stosszahlansatz at every moment of time in deriving that equation, This leads to the Hypothesis of Molecular Chaos as a posit. Take the subset of the ori inal re ion in r-s ace for which the Stosszahlansatz ave the right number of collisions in the time interval. Look at the systems those

equation, we must assume that the overwhelming fraction of those points also represent systems whose collisions in the next interval of time will also be in accord with the Stosszahlansatz. And this h othesis about the dominance of the Stosszahlansatz obeying region must be continuously

trajectories started in our initial region. These will correspond to different istri utions 0 state, ZB" ZB'" an so on. But eac suc ZB' region 0 oints started in our initial re ion will corres ond to onl art of the r -space region corresponding to a given ZB' distribution, that part

state, the largest region is that for which the Stosszahlansatz will hold, just as we assume it or t e woe region 0 -space correspon ing to a iven initial Z-state. Although this new Hypothesis of Molecular Chaos may need subtle

tistical formulation of the Stosszahlansatz, and therefore a statistical envat10n 0 t e 1netlc equat10n an eorem, eX1sts. t e envation is immune to the Ob'ections of Recurrence and Reversibili . Note, once again, that no claim as to the provability or derivability of the

Historical sketch

67

2. The Ehrenfests on Gibbs' statistical mechanics

Whereas the Ehrenfests view Boltzmann, aided by his predecessors and axwe ,as e semina In er In e e , e researc er w ose wor defines the field, even if it is sometimes ambiguously stated and needful of clarification and su lementation the Ehrenfests' critical view of Gibbs is rather unfavorable. In successive sections they offer an exposition and

Critique of Gibbs' equilibrium theory. Gibbs, according to the Ehrenfests, first recapitulates Boltzmann's investigations into the most eneral stationa ensembles and then fixes his attention on two "ve special" cases - the micro-canonical distribution, which is equivalent to

tation, the introduction of the canonical distribution seems to be an ana vtlca tnc - t at IS, mere y a eVlce to al In ca cu abon. And, the oint is, Gibbs does not even mention the subtle problems of the posits of ergodicity needed to justify the claims that these methods

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Physics and chance

librium.. First, say the Ehrenfests, Gibbs defines a function a = fp logpdqdp and shows that this function takes on its minimum value subject to ap-

energy for micro-canonical case) in the canonical and micro-canonical v e p v rm Then Gibbs suggests dividing the r -space into small boxes, taking the average ensemble density in each box, Pt, and defining :E - :EtPJog~ as a new measure of de arture of an ensemble from its e uilibrium distribution. t

and such that in the limit of infinite time, the value of :E(l) will assuredly be less than or equal to its value at the initial moment. His argument for this consists in the analo with the mixin of insoluble li uids. Further his arguments can be taken to show that in the infinite time limit, the P/s

start with an ensemble all of whose coarse-grained boxes are either tota y e or tota y empty, t en t cannot increase in time, an may ve well decrease. But, say the Ehrenfests, Gibbs has not shown, by any means, all that

value. The Ehrenfests suggest that this can be expected to be many eyc es 0 t e time nee e or a system t at eaves a coarse-graine ox to return to that box - that is, man of the" 'enormousl lar e' PoincareZermelo cycles." Nor does Gibbs' argument show in any way that the

true. Finally, Gibbs fails to show that the limiting value :E(t) will in fact e t at va ue correspon Ing to t e appropnate equl 1 num ensem e or the now modified constraints. As far as Gibbs' roofs 0, lim:E(t) could be arbitrarily larger than the fine-grained entropy of the appropriate ca~oo

y

e,

the assumption, that one can identify the ultimate ensemble as that which approximates the equilibrium ensemble although, as just noted, he really has not demonstrated this for he has onl shown at best "a certain change in the direction of the canonical distribution."

69

Historical sketch

e. Gibbs observed that we often infer future probabilities from present probabilities, but that such inference to the past was illegitimate. In their 1 0 the Ehrenfests res ond: "The enultimate ara ra h of Chapter XII [the paragraph in which Gibbs makes the relevant observa~,".L~J"'.L,,,,,"i

i

Critique of Gibbs on the thermodynamic analogies. As the final systematic comparison of how the components of the statistical-kinetic tiVe approac es 0 0 tzmann an 1 s. First, the note that the Maxwell-Boltzmann distribution of molecules as they are distributed in Jl-space is formally comparable to the canonical in the canonical ensemble. If we think of slowly modifying the con-

both resulting equations will formally resemble the familiar equation oglca eory. For a system in equilibrium, Boltzmann thinks of the system as having a representative point in r-space, which characterizes its micro-state, as correspond to the Maxwell-Boltzmann distribution. From this distribuyn i u phases in the appropriate canonical ensemble with the thermodynamic quantities. How do Boltzmann and Gibbs treat the roblem of two s stems initially each in equilibrium but energetically isolated, and then brought

propriate Maxwell-Boltzmann distribution over its molecules. Once in interaction, they are in a non-equilibrium initial state. The kinetic equation and H-Theorem show that on the avera e and with overwhe1min frequency) the combined system will evolve to the new equilibrium

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Physics and chance

"approximates" the appropriate canonical ensemble for the combined w y e v rip r nn, one the slow variation in the parameters, treat the system as always in a Maxwell-Boltzmann distribution. Using the thermodynamic analogy relation obtained earlier one can derive that entro is tentativel generalizable to -H = -I:ta,loga, where at is the occupation number

canonical ensemble at every moment. The analogue to entropy is given by - plogpdpdq, where p is the density function of the canonical ensemble in r-s ace. How do they treat the increase of entropy in an irreversible process?

in almost all motions, assume smaller values of H at later times. For Gibbs, y coarse-graining r-space and it is the quantity I:,Pt ogP" e ne takin the P's as avera e ensemble densi in the i-th box whose decrease with time represents the changing ensemble and which, in the

Concluding remarks. Summing up their critiques both of Boltzmann an Gi s, teE re ests remar t at t eir conceptua investigation into the foundations of kinetic-statistical mechanical theo re uired that the "emphasize that in these investigations a large number of loosely for-

logical point of view is serious and which appears in other branches of mec anlCS to a muc sma er extent." But these foundational and conce tual difficulties have not revented physicists from applying the basic modes of thought over wider and

will prove warranted in their optimistic use of the theory? Here the re ests are guar e In t elr prognosIs, not east ecause 0 t e notorious difficulties with the fundamental consequence of equilibrium theory - the equi-partition theorem - giving apparently wrong results

Historical sketch

71

dynamical underpinning of the theory was replaced with a quantum micro-dynamics.

v. Subsequent developments peared, there has been an intensive and massive development of kinetic theory and statistica mechanics. To hope to even survey the mu tip icity of ramifications of the theo would be im ossible here. It will be hel ful for our further exploration of foundational problems, however, to offer

gases in which more than binary collisions are relevant, and so on) and to rea ms 0 p enomena suc as magnetization 0 matter an t e lstnbution of ener in radiation that althou h treatable in eneral statistical mechanics, go beyond the case of many-molecule gas systems in various

directly relevant to our project. Here I will offer the barest outline of the lrectlon In w IC attempts ave een rna e to contInue e program of rationalizin or 'usti in the methods used in statistical mechanics, in particular in rationalizing and justifying the choice of probabilistic

equilibrium theory. Such programs of rationalization and justification are a so lntlffiate y connecte WIt t e pro em 0 exp alnlng w y t e osited probability assertions are true, if indeed they are. Or why, if they are not, the theory that posits them works as well as it does. Because the

out in greater detail.

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Physics and chance

the energy fails to be so distributed when the vibrational and rotational r I qu'-p i' eor ea s 0 totally incoherent results, because the possibility of standing waves in a cavity having unlimitedly short wavelength leads to a distribution function that diver es with hi h fre uenc . One of the origins of quantum mechanics is in Planck's study of the

is transferred from radiation to matter, and vice versa, only in discrete packets, each having its energy proportional to the frequency of the relevant radiation. This was eneralized b Einstein to the view that energy exists in such "quanta" even in the radiation field. Combined by

get the older quantum theory. Soon, difficulties of applying that theory to more genera cases than the hy rogen atom, a ong wi the desire for a eneral uantum kinematics on the art of Heisenber and an exploration of "wave-particles" duality by de Broglie and Schrodinger,

with ensembles of quantum micro-states, where these are represented by rays in a HI ert space or, more genera y, y enslty matnces. We shall, of course, not divert ourselves into the m steries encumbent in a study of the meaning of the foundational concepts of quantum

quantum mechanical micro-theory differs in its structure and its predicbons rom a statlstlca mec anlCS oun e upon an un er Ylng c aSSlca mechanical micro-theo . But we will note here one curious a arent consequence of the theory and a few ways in which the change in the

Historical sketch

73

mechanics of systems made up of a multiplicity of similarly constituted particles. For reasons that are, to a degree, made available by some results of uantum field theo it turns out that in considerin ossible states of systems made of particles with half-integral spin, we must con-

whose spin is integral,. we must restrict ourselves to symmetric wave functions. In both cases, the simple way of counting micro-states in the classical theo is abandoned. This is the method that takes micro-states obtained from one another by a mere permutation of particles from one

for half-integral spin particles, and the Bose-Einstein distribution for particles with integra spin. The Maxwe -Bo tzmann distri ution remains onl as an a roximatin distribution for s ecial cases. Philosophically, the new way of counting "possible micro-states com-

of the foundations of quantum mechanics, it will be best to leave them to e SI e. More relevant to our ur oses are some s ecial features of ensembles in the quantum mechanical statistical theory that do interfere with the

mechanics, we can have an ensemble that has no "recurrence" over time In Its Istn ution, even 1 t e In IVI ua systems In e ensem e are a overned b the Poincare Recurrence Theorem. In quantum statistical mechanics, this is not true. If we want a non-recurrent ensemble, we can

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Physics and chance

similarities to the quantum mechanical situation. Extendin the uilibrium. theo to new domains. In standard statistical mechanics, the methodology for deriving the thermodYnamic

calculated by establishing the appropriate canonical ensemble for the system in question. The form of this ensemble is determined by the external constraints laced u on the s stem and b the wa the ener of the system is determined, given those constraints, by the generalized

components, one can use the micro-canonical ensemble or the canonical ensemb e to represent the pro em with the assurance that the thermod namic results obtained will be the same. Actuall the true rationale for this requires subtle argumentation, some of which we will note later.

Z=

in the classical case, and by the analogous sum -Et/T

In e quantum mec e energy an e temperature of the s stem. By a Gibbsian thermodYnamic analogy, one can identify a thermo-

derive the other thermodynamic functions. The end result of the process IS a c aractenzatlon 0 e equll num t ermo Ynamlc propertIes 0 e stem in terms of the arameters constrainin the stem and the features of its internal constitution, such as the number of constituent components

Historical sketch

75

tions and momenta of the components, depending in crucial ways, for example, on spatial separations of particles acting on one another by otential-derived forces and in the case of electroma netic forces b forces that may depend on relative velocity as well, the actual calculation

wealth of approximative techniques, series expansions, and so on. Cases such as 1. Onsager's exact ca cu ation of the thermodynamic functions for the Isin model of a two-dimensional lattice of s innin articles interacting by their magnetic moments are rare triumphs for the theory.

method is great: dilute and dense gases of molecules; one-, two-, or ee- imensiona arrays 0 spinning partic es interacting y t eir magnetic moments· radiation confined to a cavi (which can be viewed as being "composed" of its monochromatic component frequencies); inter-

and approximative techniques.

Phases and hase chan es. One of the most characteristic macroscopic features of matter is the ability of macroscopically radically differ-

"thermo-

existence of a theory that supplements the general principles of statistical mec anlCS an as proven enormous y success In s e lng 19 t upon the nature of the change from one phase to another, at least upon the nature of the transition to the solid phase. The special feature of solidity

to be expressible in a correlation of the state of nearby molecules because 0 elr ynamlc In erac on, In a crys a one n s an as onlS lng regularity that holds to macroscopic distances. Reflecting on the fact that small arrays and large arrays show the same

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Physics and chance

show phase transitions to the solid state that are quite similar in their Y I Y u im iona i and the number of degrees of freedom that are available to its constituents are what are important in the characterization of the nature of the hase transition. The s ecifics of the intercom onent forces are much less important.

2. Rationalizing the equilibrium theory Ergodicity. We have seen how, from the very beginning of the ensemble a roach to the theo there has been a felt need to 'usti the choice of the standard probability distribution over the systems repre-

would mean to justify or rationalize such a probability distribution, as we as as in w at sense suc a rationa ization wou count as exp anation of wh the robabili distribution holds in the world. It will ut matters into perspective, though, if I here give the barest outline of the

at each stage and how it was shown will be provided in Chapter 5. Maxwe ,Bo tzmann, an E e est a propose one-version or anot er of an Er odic H othesis. In some versions, one s eaks of the d amical path followed from the micro-state at a time of a perfectly isolated system.

sort from the outside. In some versions of the hypothesis, one speaks of t e pOInt as traversIng a pat t at eventua y goes t oug eac pOInt In the accessible re ion of hase s ace. In other cases, the ath is aIle ed to be dense in the accessible region, or to come arbitrarily close to each

not demonstrable? Although the micro-canonical ensemble is provably a statIonary pro a I Ity Istn utlon, we 0 no ,WIt ou an rgo IC ypothesis, know that it is the unique such stationary probability distribution. Ergodicity is supposed to guarantee this. Ergodicity is supposed to

Historical sketch

77

point in the phase space. Then if Quasi-Ergodicity holds, the trajectory started

.

.

.

., .

.

state of a system and any other micro-state allowed by constraints, the system

.

..

.

where the avera e is enerated usin the micro-canonical robabili distribution. Finally, it is supposed to follow from ergodicity that the

portional to the size of that region of phase space, when the size is measure uSing t e unique lnvanant measure. First, A. Rosenthal and M. Plancherel show that the strongest version of Ergodic Hypothesis - that the path of an isolated system will

quasi-ergodicity - that is, to the claim that the path will be, instead, ense In e p ase-space region. seems 0 e as cu 0 prove any realistic model of a system quasi-ergodic as it was to prove it ergodic in the stricter sense. In any case, as we shall see, quasi-ergodicity, even if

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Physics and chance

o ime in a region 0 si e 0 region. u condition - metric indecomposability - is establish for realistic models. " " around ergodicity, offering weaker rationales for the probability distribu-

y r rop the rationalization program entirely, proposing instead that the fundamental probability hypothesis be taken as an ineliminable basic posit of the theo . Finally, as the culminating result of a long-term mathematical research

able to prove metric indecomposability for certain models that bear an interesting relation to realistic models of molecules in a box. But, at the same time other im ortant mathematical results at the hands of Kolmogorov, Arnold, and J, Moser provide rigorous limitations on the

As noted, we shall follow out this history in detail in Chapter 5. More important y, we sha ere exp ore in etai just what one has a right to ex ect from er odici results when the are available in the manner of justifications of the probabilistic posits, rationalizations of them, or

1. We ave seen continua re erence in our

historical surve to the im ortant fact that the s stems under consideration are constituted out of a vast number of micro-constituents. This

that the canonical and micro-canonical ensemble calculations will give e arge e same va ues or t ermo ynam1c quant1t1es. an 1S use 0 numbers of de rees of freedom of the s stem be ri orousl 'ustified? The attempt to put such claims on a firm foundation is known as the study

Historical sketch

79

cases. Furthennore, there is in this theory a nice sense of "control" over the limiting process in that one can not only prove things about the limit but et estimates on deviation from the limit in finite cases. This is something usually impossible to obtain in the situation of ergodic theory where

available. What are some of the resu ts one desires from the study of the thennod amic limit? (1) We presuppose in statistical mechanics that the effect on the behavior

due to intennolecular interaction. But what genuine right do we have to be ieve that this is so? Can we s ow at in t e t enno ynamic imit, the contribution of the molecule-box interaction to the otential ener stored in the gas goes to zero?

all distributions consistent with the specified constraints, and if we deman t at it ave t e appropnate a 1tivity properties or in epen ent com onents of a s stem then the uni ue statistical mechanical definition of entropy can be derived. Now in thennodynamics we take entropy to

the entropy. Can we show in the thennodYfiamic limit that this will be so. at 1S, assum1ng t e usua equ11 num ensem e 1stn ut10n an e usual definition for entro in tenns of the distribution, can we show that in the limit of the vast system, the entropy will have the appropriate ?

will describe a system whose fluctuations away from the dominating equ 1 num state W1 ecome neg 191 e. ereas eac ensem e provides a probability distribution over micro-states that is such that "most probable" distributions in r -space overwhelmingly dominate in prob-

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Physics and chance

thermodYnamic transformational relations to one another. (4) The existence of multiple phases in equilibrium with one another can lausibl be claimed to be modeled in statistical mechanics b a partition function that is non-analytic - that is, that fails at certain points

everywhere. The thermodynamic limit approach to this problem is designed to show that such smooth behavior of the partition function can indeed break down in the thermod amic limit leadin in that idealized case to a non-everywhere analytic partition function represent-

be made in order to derive the desired results? In the case of a gas of interacting mo ecu es, it is t e nature 0 t e InteractIon t at is crucla . anI for certain ener etic interactions of the molecules will the thermodYnamic limit results be obtainable.

action potential attracting one molecule to another. Next, there is a requIrement on t e InteractIon t at t e posItIve part 0 t e potentIa conver es sufficient! uickl to zero with increasin molecular se aration. For some interactions, proving these results is not difficult. For other

Historical sketch

81

prove the four results discussed as the aim of this theory. There is an alternative to the approach that defines the thermodynamic functions for finite s stems and then investi ates how these functions behave in the thermodynamic limit. In this other program, one attempts

extent and with an infinite number of components, but whose density is a specified finite value. When it comes to defining thermodynamic ro erties for such a s stem one looks for the a ro riate ensemble definition for quantities such as entropy density, rather than total en-

chanics is finding the appropriate restriction on initial micro-states to guarantee t at wi y ivergent e avior oesn't ensue. For such i nite s stems the ossibili of solutions to d namic evolution failin to exist after a finite time arises - say, by an infinite amount of energy being

Once such constraints are discovered, the next task is to say what an equi inurn state 1S or suc an i ea 1ze 1 nite system. One approac uses er odici as the definin criterion for a state bein e uilibrium. Although such equilibrium states for typical systems have been found,

ergodic, such as the ideal gas, are ergodic in the infinite-system limit. t er 1ngeruous c aractenzat10ns 0 equ1 1 num ave a so een proosed. A su estion of Dobrushin, Lanford, and Ruell (the DLR condition) is that we think of equilibrium for a classical infinite system as

region will be describable by a Gibbs grand canonical ensemble. For quantum systems, an 1ngen10us proposa rom u 0, art1n, an Schwinger (the KMS condition) provides a clear characterization of equilibrium state that can be extended to classical systems as well. Another

local perturbations.

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Physics and chance

r ppr q u - qUI num i u e gas. But whereas the H-Theorem - at least if the presuppositions of the posit of Molecular Chaos can be established - gives us reason to ex ect in the robabiIistic s ns n r ac t iIi i non-equilibrium initial state, detailed knowledge of the nature of that

fifth-power forces - the so-called Maxwell potential - the solutions of the kinetic equation for realistic potentials were a long time in coming. In 1 16-17 S. Cha man workin from Maxwell's transfer e uations and D. Enskog, working from the Boltzmann equation, were able to

an analytically hopeless task to solve. Instead, relying upon the phenomenological equations of hydrodynamics, they simply assumed that linear transfer coefficients existed and looked onI for solutions that presupposed such linear coefficients. The implicit assumption here is

whose future evolution obeys the familiar phenomenological regularities o y ro ynamics. T e specia c ass 0 so utions C apman an Ens og found are usuall characterized as normal solutions to the e uations. Although the discovery of such solutions can hardly be taken to give a

outrun the resources of the phenomenological theory. One can, for examp e, compute numerica va ues or t e inear transport coe dents, values that are sim I inserted into the henomenolo ical theo . One can also derive such empirically confirmed results as the dependence

Historical sketch

83

equilibrium phenomena will be susceptible to the apparatus of ensembles and their evolution, or of kinetic equations derived from this kind of ensemble d amics. Whereas equilibrium theory is founded upon a choice of the equilib-

will require an ensemble way of dealing with systems. If a kinetic equation or some other dynamical description of approach to equilibrium is to be ossible at all it can onl be in a robabilistic sense - whatever that turns out to mean. But what should a theory of non-equilibrium

initial ensemble and by the dynamical laws of ensemble evolution that are entailed y the micro- ynamics 0 t e in ivi ua systems in t e ensemble. If an initial ensemble is icked at one time the evolutiona course of that ensemble is not open to our choice, for it is fixed by the

cal evolution often involves the positing of some "perpetual randomness" assumption t at in one way or anot er, genera izes t e Posit 0 Molecular Chaos. How the ensemble evolves will de end u on its initial structure. We will need to justify or rationalize any such assumptions

imposing on the micro-dynamical laws some posit of a statistical nature, we s a ave to s ow t at t e POS1t 1S aetua y conststent W1t t e constraints on evolution im osed b the micro-d namicallaws. How should an initial ensemble be chosen? This immediately leads

to be described? In the case of equilibrium theory, the answer to this question as a ways een c ear. ne c ooses t e energy or 1tS mean one was doin canonical ensembles) or, in special cases, the energy plus the small number of additional knowable constants of motion. Then one

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further problem of choosing a probability distribution for the systems qu riu ry p available to us, although, once again, it provides direction in limited kinds of cases. We shall explore some general proposals for systematicall ickin the robabili distribution iven the macrosco ic constraints. What methods, given an initial ensemble, or even leaving the choice

equilibrium behavior? Ideally, one would like there to be methods that, given a sufficient characterization of the initial ensemble could usin the micro-d namics alone, allow one to derive a description of ensemble evolution from

or justify a kinetic equation (in this case, the Boltzmann equation) derived by other means. The actual route to inetic equations in the general case comes about throu h the ositin of some robabilistic rinci Ie that superimposed on the dynamical laws, permits the derivation of a kinetic

One group of such methods appears in the form of a direct generalization 0 t e Posit 0 Mo ecu ar C aos. Here, t e unctions t at escri e the correlations of the motions of molecules of various kinds ( ositions momenta, and so on) are posited to be simple functionals of the func-

posit or Ansatz, into closed equations for the lower-order correlations. e met 0 1S a c ear genera 1zat10n 0 t e aS1C POS1t 0 a "perpetua Stosszahlansatz' of the kind noted b the Ehrenfests. By use of this method, one can seek for kinetic equations for dense

dense gas, it has proven to be extraordinarily difficult to construct an appropnate 1netlc equat10n. , 1n act, 1t 1S usua y not on y lffiPOSsible to solve the kinetic e uation derived, in the sense that Chapman and Enskog solved the Boltzmann equation, but even to prove an ap-

Historical sketch

85

be viewed as a large number of energetically, almost independent, subsystems that interact only to an energetic degree much smaller than their intrinsic ener . Man s stems are of this nature - an almost ideal as with small intermolecular potential energy, radiation coupled by a small "

Here, one can deal with states that are the invariant states of the uncoupled system, and one can assume that the introduction of the small cou lin serves to ive rise to transitions of the s stem from one distribution over the invariant states of the uncoupled system to another. The

of such functions over those states. But once again, an assumption that transitions rom one suc state to anot er are etermine y xe constant robabilities over time becomes another wa of re resentin the system as a Markov process, and leads, by means of what is called a

Other approaches to the problem of generalizing from the Boltzmann equation 0 ow a p an enve In conception rom 1 s' treatment 0 the non-e uilibrium ensemble. Gibbs su ested that we could deal with non-equilibrium, at least in part, by "coarse graining" the r-phase space

corresponding to the initial ensemble "spreading out" in such a way that a t oug Its vo ume remalne constant, t e proportion 0 eac coarserained box occu ied b it became more and more uniform with time, heading toward an ensemble in which each coarse-grained box was

increase in coarse-grained entropy would in fact occur, a deficiency clearly pOinte out y t e re ests. ne can supp y suc a postu ate, owever. As we shall see in Chapter 6,III,3, the assumption that the s stem evolves as a kind of Markov process does the trick. The assumption, in

assumption, like the generalized Posit of Molecular Chaos, amounts to a e Ynamlc evo u Ion con nuous reran omlza Ion assump Ion a ou of the ensemble, an assumption whose consistency with the underlying dYnamical laws remains in question.

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Physics and chance

is in general no systematic way of rigorously or definitively elucidating

It is one thin to be able to describe initial ensembles in a s stematic wa

and to be able to posit appropriate randomization hypotheses so as to be

another thing again to be able to explain why the description is successful, if it indeed is. The whole of Chapter 7 is devoted to just these issues but once a ain it will be useful to ive here the briefest ossible outline of what some approaches to the justification and explanation problem

the "mainstream" or "orthodox" approaches. One of these non-standard approaches argues for the reasonableness of the principles that give rise to kinetic e uations and irreversibili throu h an inte retation of the probabilities cited in statistical mechanics as subjective probabilities. The

the perpetual interaction of the system with its environment, small as that may e, t at accounts or t e success 0 t e statistica met 0 an t e randomization osits. A third non-orthodox a roach seeks the resolution of the difficulties in the positing of time-asymmetric fundamental

explanation are sought look for their resources to identifiable physical eatures 0 t e wor ans1ng out 0 t e structure 0 t e system 1n quest10n and the structure of the micro-d amical laws in order to ound the derivation of the kinetic description.

Historical sketch

87

on the structure of the initial ensemble and on posited randomizing features of dynamic evolution. In both cases one would like to find some ounds for believin that all initial ensembles will have the re uisite feature. Here, what one seeks are physical reasons for a limitation on the

underlying micro-dynamical laws - plus additional constraints upon the ability of an experimenter, restricted in some sense to the macroscopic mani ulation of the s stem to constrain a s stem's initial conditions. Whether such resources will, by themselves, serve to rationalize the re-

as well is, as we shall see in Chapter 7, a matter of important controversy and is fundamental to the study of the foundations of the theory. When one's derivation of a kinetic e uation rests either in whole or in part upon randomization posits imposed on dynamic evolution, a rather

dynamics - at least in the orthodox approaches in which isolation from t e outsi e is retaine or t e system an in w ic one accepts t e standard micro-d amics as a iven and not as some a roximation to a more stochastic or probabilistic underlying theory - one must reconcile

structure feature of the initial ensemble. We shall see both approaches use In our etal e reconstructlon 0 t ese arguments In apter. It is in this art of the ro ram of rationalizin the non-e uilibrium theory that the use is made of a series of important advances in our

instability of solutions under minute variations of initial conditions for e equatlons, an 0 t e connection 0 sue emonstra e lnsta 1 llles with various "mixing" features of the ensembles whose evolution is governed by these equations, playa central role in the part of foundational

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Physics and chance

VI. Further readin

For a history of thermodynamics, see Cardwell (1971). Two clear books on thermodynamics emphasizing fundamentals and historically and conceptually oriented. In 0 e eory, see Isza For the theory of non-equilibrium systems close to equilibrium, two good sources are Kubo, Toda, and Hashitsuma (1978) and de Groot equilibrium thermodynamics far from equilibrium can be found in

foundational issues is Tolman (1938). Two later works covering both older and newer topics and emphasizing fundamental issues are Toda, Kubo and Saito 1 78 and Balescu 1 7 . Munster 1 6 is a treatise covering both foundational issues and applications.

of the theory up to 1910. For quantum statistical mechanics, Toda, Kubo, and Saita (1979) and Tolman (1938) are excellent. An excellent surve of the im ortant develo ments in statistical mechanics in the period prior to its publication is O. Penrose (1979), a

Toda, Kubo, and Saito (1979) and Balescu (1975) contain very accessi e treatments 0 t e p ase-transition pro em an escriptions 0 t e exact! solvable models. The issue of the thermodynamic limit is taken up with mathematical

,. rur 11 ~

..

1,...1

1'r-';..I,

.. II

Historical sketch

..

r

It;.' Ull tHC l'

11

•

;..I

•

pll

• •

r

.•

Ul UIC

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..

89 -

tion of the probabilistic posits in equilibrium and non-equilibrium theory, see the "Further readings" at the ends of Chapters 5, 6, and 7 of this book.

ment of foundational problems in statistical mechanics, the concept foundational issues. It is used informally in the dialectic designed to reconcile the time-as mmet of statistical mechanics with the timereversibility of the underlying dYnamics, although, as we have seen, its used to account for the existence of equilibrium as the macro-state to account for the approach to equilibrium as the evolution of microstates from the less to the more probable. More formally, the attempts at finding an acceptable derivation of Boltzmann-like kinetic equations all tion of a "probability distribution" over the micro-states compatible with ensemble, invoked in the later work of Maxwell and Boltzmann and rna e the core 0 the Gib s presentation 0 statistica mechanics, real y amounts to the positing of a probability distribution over the micro-states changes of such a distribution over time as determined by the underlying

just what elements are described within the formal theory. As we shall problem of understanding in a general way what probabilities consist of - IS one t at remaIns rep ete WIt controversy. at seems Inlt1a y to a simple matter, requiring only conceptual clarification in matters of detail, is aetuall a uzzlin and difficult area of hiloso h . Although it will be the aim of this chapter to survey a number of the tation of probability, it will not be my purpose here to explore this to the

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us too far afield from our main goal - the pursuit of the foundational problems in statistical mechanics. But a survey of these general approaches to inte retation will rovide an essential back round for stud in the more specialized problems in the application of probability to physics

resolution on some of the particular results obtained in the study of statistical mechanics, because, as we shall see in Chapter 7,IV,1 and 2, it is sometimes unclear where the dividin line between urel conce tual issues and issues about the nature of the world as revealed in our best

I. Formal aspects of probability 1. The basic postulates

there must be a collection of subsets of E, called events, that is closed under set union, intersection, and difference, and that contains E. In general, this collection, F, will not contain every subset of E Probability numbers, P. P assigns the value 1 to E - that is, PCE) = 1. Most imcommon, then P assigns to their union the sum of the values assigned to A an B - t at is, A n B = <1>, then PCA u B) = PCA) + PCB)

When the set E is infiniteI lar e, F can contain an infinite number of subsets of E In this case, the additivity postulate here is usually extended ever i "* j, then it is posited that PCu{A i ) = :EiPCA,). This is called CoUll a e a 1 IVI or (J -a 1 IVI. IS pos a e IS some mes questioned in subjective probability theory, although denying it leads to peculiar behavior for probabilities even when they are interpreted sub-

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Physics and chance

n i n 11 pro all ,in ro uce in this approach by definition. If PCA) 0, then the probability of B given that A, the conditional probability of B on A, written PCB/A), is just P B A P . T is n nd probability of B to the condition of A being assumed to be the case.

'*

events where each event is independent of all the others in the collection, rather than independence being merely between pairs of events. A se uence of "trials" that constitute an "ex eriment" is an im ortant notion. Here, one thinks of a sequence in which some elementary event

sequence is said to be a Bernoulli sequence. If the probability of an event in a trial differs, perhaps, from its probability conditional upon the outcome of the immediatel recedin trial but if this robabili PCA n/ An-I) is then the same as the probability of An conditioned on all

of one's probability attributed to the outcome of a given trial. In a Markov sequence, nowing t e outcome 0 w at appened just before the specified trial rna lead one to modi one's robabili for a 'ven trial but knowledge of earlier past history is irrelevant.

elementary events in E in such a way that the set of all elementary events t at ve a va ue ess t an a spec' e amount is in F, an ence is assigne a robabili . One can define the distribution function for the random variable as the function whose value at a is just the probability that the

random variable will have a value less than a is increasing at a. From the lstn utlon unction or a ran om vana e, one can e ne lts expectation. Intuitivel , this is 'ust the "sum of the roduct of the value of a random variable times the probability of that random variable." Naturally,

expectation of a random variable is its mean value. e aSlC postu ates 0 pro a llty are 0 extraor lnary SlffiP lClty. Naturall ,much refinement is needed to make thin s precise in the eneral cases of infinite collections of events and so on. Although the postulational

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next section.

orne consequences

0

sequences. Suppose such a sequence of trials is performed, with a fixed probability for the outcomes of each trial and with the independence condition holdin so that for an collection of trials the robabili of a joint outcome is merely the product of the probabilities of the out-

distribution for this random variable as fixed by the probabilities of the various outcomes on anyone tna . 1S pro a i 1ty 1stn ut10n, y t e definition of the se uence as a Bernoulli se uence, is the same for each trial.

limPClYn -

J.11

> £) = 0

00

where is the sam Ie mean of the random variable in uestion and is the expected value of it on a single trial. What this says is this: Suppose

of the sample mean from the expected value becomes, as the Weak Law eman ,sparser an sparser as e num er 0 na s 1ncreases. u 1n each run, or in a large proportion of them, the sample mean never gets within £ of the expected value, and stays there forever. That this is not

Physics and chance

94 n

at this infinite collection of repeated trials, as the number of trials goes "to infinity" in each sequence, we can take it that the probability of a se uence ettin a sam Ie mean that settles down to wi hin 0 th expected value forever has probability one, and the set of those that

tells us that there is a limiting particular distribution function to which the distribution of such sample means in this infinite collection of infinite trials will conver e. These facts about large numbers of trials (or more correctly about the

moment the reader should only note how the notion of probability plays a crucial role in the statement of these results. The "convergence of the sam Ie mean to the ex ected value" noted in all these results is some form of convergence "in probability." For future reference, it is important

we shall see in discussing the role of the Ergodic Theorem in attempts to provi e a rationa e or equ' i rium statistica mec anics, at it is possi e to rovide a close analo e of these "lar e number" results even in some important cases where, intuitively, independence of trials is not the case,

a law of large numbers can be derived in some cases even where the outcome 0 eac succeSS1ve tna 1S comp ete y causa y eterm1ne y the outcome of the initiai trial. One very simple consequence of the postulates and of the definition

one At. Then, for any event Bin F, the follOWing holds:

P(A k / B) = - - - - - "LjP(B/A j)P(A j)

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3. Some formal aspects of probability in statistical mechanics The role of probability in statistical mechanics is one fraught with puzzles, man of which will be com onents of the central issues to be discussed in this book. Here, I want only to note a few basic aspects of the for-

The class of elementary events relevant to statistical mechanics is phasespace, the class of points representing each possible micro-state of the d namical s stem in uestion. For the classical statistical mechanics with which we will be dealing, each point represents the full specification of

statistical mechanics is of vital importance for statistical mechanics in practice, but not something we wi I have to eal wit very often. The class of events F. is 'ust a class of sets of micro-d namical states. Random variables will then be functions that assign numbers to micro-states in

for the random variables in question also exist. To e ne one's pro a i ities, start wit a measure, a genera ization of the ordina notion of volume on the hase-s ace in uestion. The choice of this measure ultimately constitutes the determination of the

foundational question. The standard measures are all derived from the measure on p ase-space as a woe t at wor s y ta lng a vo ume In hase-s ace to have its size be essentiall the roduct of all its extensions in coordinate and momentum values. Usually one will be dealing

those compatible with some class of macro-quantities. The appropriate measure on t s su -space w not In genera e t e slmp e vo ume measure of the hase-s ace restricted to this lower dimension, but an appropriate modification of it. In the case of equilibrium statistical me-

dynamic evolution of the systems represented by trajectories from their lnlt1a representlng p ase-polnt. n non-equll flum cases, as we s a see in Chapter 7, such measures also playa role, but one whose rationalization is rather less well understood.

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IS easy 0 s ow zero size, regIons a are assigne ero pro a 1 i from the basic postulates that the empty set in F - the set containing no elementary events - must get probability zero. But having probability in Fwill have probability zero. In the ordinary measure for regions of the

that is such that probability zero is assigned to any set with zero size in the measure - an assignment of probability that is, in mathematicians terms "absolutel continuous" with res ect to the standard meas r makes the formal method for assigning probabilities simple. When the

non-negative, measurable, function, f, over the micro-state points of the relevant subspace of phase-space. The probability assigned to a region A in the hase-s ace is then obtained b "addin u "the values for all the points in A, or, more rigorously, by integrating the function with

appropriate region of phase-space so that the total probability assigned to a region is obtained y a measure 0 the amount of the total probabili over the whole hase-s ace that is s read in the re ion in uestion. Most commonly in statistical mechanics it is "uniform" spreading that is

ll. Interpretations of probability

A single formal theory can be applied in many different domains. The same . erentla equation, or examp e, can app y to e ectromagnetic fields sound heat or even to henomena in the biolo ical or social world. The postulates and basic definitions of formal probability theory

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function to be characterizing what we take intuitively to be probability? As we shall see in the brief survey next, this has become one of those iss es hat seems at first easil resolvable but that becomes sli e and elusive when a complete and clear explication is demanded.

1. Frequency, propol1ion, and the "long run" Consider a finite collection of individuals with a number of properties such that each individual has one and onI one member of the set of properties. The relative frequency with which a property is instanced

quencies in ordinary finite populations? One objection to this is its limitation of probability values to rational numbers onI whereas in man of our robabilistic models of henomena in the world, we would want to allow for probability to have any

moving in a confined region that we think of as partitioned into nonover apping su -regions. Sure y we un erstan t e notion 0 t e proportion of some finite time interval the article s ends in an one of the partitioning sub-regions. These proportions can, in general, have non-

somewhere in the region PCE) = 1, and for exclusive regions the proportion 0 time t e partlc e spen in t e Joint region 0 an WI just be the sum of the ro ortion of time sent in A and that sent in B. But there is a standard objection to any proposal to take such clearly

frequencies and proportions should in some sense cluster around proba I ltles, we cannot, In genera , expect pro a I lties an proportIons to be e ual. The robabili of heads on a fair coin toss is one-half, but onl a small proportion of coin tossings have heads come up exactly half the

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Physics and chance i

world more strictly identifiable with probability than the frequencies or proportions in finite reference classes. For example, it is sometimes ar ed that in the case of somethin like coin to in wh r nl of a finite number of outcomes is possible, the probability of an outcome

run" - that is, as the number of tosses increases without limit or "to infinity." But such Ion -run relative-fre uen accounts of robabili are roblematic in several ways, at least if they are intended as providing some

problem is that the limit of a series - the natural way to define probability in this approach being to use the usual mathematical definition of a uanti as a limit - can va de endin u on the order of the terms in the series. Although the finite relative-frequency approach to probability

demand such an order of events. But for events that are orderable in time or in some ot er natura way, t is is not an insupera e conceptua roblem. More disturbing is the degree of unrealistic idealization that has been

frequency view are taken as existing in the world, in what sense do the I rute sequences 0 events nee e to e ne pro a Ity in e ong-run a roach reall exist? Will there ever reall be an infinite number of tossings of fair coins in the world? If not, then have we abandoned the

to handle - the possibility of deviation between relative frequency and w at IS ta en, IntuItIve y, to e pro a I Ity. Irst, note t at e assocIatIon of robabili with relative frequen guaranteed b the laws of lar e numbers holds only when the sequence is a Bernoulli sequence, a se-

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some way in which, by using infinite sequences of trials, we could "define" the notion of probability in terms that themselves do not presupose the understandin of robabilistic notions. A further consideration in this vein is even more important. Even given

relative frequency with the probability in an individual trial. We are told only that the probability that they will be equal is one, or that the probabili of their bein une ual oes to zero. But as we have noted robability zero is not to be identified with impossibility. Even if an infinite

sity. Although the frequentist takes probability as something "in the world" ut attn uta e, at east pnman y, to co ect10ns 0 tna s or expenments, the dispositionalist thinks of a probability attribution as bein fundamentally an attribution of a property to a single trial or experiment. For the

generally take the correct probability attribution to an individual trial as some mg a IS on y re a 1ve 0 lIng 0 a na as In some spec c class or other. But such problems concerning the uniqueness or nonuniqueness of the correct probability attribution to an individual trial (or,

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Physics and chance

iii en 0 e a p P "quantity" that inheres in the individual event to some "degree" or other ranging from zero to one - that probability is taken to be is a dispositional ro e . Here robabili is assimilated to su h rn1'"'\~t"t1~C bility or the fragility of objects. Intuitively, a distinction is made between

in the form only of "conditional" properties, attributions that have their presence because of what the object would do were certain conditions met. Thus a d iece of salt is cate oricall cubical but soluble onI in the sense that if it were put in a solvent it would dissolve. Needless to

distinction that holds in a context-independent way can be made, arguing that what is for the purposes of one discussion categorical can be for other ur oses viewed as dis ositional. Dispositional theories of probability vary widely in what the disposi-

to survey the most general features of these accounts only briefiy. As we ave seen, t e stipu ation or e nition 0 a ispositiona property is usuall b means of some counter-factual conditional: how the ob'ect would behave were certain test conditions satisfied. What would such an

frequency (or proportion) of outcomes on repetitions of it. Crudely, what 1t means to attn ute a pro a 11ty 0 one- a or ea s on a coin toss is that were one to toss the coin (an infinite number of times?) a relative frequency of heads of one-half would be the result. Here, at least one

single trial is determined by some underlying "hidden" parameters whose va ues rema1n unava1 a e to us, an t e tyc 1StlC case were, as 1S frequently alleged of trials in quantum mechanics, there are no such underlying hidden parameters whose values actually determine the out-

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non-trivial probabilities are involved, the probability of the outcome being one or zero depending on whether it is determined that it occur or that it not occur. For these dis ositionalists the a earance of robabili in deterministic cases is just that, an illusion. For them, where genuine

Other dispositionalists would be loath to drop the idea that even in such cases there wou d sti be real, non-trivial pro a i ities. How would these be defined? B the relative fre uenc of the outcome that would result from repeated trials of the kind of experiment in question. But what

This leads to two problems. First, it seems to some that the dispositionalist account ere may e parasitica on an actua re ative requency or proortion) account makin it no real chan e over the latter. More importantly, wouldn't such an account be subject to the same fundamental

ability can diverge from one another, even in the "long run"? at a out t e ot er case, were t e outcome 0 a g1ven tna 1S tru y undetermined b an hidden arameter values? This is the case that many dispositionalists find most congenial, indeed some of them assert-

Here we have no actual distribution of underlying hidden parameters t t y etefffilne t e outcomes 0 t e tna s to re yon, nor 0 we ave the wor that in an individual trial the "real" robabili must be zero or one because the outcome, whatever it is, is fully determined by the

tions) of outcomes in collections of trials that are (now, in all respects) 1 e t e tna 1n question. Are these frequencies or proportions to be taken to be the probabilities? No. For the dispositionalist, the real probability - the dispositional

are supposed to be,

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i s va ue somew ere i vici i 0 e revea requency. u e manifested frequencies or proportions are not taken to be "constitutive" or definitive of what the probabilities are. n

defined by the use of counter-factual locutions. The probability has a

would now be obtained. The type of trial is now fixed as a kind by all of its features, there being no "hidden" features relative to which a further subdivision of kinds can be made into which the trial could be laced. Yet that can't be quite right either. For, as we have seen, even the

only in probabilistic terms - still hold even if it were possible but we were talking about non-actual runs of trials? One can imagine how to t and im rove the situation from a dis ositionalist stance. Think of a counter-factual being true if in some possible world like ours, but differ-

number of such other possible worlds. In each such world, a certain proportion 0 outcomes wou resu t. Ta e t e isposition 0 pro a i ity as havin a certain value if the distribution over all these ossible worlds of the proportions of outcomes is as would be described by the laws

"realistic" attitude toward "

.. " A group of important attempts to understand probabili ,still in the objectivist vein that we have been exploring in the last two sections, make

Probability

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This alternative approach is in part inspired by views on the nature of theoretical tenns in science. Early empiricist and operationalist views on the meanin of terms referrin to non-directl observable entities and properties in science, wanting to legitimize the usage of such terms

lack of associability with items of experience, frequently proposed that any term in science not itself clearly enoting an item 0 " irect experience" be strictl definable in terms of such items from the "observational vocabulary." But later empiricistically minded philosophers became

played an essential role in a theoretical structure of sentences that was tie own to irect experience at some eve,' 0 y t roug a networ of 10 ical im lications? A term then could be Ie itimate and its meanin could be made clear, if it functioned in the overall network of scientific

One component of the overall structure

inference are the fundamental components of theories of statistical 1 erence. Needless to say, there is no reasonable plausibility to the claim that there is a body of such rules that is universally accepted. The appropriate

levels. Even the apparently simpler

Physics and chance

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what one will take to be legitimized inferences to and from probability e e ining w a we mean y e probabilistic assertions themselves. And fitting the probabilistic assertions into a network of inference in this way, we can "fill in" their meanin without re uirin us to ive an ex licit definiti n of " robabi i " in terms of proportion or its limit, or in terms of these even in other

some crucial elements. Suppose two statistical schools rely on differing upward and downward rules of inference, but whose rules, when combined lead to the same inferences from ro ortions in sam les to roportions anticipated in other samples. Do they merely disagree on the ".

a disagreement about which upward rule of inference is correct? The latter seems to require some notion of another constraint on the truth of the statistical assertions somethin that oes be ond its bein the assertion warranted on the upward rule chosen. And doesn't it seem plausible

the total population, even if, as we have seen, a naive identification of t e pro a i ity wit that proportion can e c allenge ? One direction in which to seek additional elements in the world inning down the meaning we give to objectivist probability assertions is to

background scientific theories. The idea here is that although some frequencies an proportions ave a mere y acci enta or contingent aspect, others can in one wa or another be shown to be" enerated" out of the fundamental nature of the world as described by our most general and

of science when one seeks for a notion of a law of nature as opposed to a mere true genera lzatlon. ereas = rna" IS suppose to ave a lawlike status "all the coins in m ocket are co er" does not. The former generalization is inductively inferable from a sample of the cases

Probability

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"if something else were a coin in my pocket it would be copper." What constitutes the lawlike force of some generalizations? Reliance on such n tions as laws bein true not onI in the actual world but in all" h sicall possible worlds" just seems to beg the question.

of a generalized nature play a fundamental role in that they are the simple axiomatic beliefs from which other beliefs of great generality and ex lanato im ortance can be derived. The idea is that lawlikeness is not a feature of a generalization having some semantically differentiable

tions that ground our overall explanatory structure of the world, or those more restrictive generalizations whose lawlikeness accrues from their bein derivable from the more fundamental laws. The connection of these ideas with objective probability goes through

multiplied by its complex conjugate, provides, in specific physical situations, a orma y pro a I IStlC lstn ution over t e POSSI e outcomes 0 an ex eriment. From this we can formulate our ex ectations not onI of what proportion to find in some run of the experiment, but even of the

tration of proportions and frequencies around theoretically posited proba lltles as t e num er 0 tna s In an expenmenta run Increases. ut we don't claim that the probability is the actual frequency even in the totali of trials, or even in that overwhelmingly large number of totalities of

Physics and chance

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with the holistic idea that meanings can really be fixed by putting a term

populations from it? One puzzle to consider is that the virtue of this approach - its ability to distin uish robabili from ro ortion even in the total 0 ulationmay be in some sense a vice as well. Could it be the case that probability

probability and frequency, that what we took to be the probabilities in the world really had the value we supposed. But from the point of view of the theoretical notion of robabili we have been lookin at could it not be the case that that probability really did have the original value

in that view, but not in any way impossible. But do we want to let the notion 0 "w at pro a i ity is in t e wor " ecome t at etac e rom the fre uencies or ro ortions that actuall manifest themselves? Perha s the idea could be filled out that probability is that proportion of the total

As we shall see in Section III of this chapter, and as we shall again see anum er 0 times, especia y in C apter 8, were t e etai s 0 pro a ii in statistical mechanics are discussed this roblem of radical divergence of probability from proportion or frequency is not merely one ".

in its interpretation, hinge on the question of how we ought to respond to t e puzz e 0 acceptIng a t eory t at POSItS a pro a I Ity or an outcome while seemin to simultaneousl maintain that the actual roportionate outcome of the feature in question in the world we inhabit

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as a whole, in the face of his contention that equilibrium is the overwhelmingly most probable state of a system. Additional uestions abound concernin the relation of robabili construed as a theoretical property and the other features of the world

but we believe a much more interesting answer is available in terms of the composition of the salt out of ions, the nature of water on the molecular scale and so on. Similarl we mi ht "ex lain" an observed ro ortion in nature by reference to a probability that leads to some degree of

nature governing the phenomenon in question, the distribution of initial con itions 0 i en parameters in nature, an so on is avai a e. at is the relation of" robabili "to these more familiar features of the world as described by our scientific theories?

ries that will dispense with it in terms of the underlying physics of the situation. So, it 1S suggeste , "so u e" 0 sap ace or escriptions 0 atomic constitution of solute and solvent and their mutual interaction and "probability" is, similarly, a temporary and dispensable place-holder

question of just what underlying facts of nature ground the attributions et er one cono pro a 11ty encountere 1n statlst1ca mec an1CS. cludes that ultimatel "robabili " ou ht to be somehow defined in terms of these deeper physical elements of nature (as some have sug-

(as others have suggested), or even that it is, in its own terms, an 1ne 1m1na e component 0 our u t1mate t eory w 1C a so suggests 1tself as the right expectation to others), one still must get clear just what the nature of the world is, on the microscopic scale and as described by

notions such as solubility for solids. Getting an agreed answer to this quest10n a presen W1 e 00 muc 0 expec . e 1ssues surroun 1ng the importance of fundamental dynamical laws, of distributions of initial conditions, of the interaction of a system with its environment, of the

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4. Objective randomness tenns of limits of frequencies in long-run sequences of events, it was noticed that the order of outcomes in the sequence could be important for reasons that went be ond the de endence of the value of the limit on the order. A sequence of two outcomes coded by zeros and ones, for

to say that the probability of a zero outcome in a trial was one-half? If we knew the immediately previous outcome, wouldn't we be sure that the zero would or would not occur? Clear! at least for the a lication of our knowledge of frequencies and their limits, some assurance that

the important suggestion that a sequence is random if the same limiting re atlve requency occurs In any su -sequence se ecte y an e ectlve y com utable function eneratin the indexes of the selected trials. This ingenious characterization proved a little too weak, because it included

the probability of zeros in the sequence as a whole was one-half. trengt ene versIons 0 It 0 a etter)o, owever. Other ingenious characterizations of ob·ective randomness have been developed. Some rely on the intuition that "almost all" sequences ought

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of measure one) sequences be characterizable in a specific way (i.e. in some limited language), and that the random sequences be those with all the random ro erties. On this definition the random se uences are, indeed, of measure one in the set of all sequences. But the definition

way statisticians reject hypotheses. A hypothesis is rejected at a certain "significance level" if the outcome observed is sufficiently improbable iven that the h othesis is true. Effectivel characterizable tests for randomness are described, and it is then shown that they can all be

at any significance level. Ano er ingenious notion 0 ran omness uti izes t e notion of how Ion a com uter ro ram it would take to ro ram an effective computer to generate the sequence in question. Although the length of

terms of a universal programming language. Intuitively, for finite sequences, t e ran omness 0 t e sequence 1S measure y t e re ative len th of the shortest ro ram that can enerate it random se uences being those that, in essence, can be generated only by simply stipulat-

extending it fails, and several, inequivalent, alternative ways can be formu ate t at exten t e not1on 1n 1 erent ways. urt ermore, getting a reement between our intuition as to what is to count as random and what gets defined as random by the method of computational complex-

makes it implausible that we can define probability in non-probabilistic terms uS1ng t e not1on 0 0 Jective y ran om sequences. The multiplicity of definitions of randomness do not all coincide, but it is not a great surprise to discover that the initial vague intuitive notion

7,II,3, however, we will discuss in some detail other notions of randomness a ave e1r ong1n 1n e way 1n w 1C a co ec 10n 0 sys ems, each of which has a strictly deterministic evolution, can be characterized as displaying randomizing behavior when the systems are given descrip-

systems suitably constructed, the evolution of a system from coarse-grained ox to coarse-gralne ox may generate a sequence 0 num ers, c aracterizing the box it is in at a given time, that has features closely related to the features we would ex ect of se uences enerated in urel stochastic ways. One of these features will be the fact that "almost all" started in different initial conditions will be of the "random" sort. We

5. Subjectivist accounts ofprobability Whereas

undertaken that outcome will result? And what is it to have such a "degree

or not, and on our degree of certainty that the outcome will in fact tum

odds at which we would bet (act as if) that outcome were really going

in an account of our action in the face of risk. One problem any such

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axioms of probability theory. For the objectivist, this conformity follows either directly or indirectly from the facts about proportions. But why should our artial beliefs obe the axioms in articular the addition postulate?

number assigned to propositions, perhaps conditional one on the other, so that we are looking for P(i/ h), the "probability of the inference i on the h othesis h." The usual Boolean al ebra of ro ositional 10 ic on the propositions is assumed. Next, axioms are posited such as: (1) the

h and the probability of jon h; (3) the probability of i u j on h (when i (J j = is a continuous an monotonic unction 0 t e pro a ilities of i on hand . on h. These axioms of fu ctional de endence Ius some assumptions about the smoothness (differentiability) of the functional

making some conventional stipulations. Here, then, the formal aspects o pro a i ity are ta en to arise out 0 our intuitions t at some partia beliefs ( robabilities) ou ht to de end in a functional wa onI on a limited class of others.

Book" arguments have us reflect on making bets against a bookie on e outcome 0 some tna . e plC our egrees 0 partla e le In t e outcomes, and the bookie offers us the minimum odds we would acce t appropriate to those degrees of partial belief. An ingenious but simple

array of bets that we will accept, but that guarantee- that when all stakes e wlns an we ose, no matter w at t e outcome 0 e are co ecte trial. In order for us to avoid havin such a "Dutch Book" made a ainst us, our probabilities must be coherent - that is, accord with the usual

An alternative and more general approach considers an agent offered

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as if he had a valuation or utility or desirability function over the rewards and losses unique up to a linear transfonnation. It goes something like this: Su ose the a ent alwa s refers x to z whenever x is referred to y and y to z. Suppose also that if the agent prefers x to y, he will also

ciently rich collection of preferences. Then we can assign a probability to each outcome, p, and a utility to the gains or losses incumbent upon the outcome occurrin or not occurrin u and u such that if each lotte ticket is assigned an "expected value," PUl + (l - p)u 2 , then the prefer-

This approach to subjective probability is important as a component of various" nctiona ist" accounts as to w at partia e ie s an esira i ities are, as well as a crucial com onent of the theo of rational decision making. Naturally the full theory is a very complicated business, involv-

among propositions that, intuitively, can be thought of as one proposition elng as wort y 0 e Ie as anot er. ntultlve y p aUSI e axIOms are imposed on this notion of comparative believability. For example, we rna demand transitivity, so that if A is as believable as Band B as believable

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probability assignment to the propositions. Such a probability representation assigns real numbers between zero and one (inclusive) to the ro ositions. And the assi nment is such that A will be as believable as B just in case the probability number assigned to A is at least as great as

holding partial beliefs, but with our changing them in the face of experience. How should we modify our distribution of partial beliefs in the face of new evidence? The usual rule su ested is conditionalization. We have, at a time, not only probabilities for propositions, but conditional

with what we have learned through the evidence, and it generates a new pro a i ity istri ution as co erent as t e one we starte W1t. Much effort has one into ivin a rationalization for chan in robabilities by conditionalization as persuasive as the standard rationales for

Lewis has the agent confronting a bookie and making bets on the outcome 0 one tna ,an t en a 1hona ets on ot er outcomes 0 ot er trials should the first trial result in one s ecific outcome or the other. Only if the second set of bets is made on odds consistent with conditional-

ization follows from demanding that one's new probability distribution as a er t e eV1 ence e as come 1n 1 e ore the observation and if hand k both implied the evidence assertion, e. And B. van Fraassen has shown that conditionalization is the only method of

event classes. A variety of other rationalizations for conditionalizing can a so e glven. n e 0 er an, ecause con 110na 1za 10n 1S a conservative procedure, intuitively changing one's subjective probabilities only to the extent that the change is directly forced by the change in the

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they call a subjectivist approach to probability in statistical mechanics

Suppose one accepts the legitimacy of probability as interpreted as a measure of partial belief, in the manner we have outlined, What, from this sub'ectivistic ers ective is the lace of the version of robabili as "a feature of the objective world" in one of the guises previously out-

(often labeled by subscripts as probability} and probability2)' objective and subjective, just as most defenders of objective probability are perfectly ha to countenance a Ie itimate sub'ectivist inter retation of the formalism as well,

as estimates on our part of the real proportions of the world, Naturally he wi I see princip es 0 rationa i erence to an rom proportions in sam les to those in 0 ulations and hence to and from ro ortions in samples to subjective probabilities, The believer in objective probability

an outcome on the trial is taken to have a certain value, the subjective pro a i ity 0 t at outcome must ave t e same measure, At east t is will be so if the ro ensi chance is one that su oses the absence of underlying hidden variables that would, if known, change the propensi-

the theory structure fixing objective probabilities gets its connection to t e wor . It as een suggeste , or examp e, y S. Lee s, t at we can ive an inte retation to the state-function of uantum mechanics b simply taking the rule that the values thought of as probabilities upon

theory so interpreted partially fix any meanings for objective probabilistic expresslons appeanng In t e t eoretlca networ , eavlng It open, 0 course, that these ob'ective robabilities rna be further inned down b their association with non-probabilistically characterized features of the

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perhaps the distribution in the world of initial conditions over tossings, which underlie its being a fair coin.) Other sub·ective robabili theorists find no room whatever for the notion of objective probability as we have been construing it. For them,

course, there are also all those other features of the world, such as the balance of the coin, the distribution of initial conditions, the quantum state of the electron that are causall connected to the fre uencies we observe. But, from this point of view, there is no need to think of prob-

is, suppose the subjective probability given to a sequence of heads and ta s 1S a nct10n 0 y 0 t e proport10n 0 ea s m t e sequence an is independent of the order of heads and tails. Then, de Finetti shows, this agent's probability distribution over the sequences can be represented

heads, but with an unknown bias. It will be "as if" the agent generated s su Jec 1ve pro all 1es or e sequences y aV1ng a su Jec 1ve probability distribution over the range of possible biases of the coin. Furthermore, let the agent modify his subjective probability distribu-

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represen e as a is u ion ov r propensi ies or iases i gave no propensity probability zero or one, the evolution of the agent's probability distribution over sequences can be represented as the agent having his indeed converging to a propensity equal to observed relative frequency v .i i over biases, but, learning from experience, both converged to the "real objective propensity" in the long run. For de Finetti, of course, there is no such "real" robabili of heads for the coin. All that exists are the convergences to observed relative frequency, convergences themselves j

experience by conditionalization. This result of de Finetti's is generalizable in interesting ways. The key to the re resentation theorem is the s mme of the a ent's initial subjective probability, the probability given to a sequence being invariant

that such sYmmetries in the agent's subjective probability will lead to is acting as' e e ieve in an 0 jective propensity to whic observe relative fre uencies would conver e. Even more eneral results can be proven to show that agents who start in agreement about which sets of

An important concept, emphasized by B. Skyrms, of "resiliency" should a so e note In t IS context. su Ject1ve pro a 11ty IS, aS1ca y, reS11ent if the a ent would not mod' the value he ives the outcome in uestion (or at least would not modify it much) in the light of additional evidence.

no exploration of hidden variables could divide the kind of trial in quese pro a 11 0 e ou come wou Ion In 0 su -c asses In w 1C differ from the probability in the class as a whole. Resiliency is a kind of subjectivistic "no hidden variables" for the probability in question. It

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kind of objectivist resiliency (metric indecomposability) that we will explore in Chapter 5.

6. Logical theories ofprobability probability distribution. If we take conditionalization to be justified by the arguments or it, then a itiona constraints exist on how su jective robabili distributions are constrained b rationali to chan e in the light of new evidence. But is there any further constraint of rationality on

priori probability?" Pure subjectivists frequently answer "no," coherence eing given, t at one a priori pro a i ity istri ution is just as rationa as an other. Others deny this. "Objective Bayesians," as they are sometimes called,

But how can the "degree of support" or "degree of confirmation" of one propos1t10n groun e 1n ano er e e erm1ne. 1rst, 1t 1S usua y assumed that these degrees of confirmation must obey the usual formal postulates of probability theory. In his later expositions of the work,

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of possible worlds in which it is true. The degree of confinnation of proposition h on proposition e could be thought of, then, as the measure of the ro ortion of worlds in which e is true and in which h is t e well. We can get the results we want, a logical probability obeying the

gests itself -letting each possible world have an "equal share" in the total probability - leads to an inductive logic in which we don't learn from ex erience. A subtler method of first dividin the robabili even! over a class of kinds of worlds, and then evenly over the worlds in the kinds

inductive logic that raises our expectation that a property will be instance as we experience its instancing in our observed sample, a kind of inductive ro·ection from the observed into the unobserved. Trying to rationalize a unique confinnational measure as the only one

of choices, and then to rationalize those criteria of adequacy, is a task t at Carnap 0 y partia y ac ieves to is own satis action. He re ies fre uentl on "intuition" even to et that far. And findin a wa of extending the method originally designed for simple and finitistic languages

then be the rational probability with which to hold a proposition before any eVI ence came In. ro a I lt1es a er t e eVI ence cou t en a e arrived at b conditionalization, usin the a priori probabilities to compute, in the usual way, the conditional probabilities needed to know

all symmetric propositions equally when distributing probability - is a mo em Ins ance 0 one 0 e 0 es I eas In pro a I I eory, e Principle of Indifference. The idea is that "probabilities are to be taken as equal in all similar (or symmetric, or like) cases." A priori, heads has

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that provides the a priori probability where we start. But why should we believe, a priori, in equal probabilities for the symmetric case? In fact the Princi Ie of Indifference has more roblems than its lack of apparent rationalization. It is, without further constraint, incoherent.

outcome of a one coming up is one-sixth. But the die can either come up with a one or else with a "not-one." So there are two cases, and by the Princi Ie of Indifference the robabili of a one comin u ou ht to be taken to be one-half. In this case, we might resort to the fact that " " away from the counter-intuitive one-half. But what if the number of possible outcomes is infinite? Here, each indecomposable outcome has robabili zero and each interestin class of these (an interval on the real line, for example) has an infinite number of elementary outcomes

categorized has been emphasized by what are frequently generically re erre to as "Bertran's Para oxes." Imagine, or examp e, a container with a sha e so that the surface area on the inside that is wetted varies non-linearly with the volume of the container that is filled. An a priori

ent results from mtenor su ace desi ned to fix abilities as the

one that distributed probability uniformly over allowable area wette y t e Ul. at pnnclp e 0 ratlona lty, the uni ue one of the coherent ossible a riori robrational one for an agent to adopt, will tell us which ?

then apply a Principle of Indifference or symmetry over those possibilities so construe . H. ]effre s initiated a ro ram of selectin from amon the cate orizations by examining the invariance of probabilistic conclusions under

transformations. We feel, for example, that in some cases picking a eSlgnatlOn 0 one va ue 0 a quantlty as zero pOlnt as oppose to another, or picking one interval as unit scale as opposed to another, should not modify our expectations. Under those circumstances, one

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app i u, as we s a s i , w en we examine attempted applications of the Principle of Indifference (or its modern day reformulation, the "Maximum Information Theoretic Entropy h i liz in c abilities is only rarely available to us. And when it is, what we can obtain

that they are understanding probability in statistical mechanics as subjective probability, it is usually, rather, a belief on their part that there is a Ie itimate a licabili of the Princi Ie of Indifference to h sical situations that can be applied to ground the positing of initial probabilities so

m. Probability in statistical mechanics A great deal of Chapters 5 through 9 will be directed toward problems

account of the world offered by statistical mechanics. It will be of use here, though, to give a pre iminary survey 0 ow pro abi ity attri utions are embedded in the statistical mechanical icture of the world and of some of the peculiarities of probability in statistical mechanics that lead

imagine it as having been prepared out of equilibrium, again subject to some macroscoplC con lhons. e t en try to s ow t at t e com mahon of a reliance u on the detailed facts about the microsco ic constitution of the system, the facts about the dynamical laws governing these micro-

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dynamic equations of evolution appropriate to the system in question. In some accounts, the probabilistic assertions will be about probabilities of initial microsco ic states of the s stem com atible with its initial macroscopic condition. In other accounts, they will be about the probability

justification. As we shall see in Chapters 5 and 7, the just' cation needed for the robabilistic claims will va uite radicall de endin on 'ust what the claim under consideration is. And getting clear on that will sometimes

which a given correct probability attribution rests. This is intimately connecte , 0 course, Wlt t e questlons concernlng t e ratlona e we can offer if a claim we make that a robabili distribution ou ht to take a certain form is challenged. The problem is complicated in statistical

different purposes in different portions of the theory. n equll num eory, or at east In t e verSlon 0 It t at seems c earest and most defensible, the aim will ultimately be to show that a unique probability distribution can be found that satisfies a number of con-

to the usual phase space measure, is a constraint harder to justify, assummg as It oes a po on 0 e onglna pro a 11 a n u lon we mean to justify. The rationalizing ground is sought in the constitution of the system and in the dynamical laws of evolution at the micro-level. The

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usually obtained by one version or another of some rerandomizing posit ynamic ev u' 0 0 a sys e ,an ence even 0 an ensem e 0 sys ems (or probability distribution over systems subject to a common macroscopic constraint), is fully fixed by the dynamical laws of evolution governin the micro-com onents of the s stems. So a rationaliz tion of th probabilistic posit here will come down to an attempt to show the

systems. Here, probabilistic postulation is thought of much as a device to generate the correct result (the kinetic equation of evolution for a reduced descri tion of the ensemble a device to be instrumentalisticall justified by reference to the real laws governing the dynamical evolution,

But another role for probability distributions in non-equilibrium theory is even more fundamental to the theory. This is the appropriate distribution to im ose over the microsco ic initial conditions of a s stem started in non-equilibrium subjected to some macroscopic constraints. In some

ture of the approach to equilibrium of a system (its relaxation time, the orm 0 its equation 0 evo ution towar equi i rium on a macroscopic scale and so on) it seems clear that such an assumed distribution will need to be posited. But here, much is still opaque in the theory. There

agreed upon account of their origin and rationalization. Microdynamical aws, constitutions 0 systems, paces 0 systems in interacting extema environments modes b which s stems are re ared a riori distributions determined by principles of general inductive reasoning, and

distributions over possible micro-conditions.

Probabili and chism. We have outlined a debate between those who would take probability to be a mode of description of a world

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with their full specification, the outcome would either definitely occur or not occur - and those who would hold that probability exists in the world onl if there is a enuine failure of determinism - that is onI if such hidden factors fail to exist.

probabilistic descriptions not underpinnable by determinism at a lower level are defended as the correct picture of the physical world. Should the initial robabilities over initial conditions in non-e uilibrium statistical mechanics be construed in a similar manner, or should they be viewed

dition in any particular instance of a physical system? As we s a see in C apters 7,111, an 9,111,1, t is is a controversia matter. Particularl interestin in this context are attem ts to ar ue that the probability of statistical mechanics is neither the "pure chance with-

tion" variety, but, rather, a third kind of probability altogether. The argument w· e t at t e insta i ity 0 t e ynamic evo ution 0 t e systems with which we are concerned in statistical mechanics makes the characterization of the system as having a genuine microscopic dynamical state

these kinds, we will find other initial conditions whose future evolution, as escn e y t e am11ar trajectory 1n p ase space start1ng rom t 1S other condition, will diver e uickl and radicall from the tra·ecto starting from the first initial state we considered. Under these conditions,

dynamical trajectory. Rather, it will be argued, the individual systems in quest10n oug t to ave e1r states c aractenze y pro a 11ty 1stn utions over phase-space points and their evolution characterized in terms of the evolution of such distributions over phase-space.

unlike the "no hidden variables" view in that the grounds for denying any er e ermID1S 1C spec a 11 0 e sys em are qU1 e u 1 e ose that are used to deny hidden variables in quantum mechanics. In the latter case, we are presented with alleged proofs that the statistics in

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n

n

probability distribution. It is worth noting that although we will be discussing this issue in the context of classical statistical mechanics, the view takes on exact! the same as ects in the cont xt f s tis' mechanics whose underlying dynamics is quantum mechanics. The ".

be alleged to hold of the underlying dynamics.

are

occasionall to determine observable

tempts to confirm these predictions by direct surveys of velocity distributions in samples of the population of molecules. More commonly, the robabili distribution is used to calculate some value that is then associated with a macroscopically determinable quantity.

account in which it is to function. Sometimes it is easy to lose sight of w at one was intereste in s owing in t erst p ace, an to t in at a oal has been accom lished when a uanti is derived havin the formal aspects sought, but whose role in the overall account is less than

possible states. These latter averages are called phase averages. Also Important IS t e Istlnctlon 0 t e atter averages rom most pro a e values of the quantities in question (calculated usin the same set of available micro-states and the same probability distribution). The justifi-

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by thermodynamics follow the Ehrenfests in associating the curve of monotonic behavior of the macroscopic system with the concentration curve of an ensemble evolution - that is with the curve that oes throu h the "overwhelmingly most probable value of entropy" at each time. And

systems." But other statistical mechanical accounts do in fact think of the monotonic approach to equilibrium as eing representable in statistical mechanics b 'ust such an "overwhelmin I most robable course of evolution." Such conflicts of understanding point to deep conflicts about

Pro ty versus proportion. In our iscussion 0 oun views on robabili we noted im ortant doubts that robabili could in any simple-minded way, be identified with actual frequency or pro-

proportion - even in the "long run." Statistica mec anics presents us wit the relationshi between robabili and ro ortion. These are exem lifled by the paradox considered by Boltzmann and discussed in Chapter

argue that we have simply looked at too small a reference class in seekmg t e proportIon In t e wor to assocIate WIt t e asserte pro a 1 Ity. Such a suggestion is that offered by Schuetz and Boltzmann to the effect that our region of the universe is only a small portion of its extent in

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u

u ny with "idealized" systems. In mechanics, we talk of systems composed of point masses, or of frictionless systems, or of systems in which all interactions save one can be i nored. Idealization la s a rominent role in statistical mechanics, sometimes in ways that require rather more atten-

Some of the idealizations we will encounter include going to the thermodynamic limit of a system with an infinite number of degrees of freedom or to the Boltzmann-Grad limit in which the relative size of molecule to scale of the system becomes vanishingly small, or to the limit

In some cases, the role the idealization is playing will be clear and

re ative y uncontroversia. T us, or examp e, t e ro e p aye y the thermod namic limit in allowin us to move from avera e values of some quantities to the values being identified with the overwhelmingly

of the Ergodic Theorem. Here, one starts off with systems idealized with re erence to t elr structure, mo ecu es lnteracting on y y co lSlon, an then erfectl elasticall, for instance, and a s stem ke t in erfect energetic isolation from the outside world. Next, one shows that for "almost

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limit as time "goes to infinity." There is of course nothing wrong with such a notion. Indeed, it fits nicely with those accounts of probability that we saw em hasized ro ortion in the limit of an infinite number of trials. But the relation of such a probability to such things as the

played by the idealization, one can think that more has been shown than is actually the case.

IV. Further readings (955). Feller (950) is a classic text on the subject. The axiomatic foundations are explained in Kolmogorov 1950). A surve of hiloso hers' ideas on the nature of robabili can be found in Kyburg (970). For the dispositional theory, a good source is

For other versions, so-called "objective Bayesianisms," see Jeffreys, H. 1 an 7, osen antz ,an aynes On Humeanism and its 0 onents, see Cha ter V of Earman (1986). Lewis (1986), Chapter 19, is important on the relation of subjective and "

randomness," see Chapter VIII of Earman (1986). Fine (1973) has detailed an compre enSlve treatments 0 t e major approac es to t is issue.

We wish to know not only what happens in the world, but why it of the world. We want

nature of explanation. It consists, he says, in giving the "causes" of the p enomenon In ques lon. ere, e lnc u es w a lS ca e e e clent cause," corresponding most closely to our present idea that an event is roduced b continuous earlier events that roduce the henomenon. Influenced by his experience in biology, he thinks of all events as govthey occur must bring out their place in the accomplishment of some end or purpose. a lS, a exp ana ory accoun mus 0 er e na cause of the phenomenon as well as its proximate efficient cause. Bringing into la his meta h sics of chan e in which all chan e is the chan e of some "form" on an underlying unchanged "substance," he calls reference cause," and reference to the underlying unchanged substrate reference o i s rna eria cause. For a very long time we find little more offered to us by philosophers that oes be ond this interestin but still va ue reference to causation as the core of explanation. Leibniz, like Aristotle, believed that were we but ratiocination from self-evident first principles. But, he believed, given u

generalization from our experience of causation in the world. He offers 1 8

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Sufficient Reason. This is the assertion that every happening is explicable by the sufficient causation for it which, if we but search hard enough, can be found. Be uses the Princi Ie to reat effect both in foundin his metaphysics and in his brilliant critique of the substantivalist view of

too anomalous a relation to fit into the usual cause-effect pattern of the world. Later, Berkeley, in his critique of the representative realist's idea of matter offers the claim that onl mind could be "active" enou h to serve as a cause, the notion of having causal efficacy being incompatible ".

"

causation itself. Taking the basic empiricist principle that concepts obtain their meaning by serving as labels or items of sensory experience, and askin for what we actuall ex erience when we ex erience the causeeffect relation, he offers his famous critique of the idea of "necessary " that are spatiotemporally contiguous./The asymmetry of the causal order in time requires at causes a ways prece e t eir e ect in time. Fina y, and most im ortant! events are in a cause-effect relation to one another only if they are constantly conjoined, only if the cause is always accom-

connection between the events. For Bume, such an "explanation" of regu anhes IS anon-starter, or It Invo es a nohon 0 necessary connection in the world whose ve intelli ibili is dubious on em iricist grounds. Rather, the regularity is all there is to the physical phenomena.

source of the error is a projection onto the phenomena themselves of some ng at IS Instea a psyc 0 ogica response to t em on our part1 Experiencing a multiplicity of constantly con·oined events of a certain kind, our mind, when presented with the idea of the cause event, leaps

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i v' e p i r e c use on our part that by a mistake of projection of the subjective into the objective, we take as necessary connection in the world. N dl ss t sa this Humean anal sis of causation as s tio-t m r contiguity, temporal precedence, constant conjunction, and a psycho-

Hume's epistemology has been challenged by the assertion that we can indeed determine a cause-effect relationship to exist even from the observation of a sin Ie case castin doubt on the ori in of our determinations of causality in observed repeated regular constant conjunction. To this,

fit. (You may never have seen billiard balls collide before, but you have seen many collisions of other kinds of material objects. This is the source of our immediate understandin that the im act of one billiard ball causes the other to move.)

conjunction, rather than just expressing it. This is similar to the desire of the dispositiona pro abi ists - or at east some of them - to argue that the dis osition ex lains the revealed fre uen of occurrence rather than just summarizing the fact that it or something like it actually occurs. 'Some

the general nature of causation between kinds of events, argue that it is e particu ar events are instantiaa re ation 0 t e universa s, 0 w ic tions that rounds the causal relation. However how such an invocation of realism with respect to properties is supposed to answer the Humean

himself, a remark hard to reconcile with his prevalent expression of his views a out causation. At one point, Hume resorts to e su junctive mode in discussin causation s eakin of the cause as that event which had it not occurred, the effect event would not have occurred. Many have

relations are said to support counter-factuals, whereas accidental reguantles 0 not. f t is were copper it wou con uct e ectnclty is true, because it is a law of nature that co er conducts electrici ; but "If this were in my office it would weigh less than 300 pounds" is false, said of

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the cause being, essentially, as Hume suggested, that which, had it not occurred, the effect would not have occurred either? Much su estive work has been done in this direction but uzzles abound. Some counter-factual dependence is non-causal, requiring us to

the actual cause, had it not occurred, would not lead to the non-occurrence of the effect because if the cause hadn't occurred, some other preempted b it would have - re uire so histicated handlin . Most im ortant! questions arise regarding the epistemology, metaphysics, and semantic

formal semantics for counter-factual locutions, but we remain, with Hume, puzzle a out what in the actua wor grounds the "mo al" orce of causation (the necessa connection of the events the "wouldness" of the "would not have occurred" in the counter-factual analysis),

concern will be with the notion of statistical explanation. And within the iscussion 0 t at notion, our ocus wi e on 00 ing or some rameworks useful for later askin what kind of ex lanations of the henomena statistical mechanics can be held to provide us. But, as we shall see,

different guise.

guished physicist-philosophers as Mach and J)uhell). of the whole point o e netIC eory proJect. e1r 0 Jectlon to e t eoretica program was art! motivated b a phenomenalistic ob'ection to the ositin of unobservable entities and properties as explanatory devices in science.

ultimately, go back to Berkeley's attack on the representative realist's matter, an entlty or eature IS lffiffiune to 1rect 0 servatlona awareness by us, there is no good reason to posit it. More strongly, the very idea that one can meaningfully talk about it is rejected by the empiricist

,

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was a methodological howler. There is much confusion here. Although y u rap omc r: v enalist, there is nothing methodologically abhorrent in positing the molecular constitution of matter. Of course, this positing must itself be henomenalisticall understood if one remains a hen menalist but there is nothing more absurd in this than in understanding macroscopic

the only general consideration motivating the "energeticist" opponents of the molecular theory of matter and of heat. They viewed the kinetic theo ro ram as an un'ustified continuation of what the took to be an illegitimate demand that all physical theory conform to the model of

- in particular, the subject matters of electromagnetic theory and of thermodynamics - were not appropriate y ea t with by a search or some underl in "mechanism" that could be described in the Newtonian mechanical terms and the operation of which would explain the lawlike

thermodynamic laws of heat, energy, and their transformations. To deman suc a Newtonian mec anism as t e asis 0 any p ysica account, the maintained was to believe that the a ro riate conce ts for describing one part of the world had to be, of necessity, applicable to all. I

phenomena. Once these laws were found, physical science had done all t at cou e expecte 0 it.~ 0 eman in every case at t e aws found be unde inned b some mechanical model was to romote Newtonian mechanism out of its proper sphere as the correct description

As things turned out, of course, the warning against seeking mechanical mo e sop enomena prove to e ea t y m e case 0 e ectromagnetism, where Maxwell's models of electroma netic wave motion as a form of motion of components of a mechanical aether remain mere

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offered the wrong advice. Up to a point, the theory of heat and its transformations is the theory of the mechanical behavior of the microconstituents that form a "mechanism" under! in the macrosco ic thermal phenomena.

conjunction1- did lea to a genera account 0 what it is to explain henomena in scientific terms. hat is it led to an account of what it is to answer a "why" question in the only way science was qualified to

would need in order to be able to predict and control the phenomena o t e wor. we ow at t e occurrence 0 el is aw 1 e regu ar y followed by the occurrence of e2, then el's occurring will lead us to predict th~ future occur:e~ce of e2 • fnd we can sometimes bring it about that e2

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of historical explanations) where we feel that an explanation has been

the situation of the agent who was asking the "why" question. What that agent knew and assumed, what his epistemic lacks consisted in, all determined what would be an a ro riate answer to his "wh ?" (:.More important for our purposes was the objection that a putative

Although it seems appropriate to explain the motion of a billiard ball by the collision with it of another, the two motions linked by the laws of d namics it seems ina ro riate to think of the lawlike determination of the past motion by the later motion as explanatory of the earlier motion.'

it seems absurd to say that one could explain the height of the building y re erence to t e length 0 t e s a ow, t e position 0 t e sun, an eometrical 0 tics even thou h the derivation works that wa as well. Here, the idea seems to be that to explain is to give the cause or the

joined with it is insufficient for explanation. The explaining event must e part 0 t e "cause 0 t e event to e exp aine "or part 0 e "mec anism b which the event to be ex lained is brou ht about." We will return to these issues in sub-section 3 following. tions that invoke l strictJunexceptionless regularities) Rather, it is claimed, a Slm1 ar p1cture W1 0 as we or our un erstan 1ng 0 exp anatlons of events that invoke tatistical re larities Here the idea is, once a ain, that a particular event is explained when its connection to other events "

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then we can say that the event in question, at least relative to the explanatory events having occurred, had to happen, or could not have not ha ened. But if we connect an event which in fact does occur with other events by means of a merely statistical regularity connecting the

pening is connected by a statistical regularity of science to other events that happened, intuitively it "could have not occurred" even given the occurrence of the ex lanato events. Sometimes it is claimed that the structure of a statistical explanation

tistical regularity, is associated by means of the statistical explanation to a pro a i ity 0 a proposition, some .n 0 measure 0 justi e " egree of belief" in the truth of the ro osition describin the ex lained event. Here again, we have a kind of rationale for the search for statistical

is, if anything, one. Rather, it is that the resources brought to bear in a statlstlca exp anatlon cou ,In S1ml ar clrcumstances were we on t -" know what has occurred or where we wish to bring about or revent some occurrence, provide us with the resources needed to make

rational belief or action

them with desires in a

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from the explaining events.' Wouldn't this be the natural extension of the ra y r wi i . 'c r m. be some indeterminateness in how high probability must be in order that we take the event to be explained, but such a looseness in the concept of statistical ex lanation would be hardl sur risin . But there are many cases where we seem to think an event has been

low probability.lWe have explained the bursting into fire of material by spontaneous combustion when we point out that such things at least occasionall ha en in the circumstances obtainin even if the ha en only very rarely.' Here, the notion of the explaining event raising the

it made the explained event probable, but because it made that event more pro a e than it woul have een had not the explaining circumstance obtained. Here reference is fre uentl made to two senses in inductive contexts in which evidence can confirm an hypothesis. One

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tory of the occurrence of the event in question, even if, in that circumstance, the probability of the event occurring was still one-half? A most liberal version of the account of statistical ex lanation that holds to it as being subsumption of the explained event under a statistical

probability from background, positive, negative, or zero, to be permitted, and the systematization still counted as explanatory. After all, the information enerated to roduce the s stematization will have its a propriate uses in circumstances of prediction and control. And isn't that

probabilities or raised probabilities are of interest in the explanatory or prediction-control situations. One roblem that has received attention in the context of the anal sis of statistical explanations being understood as subsumptions under "

.

,

inductive inference is useful here. If we are to decide how much creence to give to a ypot esis, we mig t consu t its "pro a i ity re ative to the evidence" assumin we believe in such thin s and have some alleged method of computing them. But surely our actions should be

use the probability generated by the placing of the situation in t e "narrowest" re erence c ass generatmg re ia e pro a 1 ities. lour evidence consists of dis arate bodies of back round that fail to im I one another in any way, and that generate unlike probabilities for the

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new knowledge can result in what was an explanatory fact ceasing to be explanatory. Others ould avoid this relativi b insistin that the nl r reference class relative to which probabilities can serve as statistical exI

reference class A that generated C's with a purely tychistic probability p - that is, which was such that C resulted from being an A with that robabili and with there bein no additional "hidden variables" that could further specify the event more narrowly and thereby lead to a

with their associated probabilities might have a kind of weak legitimacy insofar as they constituted incomplete sketches of explanations, sketches that when filled out would brin to the surface the real reference class to which the event belonged and that was the proper one for statistical

bate over what constitutes a legitimate "reference class" in which to place an event an re ative to w ich to calcu ate its pro a ility for statistical ex lanato u oses. One dispute centers around the problems introduced by the fact that

positions. The "conservative" position takes only some maximally spec' c re erence c ass as egitimate y re erre to in exp anatory contexts. This class mi ht be construed either as a matter relative to the e istemic state of the explainer or in the "objective" sense as being maximally

legitimately generate an "explanatory probability." The former position wou 0 ster its case y pointing out ow in t e in uctive' erence case we would consider ourselves irrational to rel u on robabilities generated by reference classes known to us not to be the maximally

know the event in question to fall into two reference classes, neither of W lC contalne e ot er an t at generate 1 erent pro a 1 lUes or the outcome in uestion. The latter osition would em hasize the fact that bringing to light the probability of an event relative to any reference

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about the world that is the goal of the search for scientific explanations. A second set of issues centers, rather, around the appropriateness of reference classes for statistical ex lanato u oses where it is the nature of the classifying kind, rather than its epistemic or objective maximal

classes into which the event falls that make its "probability" relative to those classifications as close to one (zero) as we Ii e. The tric is simply to use ve s eciall and narrowl constructed classes that will contain only the specific event in question and as few other events as we like. " events into kinds are genuine generators of "real probabilities" and only p acing events in suc natura c ass' cations provi es e resources or enuine statistical ex lanations. But what are the legitimate "natural kinds" and what gives them their

of regularities describing the world. From this perspective, the artificial an too narrow c ass cations t at plC out In lVl ua events y more or less definite descri tions of them (or of them and a few individual fellow events) or that peculiarly link together categories - "is a cow or a type o

on again in the next section, where issues of causation and mechanism are lscusse. Finally, there is another debate between the "tychists" and the "proportionists" The former are those who would take a statistical sys-

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-c s, s in any "natural" way, that would generate probabilities differing from the original classification. The latter are those who would allow that any otherwise Ie itimate reference class can be brou ht into la in tisti explanation, ultimateness not being required. From the "proportionist"

be partitioned into smaller classes in a natural way and which are such that some of them generate probabilities for the event in question differing from those enerated b the reference class in the roffered ex lanation. In Section III of this chapter we will explore in a preliminary way how

3. Subsumption, causation and mechanism, and explanation In order to situate the conceptual problems of explanation in statistical

particular, we should look at the claim that notions of causation and mec anism are essentia ingre ients in any ana ysis 0 w at it is to 0 er a scientific ex lanation of henomena. And we should look at the related claim that these notions cannot be cashed-out in terms of subsumption

We will be only able to explore the issues here in a superficial way, but some summary survey 0 t e maln lssues 0 contention Wl e e p The debate starts with the observation, noted earlier, that in man cases regular connection is insufficient for explanation. Even in the strictly

event is insufficient to credit the other event with explanatory force. The most 0 VlOUS cases are t ose were t e putative y exp anatory event is reall an effect, rather than cause, of the event to be ex lained (the length of the shadow not explaining the height of the building, but vice

in which case neither of the first two events explains the other, despite e aw 1 e corre a lon e een em. Similar, if a bit more complicated, considerations come into play where it is the probabilistic explanation of events that is in question. If we

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doesn't bear the appropriate relation to event 2 of cause to effect. Similar cases where one event raises the probability of another but fails to causally ex lain the other are also eas to construct. Indeed for an robabilistic relation among events one may construct, it is possible with enough

propriate causal structure among the events. One can go further in the probabilistic case. Not only do various probabilistic relations seem insufficient to establish the a ro riate ex lanato connection none of them seem necessaty either. Let event 1 cause event 2 (intuitively). It is

Spontaneous combustion is an example of the former kind. Examples of the atter 'n inc u e suc cases as were a reme y, whi e owering the robabili of an illness causes the illness as a rare" aradoxical" effect of its being applied, a not uncommon case in medicine.

would allow us to reinstate the idea of causation as being fully analyzable in terms 0 re ations 0 ig or raise pro a i ity among events. But intuitive counter-exam les have been uickl constructed to each such proposal. Or, in other cases, the proposals have been found to involve,

proposal as a fully reductive analysis of causation in terms of regularity, sn t it t e case, it is argue , t at w at rna es one event exp aln another the fact that the first event caused the other, or constituted art of the mechanism by which the other was made to occur? Mustn't we take

ducible components of our ontology? Or should they be viewed, rather, as composl es - or examp e, as or ere s ruc ures 0 lngs, prope les, and space-time locations? Is the causal relation an ordinaty "extensional" relation -like "being next to," for example - so that if one object of the "

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c usa cas, it is sometimes said, driving home drunkenly can cause an accident not caused by driving home, even though the drunken driving home is identical to the drivin home. Additional problems abound concerning the connection of causation

notion over and above such projectivist accounts as Hume's? Can a notion of causation grounded in a counter-factual basis - the cause being that without which the effect would not have occurred - be made lausible against the particular difficulties such an account faces? In the case of

had not occurred we sometimes will grant that the effect might have, all t e connections among events eing mere y pro a i istic, can the modal notions still be made to fit ro erl with the causal? And even if our modal and causal intuitions can be systematically reconciled, can we

at first glance, epistemically objectionable. Do they not cut our notion of causation 0 rom an a equate evi entia asis in actua experience? A ain can we res ond to the critic who will acce t such modal notions only if they can be metaphysically grounded in the actual- that is, made, " One natural program is to simply take the causal relationship to be a pnm1t1ve,lrre UC1 e to any ot er notion an a un enta component of the full descri tion of the world. To some, the causal relation will be one that holds between individual events, with the support of general

to show the similarity of being causally connected as well. To others, the genera 1ty 1S U1 t 1n to t e causa notion, ecause causat10n 1S ta en to be an irreducible relation among universals, ro erties, or kinds. Pairs of individual events are then causally related because they instance the

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What constitutes the evidential access we have to such a primitive, irreducible kind of causality? If the causal relation holds between individual events wh is it that we cannot determine this relation b some sort of "direct inspection" of the events, the way we could determine

kinds is what we require to determine causality? If causality is a primitive irreducible relation among universals, why is it that we can't discover this relation b a mere rocess of ins ection either in this case ins ection of the universals and the relation between them? Instead, we require a

universals connected by causality. This doesn't constitute a final and irremedia e re utation 0 any positing 0 causa ity as some a ditional, rimitive and irreducible as ect of the world. But the absence from phenomenal experience of "causation itself," as opposed to constant

puzzles the causal primitivist must resolve. He must tell us how we come to un erstan t e concept 0 causation, an must provi e or us an ade uate account of 'ust how it is that we come evidentiall to determine the presence of causality in the world, as opposed to the presence of the

its presence. A qUIte erent approac to pInnIng own causation In t looks toward the facts about causal relations as the a ear in our most fundamental physical theories and in our general presuppositions about

,

,

tion of any primitive causal vocabulary. 'W. Salmon, for example, followIng up 1 eas m . usse an . elc en ac ,proposes suc an approac to causation. Causation, according to our best available physical theories, is a spatio-temporally continuous process, causal influence being propa-

propagation are limited to a proper sub-set of all one-dimensional contmuous pa s In space-time t e Ime 1 e an nu pat s . IS 1 ea 0 spatio-temporal continuity of causal influence even holds in our most advanced quantum picture of the world, quantum field theory, appearing

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space 1 e separa ions. ere are cunous non-causa corre a ons envable from quantum mechanics, as illustrated in the cases used to discuss the Einstein-Rosen-Podolsky thought experiments and the related Bell experiments can have no explanation in terms of an antecedent common ug as n-c usa in na ure - an s v s. u remain one of the greatest conceptual mysteries of the interpretation of quantum theory. So one thing we ought to say about causation is that it has this s atio-tem oral and relativisticall constrained as ect to it Going further, it is argued that one need only think of causal influence

continuity of the height of a water wave along a path in space and time is the causal propagation along the path, and no primitive notion of the wave hei ht at one oint "causin " the hei ht at the next s ace-time location need be invoked.

generate a process along a continuous space-time path that looks causal ut is not. T e propagation of the effect along the path is due, intuitively, to a remote common cause that enerates each event alon the ath of pseudo-causal propagation in tum. For example, consider the spot of

propagation of causal influence from searchlight to wall, not an influence rom spot to spot propagate a ong e pat 0 owe y e moving atch of li ht. Salmon tries to exclude these continuous chan es masquerading as causal propagation by using Reichenbach's notion of "

light spots as red, say by inserting the filter into the beam generating that spot just e ore it is generate, oesn't co or t e ot er spots a ong e ath on the wall red. It is, however far from clear that this ro osal for delimiting genuine from spurious causal processes doesn't itself invoke

isn't a case of "marking" the path of light spots at the first spot and seeing t at mar propagate own e pat 0 spots, un ess one a rea y ew that the causation was in the holdin of the filter and the ro a ation of redness along the searchlight beam? In any case, Salmon attempts to

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features. Along with considerations of spatio-temporal continuity of the propaation of a feature as an essential com onent of causali there is of course, also the fact that we take it in general that causes always are

asymmetry in time of systems that is a central problem of statistical mechanics. And we shall later in Chapter 10,111,1 and 10,111,2 explore the uestion of whether in some wa this intuitive as mme of causation can be grounded in, or reduced to, the asymmetry of the process of the

theory that seems to be most intimately associated with the intuitive aspect of causation in question is muc ess c ear than the place of the s atio-tem oral continui of causation as it fits into our eneral s acetime physics. But even that latter placing of an intuitive notion into con-

A general methodological point about approaches of the kind we have just een iscussing is important. One wants to e very wary 0 tying such a eneral and ro rammatic notion as that of causation too closel to specific features of how causation acts in the world as current science

in their very nature of definition spatio-temporally continuous in their action, t en in a wor in w iC actlon-at-a- istance governs t e mteractions of matter, no genuine explanation at all will be possible. Now, to be sure, action-at-a-distance has sometimes seemed so dis-

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regu an les invo ing ac ion-a -a- is ance can serve as exp ana lons. Newton, formulating gravity mathematically as just such an action-at-adistance force, is well known for remarks that can be construed as deh gravitational interaction, reliable as it was in its predictive aspects. But s ry. u er, e difficulties of current spatio-temporally continuous theories of causal interaction among particles - that is, the difficulties of field theories such as diver ent self-interactions of article and emitted field have led physicists to try to construct relativistically respectable action-at-a-

century were wrong as far as thermal phenomena went, their general claim - that one ought not to insist that all explanations be framed in the form of Newtonian mechanistic ex lanations - was surel correct. And the generalization of this - that it is a mistake to constrain explanation

of some form of regularity among them - is surely correct. It may be reasona Ie to emand that exp anation requires causation an mechanism. But if one oes on to fill this out with a demand that causation and mechanism have specific forms - say, of spatio-temporal continuity in

of the phenomena don't count as explanations at all on this methodological stance. An alternative immediatel su ests itself - to ar ue in a sense that causal relations and mechanisms are just regularities, but that not all

being the positing of regularity, lawlike or statistical, framed in terms of c aractenzatlons 0 events y means 0 t e concepts at p ay e most eneral and foundational role in our scientific theories. And the re larities themselves are the basic regularities of that science, or are related

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true of the most fundamental regularities of our current best physical theories (relativistically compatible field theories) that they are framed in such s atio-tem orall continuous wa s. But we would be re ared for a possibility of a scientific revision to a novel theory in which the fun-

would still believe in causation and mechanism as newly construed, but it would be a form of causation and mec anism quite di erent from that we reviousl had in mind. The intuition that the invocation of regularities is sufficient for ex-

from us, because the science invoked to account for this asymmetry 1S usua y stat1stica mec an1CS, an ecause e a ege asymmetry 0 causation is itself so frequently invoked (often unconsciously) in attempts at explaining the important temporal asymmetries of thermodynamics

case for the proposal that the necessarily causal nature of explanation as 1S ong1n 1n e s c ure ave sugges e. W1 ffi, ra er, 0 a preliminary outline of the problem that will be central to later considerations. The basic problem is that in the context of statistical mechanics,

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description and explanation. But just how does this latter level of comprehension "fit" over the presupposed grounding causal level? ll. Statistical explanation in statistical mechanics systems can legitimately be described by an underlying causal picture of the world. Here, the constitution of the system as a structure built up of micro-constituents and the lawlike behavior of that structure that follows from the laws governing the micro-constituents and their interaction,

constitution. Essentially the same picture holds whether we are dealing with presupposed classical laws, as in this ook, or with the more refined icture in which the state of the s stem is constituted b its uantum state, and the dynamical laws are those of the evolution of quantum

probability distributions, features generated out of these probability distri utions i e average or most pro a e va ues, an t e ynamics 0 these features onto the underl in causal understandin of the s stems their nature, and their dynamics.

the additional elements of statistical mechanics must be superimposed. ut In anot er sense, t ese two components are not unquestlone eatures that we must take for ranted, but are themselves sub'ect to critical investigation in the pursuit of the foundations of statistical mechanics.

ternal and interactional forces is given to statistical mechanics from the outSl e,o er structura eatures are su Ject to questlorung an open or consideration in a foundational investi ation of statistical mechanics itself. This comes about from the fact that the statistical mechanical results

and easy to justify and understand, others have a more fundamental pace In e eory an are some lffies open 0 cn lca 0 Jec lon. Many accounts of the approach to equilibrium of a system, for example, treat the system as energetically isolated from the external world. But, as

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of behavior of systems in statistical mechanics treat the systems as having an unlimited number of degrees of freedom, allowing the theoretician to work in the so-called thermod namic limit. Althou h the vast number of degrees of freedom (vast number of micro-constituents) of realistic

with care in each case. Other results are obtained by idealizing the interactions among systems as having "singular" natures, treating molecules, for exam Ie as "hard s heres" that have no interaction when se arated but that are absolutely impenetrable to one another upon collision. Here,

trying to put together the account appropriate for the highly idealized system with what can e s own to e true a out systems not so i ea ize becomes an issue. A ain it is sometimes im ortant to treat s stems as being of low density and as having their micro-constituents vanishingly

at variance with the conceptualization usually used in some crucial ways. In cases i e t east, a care u examination 0 ow e i ea ization is bein used and a com arison with the s stematic use of the alternative idealizations and their uses is important. As we shall see time and again,

dynamics onto a reversible underlying dynamics has often led to the suggestlon t at e source 0 lrreverSl 1 lty must e oun In some modification of the familiar d namicallaws, or in some understanding of them that introduces irreversibility at the level of micro-dynamics. This

In any case, it is on top of an underlying causal picture of the system at t e statlstlca or pro a 1 lStlC account 0 p enomena, essentla to t e statistical mechanical attempt to account for the thermodynamic features of the world, is superimposed. This causal picture is framed in terms of

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and e familiar thermodynamic regularities - specific like the equations of state or the hydrodynamical equations of approach to equilibrium, or general and all-encom assin like the Second Law of Thermod namics - can be "derived. "

individual system we have in mind. Sometimes it is proportion of systerns in an imagined infinite collection of systems having micro-states of a s ecified kind that is what is meant. Sometimes it is the ro ortion of initial conditions of a system satisfying some feature that is taken to be

as the Ehrenfests told us, we must always be sure that we haven't hidden some 0 e most i cu t components 0 e exp anatory pro em in an ambi uous use of robabili and items constructed from it. Statistical mechanics is usually divided into a theory of equilibrium and

in Chapter 7, the notion of "explanation" invoked in the latter, more . cu t an controversia, portion 0 statistica mec anics 00 s more familiar to a hiloso her, accustomed as he is to the standard notions of explanation as either subsumption of pairs of states of systems under

In equilibrium theory, the basic aim is to account for the regularities among macroscopiC quantities c aractenzing a system at equii num e e uations of state), and to understand such basic features of e uilibrium as the equi-partition of energy among all available degrees of freedom.

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in v ri us collection of systems whose micro-states obey some constraint. In both cases, the "probability" comes in as a measure of those proportions, and nd mental roblem of the foundations of the theo is rationalizin or justifying the probability distribution chosen. Other foundational issues

macroscopic observable. As we shall see in Chapter 5, the source of the appropriate probability measure is usuall traced to the structure of the s stem and the d namical laws of the micro-components, the crucial point being that the notion of

on accepting the posit that sets of initial conditions that are given zero probability in the measure rationalized are not to be expected to characterize actual sets of s stems in the world. As we shall see in Cha ter 5 the sense in which equilibrium features are "explained" in the equilib-

of the facts about the approach of systems to equilibrium in the world, these matters eing e to t e non-equi i rium eory. T e "exp anations" obtained in e uilibrium theo don't look at all in fact like the generative or causal explanations of how a state "comes into being" that

explanations of phenomena in physics. What we obtain in equilibrium eory is rat er a emonstration at equi 1 rium, assume to exist an to be connected with certain theoretical states must if certain lausible assumptions are made, have certain features. These follow from the

When we move to the non-equilibrium case, the aim and structure of exp anatlon ta es on ltS more amllar s ape. ere, In t e manner we are so accustomed to from other contexts, we deal with s stems initiall in one condition that move in a specifiable way to some other condition.

and why it has the features it does. We want a general account of the approac to equll num, In ee a genera account 0 w y t e econ Law of Thermod namics is true. For specific systems, we want an understanding of why their approach to equilibrium has the time scale it does

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and the presupposed micro-dynamical laws will of course be essential

As usual, there will be problems involved in trying to determine the correct relation between some probabilistic construct of the statistical mechanical basis with some observable. uanti whose behavior is described at the thermodynamic level. In the non-equilibrium context, these " " quantity to associate with the thermal quantity of entropy becomes a question fraught with difficulty, because rather different perspectives on how to understand the a roach to e uilibrium conce tuall are tied intimately to different opinions as to what to construe as the entropy of

probabilistic model to use to understand the overall approach to equilibrium. T is is e question as to w et er to understand the standard e uations describin that a roach as descri tive of the "overwhelmingly most probable course of evolution of a system" or, instead, to "

ingly most probable" states of the collection of systems at each time. As we s a see in C apter 7,11,1 an 7,111,7, espite e E re ests' critique of the former osition and a arentl incom atible as it is with the recurrence theorem, some contemporary models of non-equilibrium posit

Statistical explanation

153

way, one thinks of the distribution as specifying, in some sense, proportions in which specific initial conditions, each of which initiates a deterministic evolution from itself are realized in nature. In the other way, one thinks of the probability distribution as specifying, rather, a

containing in itself a manifold of causal possibilities for the states that it will causally determine to succeed it. The probability is then a measure of this "causal ro ensi "of this enuinel chistic initial state to enerate its followers.

of evolution they offer is that each individual system has a deterministic causal evo ution rom its initia state, an statistica mec anics provides us with a wa of characterizin how these individual evolutions in the world will be found to behave when examined as a collection or an

has sometimes been called the "real ensemble" and sometimes the "de acto" approac ,WI SImp y pOSIt t IS Istrl ution over Inltla con It10ns as a contin ent, otherwise inex licable, fact about the world. Others will seek the origin of this probability distribution in a form of causal ex-

of one system of a very special kind, the universe itself. This second approac w glVe nse to many Important me 0 0 oglca pro ems 0 its own, not the least of which is the puzzle over what kind of further explanation, if any, we could offer for the special nature of this ultimate

of a system and on alleged impossibilities of preparing initial collections o any In except t ose c aractenze In t e manner nee e or e statistical mechanical results. Each mode of trying to account for the essential initial probability distribution will lead to a rather different

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Physics and chance

1Sapproac - ea mg e m1 ia pro ail tribution as descriptive of the inner probabilistic nature of an initial tychistic-propensity state - is far less orthodox than the approaches noted. and their arguments are not easily dismissed. From this perspective, the c v v' rm1n1S ca y, a statistical mechanics is describing. Rather, it is the causal evolution of tychistic state to tychistic state of individual systems that we are attemptin to model in our theo . From this oint of view statistical mechanics does not simply presuppose, the underlying causal structure of the micro-

of its causally determined evolution. Here the argument will be that the very success of statistical mechanics, and an understanding of the grounds of that success shows us that the standard causal icture at the microlevel is a false idealization.

the non-isolation of the system from the external world and that world's intervention into the system's dynamics, fit easily into causal ideas of ex lanation. Others that t to understand the robabilities of statistical mechanics in a "subjectivist" or "logical" manner offer quite a different

Finally, all of these approaches will have to confront that touchstone e wor at is to 0 justice o any a equate exp anatory account 0 to the henomena described b thermod namic theories: Where in the explanatory account does the irreversible nature of the phenomena we

ing controversy.

ID. Further readings

of Hume. See Hume (1988) and (1955). t oroug stu y 0 exp anation as su sumptton un er a regu anty 1n the lawlike case is Hem el and 0 enheim (1948). For further ex loration, replies to criticism, and so on, see Hempel (1965).

Statistical explanation

..

155

.,.,..,

;)[aUSUCal explana110n as suusumpuon unuer a Slal1SUCal regular1Ly IS explored in Hempel (1965), Salmon (1984) is a wide-ranging treatment of statistical explanation as seen hv nhi 1

and verv useful for the

.1..

~

~ .

1

'

. v

of

L1

approaches to the problem. Another extended treatment of the notion .£ V.l

1 . 1 . 'I' .1.-

.

.1." -,

,

.lLJ

,

n.l

r1{"\{"\1'\

.u~.lLJ

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(1978), and are worked out at length in Humphreys (1989).

.

I. Autonomous e ullibrlum theo its rationalization

and

micro-canonical ensemble The existence of an equilibrium state, describable by a few macroscopic w v u w' the most fundamental thermodynamic fact for which statistical mechanics must account. A full understandin of the e uilibrium situation would require a demonstration within the context of the dynamical theory of " " the dynamics of non-equilibrium drives systems, and it is that approach Beginning with Maxwell's first derivation of the equilibrium velocity distribution for molecules of a gas, though, there have been approaches to deriving the equilibrium features of a system that at least minimize tion toward equilibrium that one hopes to show is driven by it. As we on the assumption that the components of molecular velocity in three perpen icu ar irections were pro a i istica y in epen ent, an assumption clearly recognized by Maxwell as "precarious." eral dynamical perspective, using the kinetic equation and H-theorem stationary distribution, he later developed, in the course of his probabilistic response to e ear ier criticlsms 0 t e inetic equatlon an -t eorem, his new method applicable to the case of the ideal gas. Here, one counts identifies the equilibrium distribution as the one obtained by the over-

Equilibrium theory

157

the choice of position and momentum as the appropriate dimensions for the Jl-space was clearly guided by the dynamical considerations with hich Boltzmann had become familiar. Next in this evolutionary development of autonomous equilibrium

, over possible micro-states for the systems - is introduced. The standard micro-canonical distribution as at least one distribution invariant under d namical evolution and the identification of e uilibrium values with phase averages of micro-quantities relative to this probability distribution,

demonstrating that the standard probability distribution is uniquely stationary in time. For our u oses at this oint we can view the ma'or contribution of Gibbs to be his extension of the notion of the "equilibrium ensemble"

reservoir and to systems that can also interchange numbers of molecules wi t eir extema environment. ese atter cases are to e escri e y usin the canonical and rand-canonical ensembles res ectivel . But our attention will be focused on the micro-canonical ensemble. This is

dimensions are the position and momentum coordinates of the microconstituents. lven e xe energy 0 e system, co ne your attentlon to the a ro riate "ener her-surface" in the hase s ace. Pick a probability distribution over this now restricted phase space by con-

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Physics and chance

function over the allowable phase points, calculating the average by using the constructed time-invariant probability distribution as one's wei htin function. The usual thermod namic e uilibrium relations then follow, as do the general results on equi-partition of energy among

We ought to note here, however, one useful generalization beyond the micro-canonical ensemble. In the usual case, the only known constant, macrosco ic d namic arameter of the s stem is its ener . A as confined to a box, for example, cannot be expected to have a constant

we know a fixed angular or linear momentum for the system in question. Can the micro-canonical ensemble approach be generalized to allow us to construct a d namicall invariant ensemble where the existence of constants of motion for the system over and above its total energy allows

An affirmative answer is given by the work of R. Lewis, who showed t at one cou genera ize e micro-canonica ensem e in just is way, resultin in anta-micro-canonical ensembles. These are sub-s aces of the original phase space of lower dimension than the energy hyper-

probability distribution over them provably invariant in time under the ynamlca evo utlon 0 t e system. Althou h this reci e for the solution of the e uilibrium roblem is clear, it is the rationalization of it that is our primary concern. Clearly,

"end state" toward which a system tends evolving from an initial nonequl 1 num startIng pOInt. ut any ope 0 s oWIng at t e statlstlca mechanical treatment of e uilibrium is a ro riate for e uilibrium so construed must rest in the understanding of the full non-equilibrium

features is indeed the right way to calculate equilibrium features of a system su Ject to appropnate macroscopIC constraInts an Ul t up 0 micro-constituents in a particular way? And if we are already convinced that the recipe is reliable on the basis of its success in practice, what

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159

2. The Ergodic Hypothesis and its critique

What are some of the questions we must answer in order to be fully s tisfied that we understand wh the micro-canonical reci e works? (1) Surely it is reasonable that the probability distribution we use to

would expect the statistical surrogate by which we calculate equilibrium values to share this temporal invariance property. But just exactly how does the d namical invariance of the micro-canonical robabili distribution function to rationalize, justify, or explain its use?

variant. Couldn't there be other, inequivalent, probability distributions that were invariant in time as well? Is t ere any way in which we can demonstrate that the micro-canonical robabili distribution is the un' ue distribution invariant over time?

have an average value for a quantity even if no individual in the popuatlon as e eature in questlon at t at average va ue. popu atlon 0 e ual numbers of five foot tall and seven foot tall ersons has an avera e height of six feet. What is the connection between average values calcu-

A rich and fascinating, if often puzzling and frustrating, route to find e answers to t ese questlons as its egmnings in e suggestions 0 Maxwell and Boltzmann that the evolution of an individual system followed over all time would be such that its path would go through every

ought to be quite cautious in invoking imperfect isolation of the system as a ratlona e or one s pro a llStlC assumptlons in statlstlca mec rucs. But we will also see why such a resort to the imperfect isolation of the s stem has roved to be a ealin to man theorists. Suppose we make the bold conjecture that an isolated system is such state, eventually go through every micro-state possible for the system in questlon given its macroscopiC constramts. at wou 0 ow rom this so called Ergodic Hypothesis? The trenchant ar ment of the Ehrenfests shows us that iven the Ergodic Hypothesis - if !(q,p) is any function of the phase (micro-state)

for all other points. By the earlier result, the infinite time average of g

tion of its life history with its phase point in a given region that is pro-

Equilibrium theory

161

e more 0 ows in a s raig rw r w y. si r n 'me invariant probability distribution on the allowed region in phase space. Consider any region of non-zero size. The probability assigned to this m

assign to the g(q,p), which has value 1 for points in the region and value average 0 g v r i ni i 's, iv r ic ypo esis, the size of the region. So if the Ergodic Hypothesis is true, there is only one invariant probability measure to consider, at least if we insist that e ions of measure zero receive zero robabili . So the roblenl of the uniqueness of the invariant measure has been substantially resolved.

of the infinite time limits of phase functions, something that must in fact be demonstrated. But its real difficulty is much more severe. In its strict Boltzmannian form it must be false. In 1913, A. Rosenthal, using newly developed work in the topological

phase space. A distinct argument by M. Plancherel, appearing in the same year, and utilizing a subtle argument from measure theory, proved the same result. The Er odic H othesis in its Boltzmann version cannot be true.

through every point in the phase space over infinite time, but rather at a trajectory, starte at any point wou ,in e ness 0 time, come arbitraril close to eve oint in the allowed hase s ace. In the mathematician's language, the conjecture was that each trajectory in phase

of time average with phase average that followed the ypo eS1S an at groun e t e ot er the H othesis was first con·ectured. The difficulties encountered led to skepticism in general about the

standard measure - the micro-canonical probability distribution - as a

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Physics and chance

serve any useful foundational role in the theory.

3. Khinchin's contribution A program of research carried out by A. Khinchin, while not circum-

thesis, did initiate a series of results that in the longer run, played an

It can

u . But how can we be assured that!Cq,p) differs from the phase average of !Cq,p) only with low probability? Here, Khinchin relies on the special nature of the's we must consider in statistical mechanics on the particular constitution of the system out of its microscopic components,

First, he notes that the only functions !Cq,p) we need be concerned with are those whose averages we identify with some thermodynamic uanti . The all turn out to be so-called sum functions - functions of the total phase of the system that are sums of functions of the phases of

energy is the sum of the energy of its component subsystems. With these restrictions and assumptions in place e is ab e to show that given the micro-canonical robabili distribution the robabili of C ~ differing from its phase average by a given amount does indeed rapidly

we can be assured, if the system is indeed ideal and if we restrict our attention to suita y restricte p ase nctions, t at given e microcanonical robabili distribution the robabili of the time avera e of a function !Cq,p) differing by much from the phase average of !Cq,p) will

Equilibrium theory

163

First, there is the restriction of Khinchin's results to the case of the c n i u u because it is the realistic case of systems with intercomponent interactions contributing to the total energy that is the real concern of statistical mechanics. Here the more recent results on the thermod namic limit obtained in rigorous statistical mechanics have gone a long way to weaken-

interaction. But there is a much more disappointing aspect to the Khinchin results. The oint of the Er odic H othesis was to allow us to iden' the phase average of a function of phase with its infinite time average. The

themselves being computed by means of the micro-canonical probability istri ution w ose rationa e was in question in t erst p ace. Suppose there is onl "zero robabili "accordin to the micro-canonical robability distribution that the time average of a phase function will differ "

nature? Certainly not. Suppose there is some global constant of motion o e system a ove an eyon ltS constant energy t at we ave 19nore (such as the constant an ular momentum in the astronomical case). The system in question will have its motion confined to a sub-space of phase-

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Physics and chance

pro all ero in e micro-canonica pro ali is 1 u lon. e cannot conclude, then, that having such "zero probability" is grounds for being considered irrelevant in our physical expectation. Khinchin will function prominently in some of the ways in which the

II. The Development of contemporary ergodic eory 1. The results

0

von Neumann and Birkho

Results obtained by J. von Neumann and G. Birkhoff initiated the course of research leading to contemporary ergodic theory. The initial suggestion of von Neumann's was to look for sufficient conditions on the d namics of a system to prove directly the aimed-for end result of the

point was to avoid the demonstrably false Ergodic Hypothesis and the quasi-ergodic substitutes suggeste or it and to try to prove directly, from some a ro riate condition on the d namical structure of the s stern, the sought-for identity of averages. First, results were obtained by

and also more directly usable for the intended purpose of ergodic theory, I s a co ne my attention to it. Birkhoffs first result fills a a we noticed earlier in the Boltzmannian approach. We noted that the Boltzmann argument, as formulated by the

the allowed phase space using the invariant probability distribution to compute t 1S P ase average. uppose tea owe p ase-space can e broken into ieces such that (1) there is more than one iece, (2) each piece has non-zero measure - that is, the probability that a point is in the

165

Equilibrium theory

gore -. system u

r

composa . uppose e p ase space region or a can be split into two (or more) regions A, B, ..., that are of non-zero c

.,

. ,

micro-state is represented by a point in one of the regions always evolves in such

.

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. .

.

, that is a function of the micro-state of the system whose time average in the . . . . assign each system in A the invariant quantity 1 and each system in B the invari-

.

.

.

,

of the quantity will be p. But for A region systems, the time average of the

.

.

.. .

metrically indecomposable.

for a set of points of zero probability, the phase average of an integrable p ase nction over e a owe p ase-space, uS1ng e 1nvanant pro abili to com ute this avera e, will e ual the infinite time avera e of the function along the trajectory starting from an initial point in the phase

insight by considering the notion of a global constant of motion of a system. uppose t ere 1S some p ase nct10n g q,p suc t at start1ng from any given point, g(q,p) remains constant along the trajectory from that point, no matter how far into the future we take the trajectory. Then

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Physics and chance

our uniform probability distribution and computing our average values. v u nw can define the region of points for which g(q,p) is ~ c, for c in the range [a,b] and those for which g(q,p) < c. These two sets will then metrically decom ose the re ion. So the condition of metric indecom osabili is essentially equivalent to the condition of there not being any neglected

Notice also the way in which the Birkhoff result (sometimes called the Pointwise Ergodic Theorem and that we will call, henceforth, the Ergodic Theorem avoids the necessi for an hin like the Er odic or uasiErgodic Hypothesis. It is now metric indecomposability that is the nec-

time average and the phase average. There is a condition equivalent to metric in ecomposa ility, thoug, at rep aces e 0 Ergodic and QuasiEr odic H otheses for metric indecom osabili is e uivalent to the condition that given any set of positive measure in the phase space, the

on a specified

is, if the probability of an outcome in a trial conditional on the outcomes o ot er tna s is t e same as its uncon itiona pro a 1 ity on a slng e tria - then one can rove that in the limit as the number of trials oes to infinity, the relative frequency of an outcome will, with probability one,

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Equilibrium theory

Figure 5-2. The ergodic theorem. Let a system be started in some micro-state .. . . ,

constraints. Let R have a definite, non-zero, size in the phase space. Then, when

.

., . .

.

micro-states of size zero, the trajectory from the initial micro-state a will even-

Hypothesis were true. Suppose the allowed region of phase space is metrica y in ecomposa e. en, any pro a i ity measure t at is invariant in time and that is such that it assi ns measure zero to all those sets assigned probability zero by our initial invariant probability measure,

and our invariant

The mathematicall rofound results of the von Neumann-Birkhoff theorems are, alas, not very useful by themselves to a program of rationaliz-

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Physics and chance

a program that required thirty years to be brought to its successful con-

system. The core idea is that of the instability of the trajectories in phase-space of the s stem. Here instead of in to rove some con'ecture about single orbits, such as the Ergodic or Quasi-Ergodic Hypothesis, one stud-

both in the future and past directions of time, then it will be possible to prove metric indecomposability. That is, one will be able to show that not more than one collection of hase oints exists that is of more than zero measure, and that is such that the orbits started from a point in it

of course, the entire allowed phase space except, perhaps, for a set of points 0 measure zero. Because we will need to review the nature of this instabili or divergence condition in the trajectories when we examine the rationalization

in the important standard cases proves to be rich enough to prove more powe u resu ts a out t e "ran omness" 0 e ynamica motion as well. These more owerful measure theoretic results rove to be vital in going beyond the rationalization of the equilibrium theory to gaining

The problem of demonstrating ergodicity has, however, now been pus e ac one more step. or now, 1t 1S necessary to emonstrate, from the d namical laws and structure of the s stem the a ro riatel strong divergence of nearby trajectories. A lengthy development of im-

Equilibrium theory

169

an interesting dynamical model by Ya. Sinai. The work begins with the stud of the eodesics curves of minimum curvature and len th on a curved surface) on surfaces of constant negative curvature. These have

"free" motion in that space. One can prove that such "geodesic flows on surfaces 0 constant negative curvature" are indeed ergodic. Sinai was able to show that the motion of s herical articles confined to a parallelpiped box - spheres that move freely except when in collision

particles impinging at nearby points on a given spherical particle are, eir ivergent re ections, soon trave ing trajectories quite ecause 0 distant from one another. As a result tra'ectories that are uite close at one point are soon wildly divergent from one another. (See Figure 5-3.)

3. The KAM Theorem and the limits of ergodicity

Although it has proven possible to show that "hard spheres in a box" constitute an ergo lC system, lt as a so proven POSSl e to s ow at other d namical s stems have features exact! contra to those re uired for ergodicity.

170

Physics and chance

gravitational interaction of the planets with each other, then they will s v i ssu awn a e mutual interaction of the planets with one another will not, in the fullness of time, throw one or more of the planets out of their stable elliptical orbits sendin them either crashin into the sun or "off to infini ?" This was a classic problem of Newtonian dynamics as early as the

tions is impossible. Instead, one resorts to solving the equations by means of series approximations. Early "proofs" of the stability of the solar system suffered however from a severe defect. It roved im ossible to show that the terms of the series being used to approximate the solution to the

The fundamental difficulty is the problem of resonances, or "the problem 0 small denominators." The terms in the series expansion will be fractions fractions whose numerators will decrease in size as one oes out in the series. But the denominators will typically contain a factor of

terms where nand mare

The physical reason behind these terms is intuitively not hard to understan. e p anets w· exert e maximum istur ing e ect on one ano er when the are nearest. These ositions of near attraction will occur periodically. It is when n traversals of the orbit of one planet equal m

Equilibrium theory

171

Figure 5-4. The KAM Theorem. The closed curve S represents a system that

.

.

.

.

example is a planet that, unperturbed, repeats a closed orbit forever. The KAM theorem states that for s stems satis in certain conditions a small enou h perturbation of the system (say, of the planet by the gravitational tug of another laneO will result in an orbit that althou h it will not in eneral an Ion er be closed, will be confined to a finite region in phase-space, indicated by the tube T surroundin the initial curve S. Such a s stem cannot then be er odic and wander over the available hase s ace.

oincare an

proximations to the equations of motion under perturbation as to obtain series w ose convergence cou e guarantee. aSlca y w at t e ........,........ Theorem tells us is this: Su ose a s stem has been obtained b introducing a perturbation into a system that when unperturbed, followed a

there will be a set of initial conditions that will define trajectories such t at a 0 ese traJectones Wl e con ne to a reglon 0 p ase space that contains the original unperturbed multiply periodic tra'ectory, but that is not the entire allowed phase-space. This region of trajectories that

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Physics and chance

KAM results to hold. The case of the Boltzmann model of hard spheres

there is plausible theoretical reason to believe that more realistic models of the typical systems discussed in statistical mechanics will fail to be er odic. Further support for this claim comes from the ingenious computer

computer a vast number of initial conditions for the system. Then one ollows out the trajectory generate y each initial condition. The hope is to find b this method those re ions of initial conditions that ive rise to stable trajectories and those that give rise to the wildly wandering

results when a method, again due to a technique of Poincare, is used. A sing e two- imensiona su ace in e p ase-space is c osen, an t e oints of intersection of a tra'ecto started from a oint with that surface are calculated. If the trajectory is a simple periodic one, it shows up as

gions whose existence is assured by the KAM Theorem. Outside these regions t ere are regions were t e motion seems to e ran om Wit individual tra'ectories wanderin all over the re ion. But are these reall ergodic regions? It isn't completely clear. In the case of those systems

173

Equilibrium theory

A

B

c in phase space followed by the point representing the micro-state of a system is

.

,

.. ,

.

..

with a Poincare section will be one (or more) simple points, as in A. If the system

·

"

, ..

.

.

the phase space, the intersection with the Poincare section will be a curved line

·

..

....

.

section that seems to fill a region on the section as the number of intersections · .

follow. But in the cases where the KAM Theorem gives a region of stable

ness of trajectories in the computer simulations could be compatible Wlt t ere Stl elng metrlc ecomposlt10ns 0 t e ran om reglon. e existence of additional constants of motion that are, however, wildl non-analytic or "unsmooth" in the phase parameters could give us such

grip on the appropriate notion of randomness or chaos to apply to the reglons outSl e t e reglons 0 sta 1 lty lS one t at on y now lS getting under way and in which little progress had been made. The problem of trying to understand what physical situation causes

that the instability is the result of a multiplicity of resonances in the sys em. ac resonance Wl e ec a s rong lS r ance 0 e raJec ones in some regions of the phase space. It seems that the overlapping of regions affected by distinct resonances is responsible for the chaos ob-

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Physics and chance

a re i gri n s c sa an uns e regions or even the simplest cases. A pattern of regions of stability containing within them regions of instability that contain within them islands of stability, n n h m f in even very simple cases.

v

v

system of the kind treated in statistical mechanics, over and above the special cases like that of the hard spheres in a box that are provably er odic can be ex ected to be an er odic s stem. It seems clear that at this point resort will be made to the vast number

that the multiplicity of interactions among them will result in ever smaller regions of stable orbits and an ever greater fraction of the allowed phase s ace bein occu ied b orbits of the a arentl er odic sort. Com uter simulations do indeed suggest just this sort of diminution of the stable

limit of an infinite number of interacting systems, the regions of stability s rin to zero size, in t e ami iar measure, an the entire remaining hase-s ace consists of enuinel er odic tra·ectories. Even this result would leave us with less than we might desire, because the systems with

But we cannot even obtain this result of "ergodicity in the thermoynamic imit or rea istic interactions, at east at t e present state 0 mathematical develo ment. Some theoreticians have in fact ex ressed doubts that the result is even true. The best we can hope for at present

system is "very probably ergodic." And because our notion of probability ere presupposes t e use 0 t e stan r measure 0 pro a llty, even this result re uires that at least that amount of probabilistic thinkin has to be introduced into our rationale of the standard probability measure

Equilibrium theory

175

ID. Ergodicity and the rationalization of •

Summarizin the mathematical- h sical results of the last few sections we can say the following:

mathematically well-behaved - integrable - function of the phase point, each of the following will then be true: (a) For all except a collection of oints of robabili zero in the standard robabili measure the infinite time average of the phase function over the trajectory starting from

phase average for all points except possibly a set of them of probability zero. Cc) The proportion of time a system spends in a region of phases ace as one traverses a tra'ecto will a ain exce t for a set of initial points of probability zero, in the limit as time goes to infinity, be the

thermodynamic limit the systems will once more be ergodic, the regions o sta ity s m ng to measure zero an t e remairung region eing enuinel er odic, it is not clear that this will in fact be so. There is ood reason to think, however, that large finite systems will have small regions

standard way, will be at least ergodic-like. n t e next sections Wi assume t at we are ea ing Wit a system that is genuinely ergodic. Here, we will be concerned primarily with the following question: Given that a system is ergodic, how can we legiti-

176

Physics and chance

1. Ensemble probabilities, time probabilities, and measured uantities One proposal for utilizing the results of ergodicity to account for our experience with equilibrium systems has the advantage of being remarkabl strai htforward. This a roach focuses on the fact th our m surements of macroscopic thermodynamic quantities take time. The

in such microscopically appropriate units as the mean time between molecular collisions, for example. Because this is so, can we not argue that a macrosco ic uanti should be identified with the avera e over an infinite time of its appropriate microscopic phase quantity? In the case

of that microscopic phase function that keeps track of the transfer of momentum to the walls of the container by molecular impact? Then, by the er odic theorem we can for almost all microsco ic states of the as identify the infinite time average of the phase quantity with its phase

idealization. For we have quite good reason to believe this is false. In ee , patent y a se. lour macroscopic measurements cou a be Ie itimatel construed as infinite time avera es then eve macroscopic measurement would have to result in the equilibrium value for

rium, nor a macroscopic ability to track the approach to equilibrium by o OWing t e vanation in t ese quantities as ey approac e elr na e uilibrium values. But of course we can macrosco icall determine the existence of non-equilibrium states, their features, and the laws that

Equilibrium theory

177

sea s, u averages. There is, however, a subtler way of utilizing the results of ergodic th 0 - in articular its abili to ident' ensemble uantities with temporal quantities of an individual system - to understand the success

Let us return to the picture of a system isolated energetically over a time from past infinity to future infinity as proposed by Boltzmann in res onse to some of the criticisms of his earlier versions of the HTheorem. In this picture, a system over infinite time moves from micro-

micro-state described by the Maxwell-Boltzmann distribution. Excursions from this "most probable" state 0 occur, but the greater the excursion of the s stem awa from this dominatin most- robable state the less frequently it occurs. The picture is, remember, completely time-symmetric,

If the system is ergodic, we can, at least to a degree, back up this Bo tzmannian picture wi an argument t at t e picture is in some sense a correct re resentation of the facts about the s stem. First we must note that the ergodic results will tell us only that the Boltzmann picture holds

in micro-states from a set of probability zero in our usual probability picture. But elng a mem er 0 a set 0 measure zero IS not elng an im ossible state. For the time bein ,however, let us focus on the "hi hi probable" systems, leaving the qualification to be attended to later.

ate to the size of that region relative to the total size of the accessible p ase space In t e stan ar measure. ut w at assures us t at t e SIze of a re ion of oints that re resent s stems close to e uilibrium will, in fact, be large in the accessible phase space as a whole? Not ergodicity.

thermodynamic limit. Here, we restrict our attention to those phase functlons t at 0 ey t e sUlta e structura con Itlons eman e y t e Khinchine program, the hope being that we can show that all of those phase functions that we relate to thermodynamic macroscopic quantities

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..........L.L'U'u rm 'c 'm', i is useful to note that the rate of convergence of quantities to the limit quantities is something one can compute in the theory of the thermod namic limit. And assurance that normall macrosco ic s stems are large enough to appropriately idealize them by the thermodynamic limit

distribution converge to one sharply and symmetrically distributed around the overwhelmingly dominate most probable value, which is then identifiable with the mean or avera e value ives us a clear unde innin to the Boltzmann picture.

world, Ergodicity, where it can be demonstrated, does certainly provide us with something of deep conceptual interest - that is, a proof that for almost all s stems one formalisticall robabilistic notion size of re ion of phase space in the standard measure) is provably equal to another

about the nature or origin of equilibrium states does that allow us to answer? It does not seem lausible to ex ect er odici to la a role as a component of a causal explanation of the equilibrium condition of a

though, we shall see that hopes of a complete statistical-causal explanation 0 t e ong1n 0 t e approac to equ11 num W1 not e y gratified. On the other hand, the derivation of the er odic results from the pure dynamics of the micro-components does give us, in some sense, a "

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we experimentally found states by randomly selecting them from a system maintained in isolation over an infinite time, then we could, presumably, subsum tivel account for an observed state bein of a certain macroscopic nature by making reference to the time proportion of the system

to equilibrium sort. But of course we really don't experimentally fix on equilibrium as that state in which a system isolated over infinite time sends nearl all of its time. Rather it is that state uickl a roached by non-equilibrium processes and that maintains itself over what are

spent in a region of micro-states over an infinite time is a formally probabilistic notion. It is, in t e case 0 ergo icity, a well- e ne dis ositional notion even from the actualist oint of view because it is derivable from the underlying actual lawlike propositions governing the

sure closer to "what we physically determine," in the sense of allowing us to see t at t e measure equi i rium quantitles are not just t ose overwhelmin I robable in an a riori resented robabili measure. They are also overwhelmingly probable in the sense of being the time-

Here we see a case both of probability as legitimatized disposition and o statlstlca exp anatlon as s oWlng t e assoclatl0n etween w at occurs and some idealization. So Ion as we kee ourselves aware of what such an "explanation" of the nature of equilibrium isn't - not a causal ex-

been provided. Ergodicity does serve to satisfy t at pro a 1 ltles as proportlons ln lffiaglne ensembles of identicall pre ared systems be brought down to earth by being shown to be legitimated representatives of probabilities as propor-

In the last section, we utilized the consequence of ergodicity that stated that for almost all points, in the limit of infinite time, the proportion of

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upon the simple corollary of the theorem to the effect that any probability measure that assigns measure zero by the standard measure and that is invariant in time under the d namic evolution of the s st ms in th ensemble must agree with the standard measure on the probability to be

state of a system. Its empirical origin is in terms of a system whose macroscopic thermodYnamic variables remain the same over observable eriods of time idealized to invariance over an infinite time in our theoretical picture. But, of course, in our statistical mechanical picture

greater detail in Chapters 6 and 7, the probabilistic picture that takes the place of the thermodynamic lawlike approach to equilibrium of a system is that of an ensemble of s stems ori inall re ared b fixin certain macroscopic constraints, evolving to a new ensemble when these macro-

ensemble having their micro-state move out of a region of micro-states ue to t elr ynamlc evo utton, an 0 ot er systems w ose micro-state was initially outside the region having their micro-state enter the re ion because of their dynamic evolution. But the proportion of systems in the

It is easy to

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,

1

1

the same probability even if their size in the standard measure is the same.

standard measure renormalized there and give probability zero to all t e ot ers. e regions 0 t e piece in w lC t e pro a llty a een concentrated would have their robabili remain invariant as before, and the subregions of the other pieces would have invariant measure

the ensemble is metrically indecomposable, we can then be assured that t ere is 0 y one pro a llty measure t at is lnvanant in time, at east as Ion as we confine our attention onl to robabili measures that assign measure zero to every set of points assigned measure zero by the

tions of dynamic invariance and "absolute continuity" with respect to the stan r measure. At this point, we can then reinvoke the thermodYnamic limit results used before to assure ourselves that having chosen the standard prob-

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ity distribution, we can be sure that it is "overwhelmingly probable" that rm y s. u u us, v s number of micro-components in the system, the special nature of the phase functions used to reduce the macroscopic quantities, and the special structural features of the s stem re uired to rove the thermod amic limit results. 3. The set of measure zero problem

At this point, we ought to deal with a problem that has repeatedly surfaced in our treatment of the use of er odici to rationalize e uilibrium statistical mechanics. If a system is ergodic, then phase averages of phase

regions will equal, in the infinite time limit, proportions of duration the system spends in the region, except, possibly, for a set of measure zero of initial hase oints. The standard robabili measure will be the unique invariant measure, except, possibly, for other, non-equivalent

sets of probability zero in the standard probability assignment can be ignore? That a set has robabili zero in the standard measure hardl means that the world won't be found to have its total situation represented by

on will not equal the phase average of that quantity. Such an assumption, a pnon as 1t may e, 1S certa1n y a wea er, pure y stat1Stlca, a 1tlon to the underl in d namics than was the ori inal assum tion of the full standard probability measure as correctly representing all probabilities. ?

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s s u assume that there is no likelihood above zero of a state of the relevant kind being encountered in the world - usually rely upon ways of connectin measure zero in the standard measure with other features of the set.

neighborhood of that point (that is, open set containing that point), there are points not in the set contained in that neighborhood. This suggests the idea that an s stem of that kind would be instant! driven out of the relevant set by even the slightest interference from the outside.

But as D. Malament and S. Zabell point out, one must be careful with such arguments. Some sets that fail to have interior points are such that we certainl do not want to attribute zero "real" robabili to them. The set of points having at least one coordinate with an irrational value is also

measure one in the standard measure. Malament an Za e suggest an a temative property associate havin measure zero in the standard measure to focus on. Su ose we demand that probability have a continuity property - that is, if one set

should be close to the probability of its being in the first. The idea seems to e at a met 0 0 prepanng an ensem e - t at 1S, 0 Xlng constraints so that one uarantees that a s stem will be in a iven set - ou ht to have this continuity property. The condition implies that sets of mea-

realize, we can sometimes so prepare systems as to guarantee that they W1 e 1n a set 0 measure zero. y ng t e energy 0 a system, we arantee that its hase oint will be on the ener her-surface, a set that has zero measure in the volume measure of the full phase space. In

started with, and such lesser dimensional sub-spaces are always of measure zero. Malament and Zabell res ond to this b ointing out that one could then simply restrict one's attention to the phase space defined by fixing

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wou en guaran ee a one oug 0 assIgn pro a I I e sets of measure zero in that refined standard measure. One can go even further, because we are working in a context in that there are no global constants of motion that vary continuously over s an s 0 mo ion, e region 0 p ase space wou posable and the system not ergodic. Yet we must still be cautious. What exactly do constraints on our ili to limit an en m Ie h v do wi 1 ........" h,., world? Even if it were the case that we could not control systems in such .l.l.l"~U"'u

w have grounds, over and above purely postulational ones, for assuming that we have a right to take as zero the real probability that a system in the world could be in the deviant set? I think that this uestion is mor easily discussed in the context of the rationalization of non-equilibrium i

n

Ul:

grounding the postulates of statistical mechanics are most fruitfully treated. Let me mention here, though, in anticipation of the fuller discussion in Cha ter 7 a few salient facts. We shall see that our abili to confine the state of a system to a particular region of phase space by purely

even such plausible principles as the continuity principle invoked by Malament and Zabell to rationalize attributing "real" probability zero to sets of measure zero in the standard measure. We shall also see that claims to the effect that certain classes of points in phase space are so

systems that we could view as a prepared ensemble ought also to be accepte on y wi caution. In C apter 7,11,1, we wi examine some cases where interference from the outside can be demonstrabl reduced to such a degree that modes of preparation we might have thought made

The suggestions of Malament and Zabell discussed, and numerous ot er suggestIons rna e In t e context 0 tryIng to rationa Ize pro a IStIC assum tions in statistical mechanics make use of im ortant to 010 ical features of the space of points representing the micro-states of systems.

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topological concepts are formally defined, we can simply note that such notions as the continuity of a curve, the "neighborhood" of a point, the "0 enness" or "closedness" of a set whether it does not or does contain all of those points that are the "limit points" of "converging" sequences

can define a topology out of the metric in a simple way. The most common standard formulation of topology defines all other topological notions from the notion of 0 en set taken as a rimitive. The 0 en sets can be specified as being all those generated out of a proper sub-set of

given point. To say, then, that a point has points of a specified kind in any neig or 00 0 it ecomes e cairn t at points 0 t e specifie kind can be found as close as one likes to the iven ont. Just as measure theory allows for the specification of a set that is

Formally, these are all sets that can be generated as countable unions of sets t at are "now ere ense" in t e space in question. at is, t ey are countable unions of sets whose com lements contain a dense subs ace. The sets not of first-category are said to be of second category. The "

measure zero in the reals in standard measure theory. Many notions app lca e In measure t eory an e stu y 0 spaces under transformations from a measure theoretic oint of view, the notions we have explored in talking about ergodic theory, reappear trans-

minimality. There is, in fact, an important "duality" theorem to the effect t at woe c asses 0 t eorems In measure t eory are converte lnto to 010 ical theorems when the a ro riate substitutions are made of topological notions for measure-theoretic notions, such as substituting

aspects of the non-equilibrium theory as well) be grounded in topologica acts a out e e aVlor 0 t e traJec ones mappe out In p ase space as systems evolve according to the laws of the micro-dynamics? Some explorations in this question have been made. There are a number

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e ese resu s rue. an we s ow a or appropria e y i ea lze systems they are true? And even if true, could they be used in the justificatory way one intended without debilitating criticisms being raised ?

One might hope to show, for example, that for a class of states all of " c n eg ry - are suc a systems following those trajectories have their states in the set of near equilibrium states almost all of the time - that is, with time average one in the limit as time oes to infini . Alas rovin such results for reasonably idealized physical systems is generally a near impossible task. Again,

is either of the first-category or has its complement of first-category, is also usually an impossible task. In answer to the third uestion noted earlier it is sometimes the case that we can be pretty sure that the plausible topological result we have

section was to explain why we ought to believe that sets of measure zero in the standard measure could be ignored. Is it a basic posit of statistical mechanics that such a rocedure is Ie itimate or can this osit be grounded on something seemingly less arbitrary? One suggestion might

open to our "choice" than choosing a measure absolutely continuous wit t e stan ar measure. But, a as, t ere is a resu t to tee ect at the real line can be decom osed into two sets one of measure zero and the other of first-category. It would seem, then, that being of the first-category

measure-theoretic or probabilistic point of view. For anot er examp e, we nee on y re er ac to t e notion 0 quasier odici mentioned in the historical surve of Cha ter 2 and earlier in Chapter 2,1,2. There we noted that the original hypothesis that each

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quasi-ergodic was not sufficient for infinite time averages for each trajectory to equal the phase average of a quantity over the accessible region of hase-s ace. And it was this latter claim that the er odic osit had been introduced to justify in the first place. Here we see a topological

Other, weaker, results can be obtained, however. Given a topology, the open sets are sets that have interior points - points surrounded by nei hborhoods all of whose oints are in the iven set. It is lausible to think of such sets as "larger than zero in size" and to demand of a

nate some possible measures in the abstract sense, and may even pin one down to t e stan ar measure as t at one measure unique y regu ar relative to the chosen to 010 . Here a ain is a su estion for alleviatin some of the arbitrariness in choice of measure that afflicts the usual

demand regularity of a measure, given our intention to think of measures as some in 0 in icator 0 t e "rea proportion" 0 cases in w .c micro-states occur in the world? And how non-arbitra is our choice of a topology, and the associated metric distance function in the phase

that spends almost all its time near equilibrium), then we are right to say t at any system, no matter ow prepare or se ecte ,W1 Just a y e thou ht to have that feature. But such ar ments depend u on im licit assumptions that the metric used, usually the familiar standard metric of

"arbitrarily close" systems in the given metric cannot be distinguished by any p YS1ca process W1 1n our con ro . But, as we noted in the comments on the arguments of Malament and Zabell, questions of just what we can and cannot do to select out systems

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equilibrium theory in Chapter 7. At this point, we ,need only remark that any attempt to get around issues of the "arbitrariness" of the imposed measure used in the usual foundational formulation of statistical mechanics that relies upon some deep association of the standard measure with

we must show how the topological feature invoked is to be presumed relevant to the foundational issue in question. Then we must show that the choice of to 010 and associated conce ts such as "arbitraril close" for phase space points, can be backed up with a demonstration that this

4. Ergodicity and equilibrium theory in the broader

for a very limited class of systems, full ergodicity can be demonstrated. ntIs case, we ave, neg ectlng e pro em 0 a POSSI e c ass 0 exce tional micro-states of measure zero, the result that the standard probability measure is uniquely invariant and hence uniquely suited to

kept isolated for doubly infinite time, the proportion of time spent by t e system In a regIon 0 mIcro-states IS proportlona to t e SIze 0 at re ion in the standard measure. Invokin the vast size of the s stem and the special nature of the phase functions used to compute thermo-

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ergodic-like region outside the regions of stable trajectories become overwhelmingly dominant as the size of the system increases are hinting at some kind of enuinel ri orous er odic-like result in the thermodYnamic limit.

standard equilibrium ensemble and the standard technique of calculating equilibrium values by means of phase averages, we have stayed away entirel from the roblems of characterizin non-e uilibrium states and their dynamic evolutions. For this reason, we have made no attempt at

..

" Nor have we gained any insight, of course, into the details of the process were y systems evo ve into equi i rium, suc as t eir stan ar rates 0 a roach to that state. Most cruciall we have made no attem t at all to understand the peculiar time-asymmetry of the process summarized by

topics from being investigated, it also limits the kind of account we can give 0 w at we are invest1gating. We can, in t e equ i flum approac , offer some de ree of "transcendental" rationale of the e uilibrium robability distribution, by showing it uniquely invariant. And we can show

causal or statistical-causal account of the origin of equilibrium, because 1S, pe orce, requ1res re erence to t e states out 0 w 1C equ11 flum evolves and statistical eneralizations over their likelihood of occurrin . Nor can we even offer an explanation of equilibrium of the simple

states will require fitting equilibrium into the much broader context of all states 0 systems, equ11 num an non-equ11 flum, an t e1r ynam1c interrelations. The reference class of all states over doubl infinite time of an ever isolated system is an interesting idealization, but of much

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eory

u e u eful to explore at this point. Jaynes suggests that equilibrium statistical mechanics can be viewed as a special case of the general program of s stematic inductive reasonin . From this oint of view the robabili distributions introduced into statistical mechanics have their basis not so

basis of probabilities at some point, but instead in a general procedure for determining appropriate a priori subjective probabilities in a systematic wa . a nes has ar ued that these results can be extended to nonequilibrium cases as well, a matter we shall explore in Chapter 7,111,3

to designate someone who argues that the inductive process ought to be viewed as the updating of subjective probabilities we hold prior to an ex eriment b utilizin the results of the ex eriment. The u datin is done by the use of Bayes' Theorem that allows us, by conditionalization,

But where do our initial subjective probability assignments come from? Many Bayesians wou 0 t at t ey are simp y re ective 0 our proensities to believe. Insofar as the are coherent robabilities - that is insofar as they obey the formal axioms of probability theory - they are

loss is possible but no gain is), and this is sufficient to demand that the pro a i ities 0 ey e orma princip es. But no stronger constraints 0 rationali can be demanded. The "Objective Bayesians" demand more. For some of them, only a

abilities is a generalization of the famous Principle of Indifference. That ru e te s us to asslgn equa pro a 1 llles to a symmetnca y equlva ent outcomes. An exam Ie is the assi nment of e ual robabili 1/6 to the outcome of any specific face of a tossed die ending up topmost. The

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Gibbs' construction of the standard statistical mechanical probability distributions. They, remember, were the distributions that maximized the fine- rained ensemble entro as Gibbs demonstrated. The su estion now is to do this: Let there be some appropriate measure over the con-

values that maximizes the ensemble entropy calculated in the Gibbsian manner. This assignment will be the generalization of the probabilities determined b the Princi Ie of Indifference in the case of a finite number of basic outcomes, and it will specify one's rational choice of a priori

Shannon on the foundations of information theory. Consider any probability distribution over a finite number of possible outcomes. Shannon shows that the entro of that robabili distribution -~ 10 is the unique function that is additive for independent distributions - that is

when we have no grounds for expecting one outcome over another. And at is t e case w en t e pro a 1 lty istn ution we assign to t e outcomes is uniform. We now have what looks like a justification for the standard probabil-

entropy. May we not then justify this choice not on some grounds of a specla postu ate a out t e requencles 0 occurrence 0 micro-states 0 s stems sub'ect to macrosco ic constraints but on the rounds instead that this is just the choice of a priori probabilities always justified on ?

The injunction to "spread a priori probability uniformly" over a contmuum 0 POSSl e va ues is, a as, vacuous. or unl orm y Wi vary de endin u on the measure we impose on the set of values. If the values are in the range [0, 1] we will get quite distinct a priori probability

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a 0 i nc n e in e e re aining a - an spread a priori probability uniformly relative to the new, unusual measure. A means of resolving this very familiar ambiguity was suggested by H. x probability is to be assigned. On the basis of the nature of that experipup xp v u ,11 a 've to which the probabilistic consequences of the experiment must remain invariant. Suppose, for example, that we are trying to guess a parameter that is a sin Ie real number. Our ess should remain the same no matter where we arbitrarily fix the zero point of measured and guessed values.

This fixes the unique "uniform" a priori probability as that uniform in the usual linear measure from minus infinity to plus infinity. Actually this is a so-called im ro er rior because it reall isn't a normalizable robabili measure. Every "well posed problem," in Jaynes' terminology, should

As a general rule of inductive inference, this "objective Bayesian" approach has its problems. These include difficulties in reconciling the rule for a riori robabili with conditionalization in the case of multi Ie experiments, the problem of uniqueness of results given possibilities of

abilities. We shall not explore these here, but instead focus on the application 0 e program to statistica mec anics, an in particu ar at t .s oint to the e uilibrium case. Suppose we have a system in equilibrium, and we wish to guess at its

trollable constants of motion and fixed macroscopic constraints. Because e system is in equ' i rium as time goes on, no . g macroscopic c anges. Hence our "i norance" of the exact micro-state remains the same over time. So the probability distribution we attribute to micro-states should

one with respect to the invariant panta-micro-canonical measure. Hence 1t 1S raUona 1ze y t e pnnc1p es 0 0 Ject1ve ayes1an1sm. Clear! the ar ument here is hard to distin uish from the ar ment, familiar since Boltzmann, that equilibrium, being an invariant state over

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tion was the uniquely temporally invariant measure was required to fully justify choosing it. This was one of the motivations behind the Ergodic H othesis of course. What does the ob'ective Ba esian res ond to this demand?

infinite averages with measured quantities using the time duration of measurement processes as justification is fallacious, As we have seen, however er odici when it is demonstrable does ive us a demonstration of the uniqueness of the invariant probability distribution, subject to " priori assumption, or at least an assumption that, as we have seen, requires resort to non-equilibrium theory for its justification, it still reduces what we must acce t a riori in our robabili assum tions and should be of interest even from the objective Bayesian approach,

tions deemed appropriate by objective Bayesianism relative to the constants we are aware 0 w' ea to e wrong pre ictions a out equ' i rium values. Now a es oints out that this will in fact su est to us that there are remaining constants of motion to be found, Indeed it is true that false

we have always found them. But several things need to be said, The first is at' t ese constants 0 eXIst, our Initla pro a i Ity assignment was wron su estin that there is an ob'ective ri htness and wron ness about a priori probability distributions. But from a subjectivist point of

priate invariance characteristics relative to the knowledge that we had. Even 1 t ere are no contra a e macroscopIC eatures 0 e system we have i nored, either because the s stem is er odic or because the regions of non-zero measure in which phase averages don't equal time

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might be answered in the fuller context. r . m j v v w is sugg stion is that statistical mechanics is just inductive reasoning, and "inductive reasoning usually works." He continues, "To push the question one ste further baskin 'wh does inductive reasonin work?' would lead us beyond science and into philosophy. In science, one ordinarily as-

But this is not very persuasive. To shunt the question out of science into "philosophy" where, presumably, it can be dismissed by scientists, seems a eculiar move for an ex onent of a view that assimilates an important branch of theoretical physics to "pure inductive logic." If in-

tive reasoning, work?" is precisely a demand for a scientific explanation. Even Hobson admits that when "inductive reasoning" doesn't work it is a scienti c ex lanation that we seek - that is "what controllable constant of motion have we missed?"

for the standard probability distribution. And, admittedly, even this is unavai a e to us in e case 0 t ose realistic systems or w ic we can, b lausible extra olation of the KAM Theorem reasonabl infer that the system is not ergodic in the full sense.

And we shall use the suggestion made in that context as rationale for the approac . We s a a so once more contrast is attempt to view statistica mechanics as a branch of eneral inductive reasonin with the more orthodox attempts to see it as a physical theory acceptable only on

For the classic treatment of e uilibrium ensembles see Gibbs (1960). Balescu (1975) and Munster (1969) are typical modem treatments.

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Ehrenfest (959). For the refutations of the Ergodic Hypothesis, see Rosenthal (913) and Planchere1 (913). The discussion in Farquhar (964), . 6- is illuminatin . For the Quasi-Ergodic Hypothesis, see Farquhar (964), pp. 77-78, and

For Khinchin's use of the large numbers of degrees of freedom and the laws of large numbers to evade the need for ergodic posits, see Khinchin 1 4 . For comments see Truesdell 1 61 . Farquhar (964) is an excellent work on the older ergodic theory (that

. .

.

On contemporary ergodic theory, two very brief sketches are the appendices to ]ance1 09 9) and Balescu (975). Arnold and Avez (19 8) is an e1e ant and mathematicall so histicated reds of the basics. Sinai (976) is a good introduction for those with some mathematical sophis-

A brief sketch of the KAM theorem and its importance is in Arnold and Avez 19 8 , C apter . A more extensive iscussion inc u ing t e aplication of the results to roblems in d namical astronom is Moser (973).

problem for realistic systems, see Wightman (1985). For teo servatlons 0 some p 1 osop ers on ergo ic t eory an its ex lanato role see Sklar (973) and Malament and Zabell (1980). Other philosophical pieces that should be consulted are Quay (1978), Friedman

with a statistical, microscopic theory. t 1S an expenmenta act t at many non-equ11 num states 0 matter (and radiation) can be characterized in terms of a small number of field arameters - that is assi nments of values of a h sical uanti b means of a function from locations in the system to numerical values. In the equilibrium situation by a kind of generalization. Thus, such kinematic an ynaffilca quantities as energy, pressure, an vo ume are carne over, except that an intensive quantity like pressure now becomes local ressure at a oint. And such urel thermod namic uantities as temperature and entropy are generalized to local temperature and local

time, as if it were in a temporary, local equilibrium condition. e s a es so escn e are 1n ynaffilc ranS1 10n. we 1scover experimentally that we can frequently find laws governing the dynamic evolution of the s stems. These ran e from the sim Ie heat trans ort equation relating heat flow to temperature gradient, to the hierarchy of

19

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of material diffusion appearing in diffusion equations. These have definite values fixed by the constitution of the system. In the situation where a near e uilibrium s stem is evolvin to e uilibrium, we frequently discover important "reciprocity" relations among

gradients on heat flow. Next, there are the dramatic experimental facts revealing the existence, in s stems far from e uilibrium of stead -state flows. Such henomena as hexagonal flow cells in fluids maintained under thermal gradients,

dynamics. There is also the fact that these steady-state systems can exist in multiple phases, with transitions etween t em at critica va ues 0 e thermod namic uantities. This interestin arallel to e uilibrium hases and phase-transitions also requires explanation in our statistical, micro-

ibility of non-equilibrium processes. This is the most important qualitative act 0 a a out t e non-equi i rium situation, t e tempora y one-si e drive of non-e uilibrium s stems to " oal" e uilibrium states. There is reason to believe that we can extend the role of entropy, which appears

Here, we talk of the entropy in a region of a system at a time, and try to c aractenze t e unl uechona c ange 0 e system y a pnnClp e 0 local entro ic increase or roduction as summarized b somethin on the order of the Clausius-Duhem inequality. Under special circumstance

minimal for the actually observed dynamic flow. a I lty to macroscoplca y c aractenze t e lven t IS nc an rul non-e uilibrium situation, what can we ho e to do in the wa of accounting for its existence and success by underpinning this macroscopic .

?

scopic account? Certainly the underlying dynamics of the microscopic cons I en s, el er c aSSlca or quan e l Y a so e cons I u Ion of the system out of its microscopic constitutions including the specific laws of interaction between them - for example, the specific form of

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And, we expect, we will need to invoke some principles for determining i u vip 1 1 a pa icu ar micro-state compatible with that macroscopic constraint. We also expect, and we shall see how true this is, that many of the most fundamental issues in roundin the non-e uilibrium theo will rest u on the rules for assigning such a probability distribution and the explanatory grounds

of the statistical micro-theory? We would like to be able to specify in the grounding theory's terms just what conditions are necessary and sufficient for there to be a descri tion of the s stem in a small number of macroscopic parameters. And we would like to be able to determine just

requisite kind are encompassible in the macroscopic-thermodynamic mode. We would like a characterization in the roundin theo 's terms of just when a dYnamical evolution equation is possible and a derivation

ideally like to be able to derive a general solution of the dynamical equation 0 evo ution. An we wou i e to e a e to re ate t is genera solution both to the nature of the macrosco ic evolution e uations and to their most general solutions. Insofar as the dynamics results in multi-

fundamental theory as well. o e egree t at t ese envatlons 0 t e equatlons 0 evo utlon, t e values of their arameters or the nature of their solutions re uire the positing of principles over and above the basic laws of micro-dYnamics \

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why we can with success make such posits. Finally, we would like a characterization in terms of the statisticalmicrosco ic theo of the all-im ortant time-as mmet of macrosco ic phenomena. Insofar as this can be characterized in terms of entropic

statement of the conditions necessary and sufficient to guarantee its non-decrease. Insofar as we can go beyond this to a non-equilibrium entro erha s of the field varie and to a rinci Ie of entro ic nondecrease for this generalized entropy, perhaps of the Clausius-Duhem

Here, once again, it will not be enough to state the conditions necessary to guarantee the existence of an entropy and to guarantee its asYmmetric behavior in time. We would like in addition an ex lanato account of the reason why such posits as are necessary to gain these

ll. General features of the ensemble approach

1. Non-equilibrium theory as the dynamics of ensembles clear. Macroscopically, a system is prepared in a state that is unstable and t at evo ves lnto lstlnct states In a progression towar a state t at is macrosco icall unchan in the e uilibrium state. But the develo ment of kinetic theory through the responses to early critiques makes it clear

r -space,

the phase space in which points represent the entire microSCOplC ynamlca state 0 t e system. e lnltla macroscoplC preparatlon restricts our attention to a limited region of this phase space. We impose a probability distribution over the set of micro-states compatible with this

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procedure will involve, of course, some appropriate averaging done with the probability distribution over the ensemble. The ultimate aim will be to derive some e uation in the macrosco ic uantities that described the monotonic approach to equilibrium through time of the idealized system

equation, and which extracts from the ensemble evolution some feature of it that has the appropriate time-asYffiffietric, evolutionary quality. Here it is worthwhile makin a cou Ie of eneral observations that hold irrespective of the details of the theoretical construction and inde-

classical and quantum statistical mechanics, the systems investigated are usually such t at t e Poincare Recurrence Theorem holds. Thus we can ex ect an individual s stem in the ensemble ke t isolated for a sufficiently long time to recur to within any small degree of approximation

at which they occur, their individual "almost return" to a state at a given initia time is compati e wit t e ensem e never sowing suc an "almost return" to its initial confi uration. It is interesting that this argument fails in a statistical mechanics based

question has only a finite number of degrees of freedom, which, for parUc e systems, means a nlte num er 0 ffilcroscoplC components. n the uantum case, one can onl avoid recurrence for the ensemble b making the further idealization to a system with an infinite number of

Describing non-equilibrium

201

or level in the quantum context. Next, we must be careful in the way we understand the use of the n m I vol tion to enerate a time-as mmetric kinetic e uation. Here we must remember the careful explication of the probabilistic interpre-

is usually not supposed to represent the "most probable" pattern of evolution of an individual system. We know that cannot be the case because all exce t exce tional s stems will show non-monotonic recurrence behavior. Rather, the usual picture is this: We start at a time with

tify the state achieved at that time by the ovetwhelmingly greatest number of systems. We plot the succession of such dominant states. It is this "concentration curve" of the ensemble evolution that is su osed to be represented by the solution to the kinetic equation. The monotonic, time-

recurrence behavior, so long as the recurrences of the individual systems in the ensem Ie are proper y uncoor inate wit one anot er. But, we should note some ro rams for rationalizin the non-e uilibrium theo will in fact propose the alternative picture of the statistical theory as

ponent micro-systems of a given system, even in the classical case. To comp ete our account 0 w y systems s ow apparent y aw i e evo utiona behavior of their macrosco ic arameters we would like to show that the probability of these parameters having at any time the value they

and the appropriate means of going from ensemble structure at the time e inc in to parameter va ue. ere,o course, 1t W1 e arguments 0 varie that are either im licit! or ex licit! brou ht to bear. A ain, there will be alternative rationales that utilize the large numbers of degrees of

meter value is calculated from the ensemble. For example, there is the assumption t t t e quantities 1n question W1 e appropnate sum functions" of the values of the d amical state of the microsco ic components, or some appropriately well-behaved generalization of a sum

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u u' i iumca e, one would have to give a proof to the effect that if the intercomponent interaction is sufficiently well behaved, then, in the thermodynamic limit of th n m er of com onents an size of th s t m in with density held constant, the use of law of large number theorems will

probability distribution for the parameter value calculated for an individual system in the ensemble will in fact cluster overwhelmingly around the mean value that we calculate from the ensemble. It is this ar ent not usually rigorously carried through but, instead, presupposed, that

parameter, in the sense of "probable evolution" derived from the Ehrenfests and just outlined.

2. Initial ensembles and dynamical laws outlined then comes down to this general procedure: First define an initial ensemble, that is an initia region 0 points in the r-space phase space for the s stem alon with a robabili distribution over the oints in that region. Follow the evolution of that ensemble as determined by the traject-

scopic observables to derive from the kinetic equation and its solutions t e appropriate macroscopic equations 0 evo ution at represent e ex erimentall observed non-e uilibrium behavior. But, alas, carrying out this general procedure is very far from straight-

Describing non-equilibrium

203

tack has been to seek generalizations of the familiar micro-canonical and canonical ensembles of equilibrium theory. The generalization of the micro-canonical ensemble is resumabl the one to a 1 in cases of isolated systems and the generalized canonical in the non-equilibrium

often used for its computational simplicity, with a "thermodynamic limit" type argument invoked to rationalize this because the results of microcanonical and canonical will a ree in that limit. To characterize an ensemble, we need to pick a set of macroscopic

assigned values (or probability distributions) over the selected macroscopic quantities. But how are the macroscopic quantities to be determined? There are basicall two a roaches. One relies u on the observational-experimental facts we have about non-equilibrium systems.

quantities. T erst approac requires some insig t into w at macroscopic eatures of a s stem can be ex erimentall determined. Here it is not some general theory of this that is being used, but rather our experience that

of interest. Another closely related "empirical" approach relies upon our 0 servationa experience t at te s us at in many cases we can determine at the macrosco ic level d namical e uations of evolution that describe the lawlike behavior of many systems. The macroscopic

evolution on the macroscopic scale. e ternative approac to c aractenzatlon 0 appropnate macroscopIC arameters for initial ensembles relies on the observation of the wa in which, in typical cases, information from the ensemble is extracted to

function!n(ql' .. qn,Pl ... Pn) is the function that gives, for the ensemble at a tlme, e set 0 a con Itlona pro a Iltles a out ocatlons 0 articles in osition and momentum ranges. That is, it will contain such information as "the probability of finding a particle in the range q + t1q,

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Physics and chance

abilities that the particle is in that region conditional on each possible ju v r u c i uc ess i orma ion than the list. Similarly,.h contains only probabilities conditional on the location of one other particle; etc. Each h+1 contains more information than but less than The point here is that in general, macroscopic parameters will be del'

suggestion is, then, that we take as macroscopic parameters for characterizing initial ensembles only those quantities that are to be associated in our statistical theo with uantities definable b where i is less than n, the total number of particles. If one goes to the infinite system limit

macroscopic parameter that may be useful for a general theoretical study o initia ensemb es. In practice, to study systems we must resort to our em irical ex erience as noted earlier in decidin which macrosco ic parameters, with their particular association with an /" are to be used.

statistical mechanics in Chapter 9. Suppose t e macroscopic parameters ave eir values or ran es over values (or a robabili distribution over values as in the generalization of the canonical ensemble), have also been chosen.

and one can characterize the probability distribution as one with "minimal corre allon a ove t e spec callon 0 t e macroscopIC parameters. n other words, althou h the ensemble is not, in one sense, uniform over the allowed phase space, once the constraints have been established that

Describing non-equilibrium

205

equilibrium that becomes non-equilibrium because of the instantaneous change of a constraint. For example, consider a gas initially confined to the left-hand side of a box 'ust after the artition has been removed. Our assumption that the system was in equilibrium when the partition was

partition was removed. Of course, after the partition is removed it is not uniform over the new y enlarged available p ase space. In ractice two kinds of initial ensemble are considered. One kind is that described, where a system that was in equilibrium no longer is

temperature distribution, density distribution, and so on. Here, the uniormity assumption is ut' ize in e assumption t at t e appropriate initial ensemble is the "local e uilibrium" ensemble. This ensemble is the ensemble generated by taking the distribution of probability in regions

field quantities (local density, local temperature, and so on) associated wit at region 0 t e system. Many non-equll flum systems are emiricall discovered to be such that even if the are not local e uilibrium systems to begin with - that is, they are in initial stages where even local

investigation as we proceed. ow lt wou seem t at aVlng spec e our mlt1a ensem e our work is done. For the evolution of the ensemble is completely fixed by the evolution of each of the systems in the collection. And their evolution

the system and no purely stochastic underlying dynamical laws. u e way m w lC lne lC equa lons are usua y eflve lS no y a study of the effects of dynamical evolution on the initial ensemble. Instead, use is made of an apparently independent statistical assumption,

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Physics and chance

meters obtained up to that point. That is, if the ensemble has developed we can pre ic i r: ev u ina we were s a ing wi a ran om system having those parameter values. The past history of the system how it got to the point it is at - is taken to be irrelevant for our future statistical redictions. But it is not at all clear that such a "rerandomization" posit is even

condition. We shall see two approaches to resolving this problem, and we shall see each approach introduce its own need for foundational roundin . One approach attempts to derive the legitimacy of the rerandomizing

evolution that govern each member of the ensemble. The other approach tries to show, from the laws of dynamic evolution alone, and without use of a osit about the nature of the initial ensemble that in some appropriate sense the rerandomization posit is legitimate. This approach

In other cases, it proceeds by trying to show that even for finite times the reran omization wi e egitimate, at east wit respect to consequences drawn from it for some macrosco ic variables. How these rocedures work in detail we shall explore in Chapter 7.

the initial ensemble and of explaining why it holds when it does. In the atter .n 0 justi cation 0 reran omization, t ere wi e questions 0 several kinds. One series of uestions arises out of the fact that there are severe limits on the conditions under which the results hold, limits that

there remain issues that can only be resolved by once again facing the pro em 0 t e nature 0 t e onglna ensem e, Its Just cabon, an t e ex lanation of wh it is the wa it is. This is so because the results obtained from the dynamical evolution equations alone will not allow us to infer

Describing non-equilibrium

207

c equilibrium at a specific rate or in a specific manner when the system is characterized to a given degree of accuracy in certain chosen macroscopic terms. All of these issues will a ain be ex lored in detail in Cha ter .

m. A roaches to the derivation of kinetic behavior 1. The kinetic theory approach

The first scheme for dealing with non-equilibrium that we will outline

that are generally in free motion but that interact by means of an intermolecular potential. Here, the aim is to go beyond the dilute gas considered in the derivation of the Boltzmann e uation and to look for a general procedure by which the Boltzmann equation and its appropriate

deriving kinetic equations for the more difficult cases, but also of sorting out an rna ing exp icit just exact y w at reran omizing type statistica assum tions are needed in each case to derive the desired kinetic e uation result. This is the first stage of what needs to be done if one is to

A broadly applicable schematism is called the BBGKY approach, after

Bogo yu ov, Born, reen, Kir 0 0 , an Yvon, a 0 w om 0 ere s stematizations in this vein. Here the formal scheme is usuall accompanied by an informal rationale for it. This doesn't constitute a real

tion that gives all the information about the probabilities of a particle emg oun in a particu ar region 0 position an momentum space iven the distribution of all the remaining n - 1 particles. It is the complete description of the ensemble distribution, and giving this function

Physics and chance

208

In egra mg over remaInIng vana es, In a manner exac y ana ogous 0 the way in which one can obtain the total probability of a 1 coming up on one pair of dice by adding the probability of 1 on that die given a 1 other die, and so on.

c e u ion ivin e rate of change of Is will have on its right-hand side only components depending on Is and on Is+l' If we could solve these equations individuall we would then have a route to derivin a kin tic e ation. Why? Most thermodynamic quantities we wish to calculate in the usual 1

the gas in local regions, and tracking the evolution of It will allow us to track the evolution of the density distribution of the gas as it goes to e uilibrium. For calculatin transfer coefficients will enerall be enough.

Boltzmann's original formulation, his evolutionary equation governed f, the Jl-space distribution nction or the istribution 0 a single molecule in a as. In the ensemble transformation of the Boltzmann e uation it is It, the probability derived from the ensenlble of finding a molecule in a

Describing non-equilibrium ,

2

1

209 1

a ...'-" ...........

shall see in Sections 2 and 3, each of the approaches to describing nonequilibrium follows this general scheme of (1) finding a partial description to the ensemble that is ade uate to allow us to calculate the macrosco ic quantities we are interested in, and then (2) trying to formulate a closed

Bogolyubov's assumption is that we can distinguish three "time scales" in the evolution of the gas. The shortest is the time of the magnitude of the avera e time between collisions for the molecules. Here it is assumed that the f,'s can be arbitrary and can vary very rapidly. Next, there is the

are dealing with time scales large with respect to the collision time (but still small with regard to t e time sca e 0 e macroscopica y 0 servable chan e of the thermod namic features of the as) we can assume that .h, for example, is reducible to just It x It - that is, that the probability

closed it equation by expanding the solutions in terms of powers of the concentration 0 t e gas. ere ere les upon very success tec nlques from e uilibrium theo that derive the ideal as constitutive e uation in low-density limits and then derive modifications to it by treating the

increased. Making the appropriate expansionary assumption and going to e ute gas lmlt, one can convert t e c ose equation or 1 Into t e ensemble version of the Boltzmann equation. Alas, the derivation of the equation at this stage is hard to make rigorous,

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Physics and chance

hoped-for generalizations of the Boltzmann kinetic equation for gases denser than those for which the Boltzmann equation works. When one allows for exam Ie collisions in which three molecules become close in position and momentum space, many intractable correlation effects come

hope for a non-equilibrium expansion technique to parallel the very successful density expansion techniques applied in equilibrium theory turned out to be overl 0 timistic. From our point of view, the crucial step in the derivation is of course

It is the source of the closed kinetic equation and the origin of its timeasymmetric nature an the time-asYmmetry of its solutions. It is this step that cries out for ri orous understandin . Exact! when is it Ie itimate to

make such an assumption? And why is it legitimate when it is? 2. The master equation approach

T e master equation sc ema or ea ing wit non-equi i rium is app icable in situations where the s stem can be considered to be com osed of a large number of component systems that are coupled to one another o

1,

examples might be gases that are nearly ideal, simple harmonic oscillators coup e y a wea energetlc mteracbon or, a common examp e, ra labon confined in a cavi in which the different modes Cfre uencies) of radiation are coupled to one another weakly by a "dust mote" that weakly absorbs

the array of almost, but not quite, independent modes. Thus, for example, we mlg t start Wl ra latlon lstn ute over a range. 0 requencles m va in amounts in a cavi , and ask how, over time, the fre uen distribution changes as radiation is absorbed and emitted by the dust

a region of modes will be proportional to the spread of that region, and t at t ere Wl e a constan actor t a glves t e pro a 1 lty 0 energy from that one mode making the transition in a unit of time to any other region in question. A similar assumption is made about the probability of

Describing non-equilibrium

211

This sa s essentiall that the distribution of ener chan es over time in the following way: In an infinitesimal time, an amount of energy

designated mode. And an amount leaves the mode depending upon how occupied it is and the constant probability of a system in that designated mode transitin to each of the other available modes. Once again this is a kinetic equation that tells us that the change of the

is irrelevant in determining its future evolution. As usual, all of this is to be understood, 0 course, as a kinetic equation or t e ensemble, not for an individual s stem. The e uation has as usual the a ro riate timeaSYffiffietry built in and, again as usual and as is desired, the appropriate " the equation rests? Here again, a "picture" can be constructed that provi es an un erstan ing 0 t e nee e assumptions. But we must e careful in understandin what the icture rovides. It makes it clearer to us what the nature of our basic assumption is. It in no way demonstrates

initial ensemble. Nor does it tell us what initial ensembles will in fact give nse to ese resu ts. The icture resented is one in which the interaction of the s stems is turned on for short periods of time and off for short periods of time

begins. But the assumption that in the "off" periods the system goes to equ11 num a ows us 0 ran om1ze a eac new mteractlon peno over all possible initial conditions for the components in each mode. Here we use the usual equilibrium probability distribution for each mode

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Physics and chance

tions is clear. In both cases, irreversible approach to equilibrium shows itself by considering a function that reveals only part of the total ensemble robabili distribution. In the BBGKY a roach this was rather than 1m and in the present case it is the distribution function that restricts

a continuing rerandomization posit is used to derive our kinetic equation. In the BBGKY approach, it is the assumption that the It's can be construed as roducts of . In the master e uation a roach it is the assumption of a constant probability of transition from mode to mode.

3. The approach using coarse-graining and a Markov

process assumption A very general derivation of a time-asymmetric approach to equilibrium

of the method is matched, unfortunately, by the difficulties encountered in nnglng lt own to eart so as to rna e lt app lca e to e envatlon of a s ecific kinetic e uation in articular cases. But from the foundational point of view it provides the clearest understanding of what we are

Let us look at a situation that is describable in a generalized microcanonlca ensem e vern. system lS su Ject to certaln rnlUa constrarnts. An ensemble is constructed consistin of oints in r -s ace that re resent all possible micro-states of the system compatible with the initial con-

world. For example, the partition confining the gas to the left-hand side o an lnsu ate ox lS remove. ow oes t e ensem e 0 pOln s representing all possible initial micro-states evolve? We know, by Liouville's Theorem, that its total phase-space volume

Describing non-equilibrium

213

closely as time goes on? The suggestion is to divide the newly expanded phase-space into " oxes" subsets that are small relative to the total size of the now available phase space but that are still large enough to contain a substantial

measure, the fraction of ensemble points in a coarse-grained box will be proportional to the size of the box. This is because in the microcanonical case usin the standard measure the e uilibrium ensemble for the newly fixed constraints is uniform over the newly available phase

becomes more and more proportional to the size of the box. In this sense, it can "approac" e new equi i rium ensem e. As a means of re resentin the a roach to e uilibrium we observe for individual systems, the approach is rationalized by the familiar Gibbsian

is to locate the state as being in some region of phase space, say the regions were e ensity in some macroscopIC region 0 space is wit in some small variation from a s ecific value. An ensemble of s stems that had approached the equilibrium ensemble for the new constraint values

ensemble relative to the new constraints. This argument is one of some e lcacy, an we s a ave to return to It In etal In apter, when we discuss the rationalization of non-e uilibrium a roaches. For the moment we will let it ride.

is fully determined by the micro-dynamics governing the trajectory in p ase space 0 owe y any system starte at any pomt. at te s us that the distribution re resented b proportion of the ensemble occu n a coarse-grained box will, for a given coarse-graining, approach the size ?

, making a posit about the continual rerandomization of the ensemble. ocus attentIon on a SIng e coarse-gralne ox, ox t. maglne an ensemble of phase points distributed over i with the usual uniform microcanonical probability distribution. In a given time a certain proportion of

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Physics and chance

tional probability is a constant

.

.

asking what proportion ..wij that can be found .by. .,

observation moment. The probability that one of the given ensemble points will ... . . . .

,

,

of going from i to j multiplied by the fraction of box i filled by the ensemble

look at our original evolving ensemble. At a given time, a certain proportion 0 its points wi ave occuple ox i, 0 owing t e trajectories from the initial oints determined b the under! in micro-d namics. Assume that in the next time interval the same proportion of these points If

sumption, the so-called "Markovian" postulate, we can once again prove e ensem e OccupYlng a at t e nchon at escn es portlons 0 iven coarse- rained box will obe a kinetic e uation that has a uni ue stationary solution, the solution corresponding to a coarse-grained uni-

master equation of the previous section, an equation that equates the c ange In occupation 0 a coarse-gralne ox at a hme to a sum comosed of two factors: (1) the fixed transition robabilities from box to box acting on the proportion of systems in other boxes, giving the numbers

Describing non-equilibrium

215

pr p number of systems that leave the designated box. Another useful equation derivable is the Chapman-Kolmogorov equation. This tells us that we c n com ute the ro ortion of one box that arrives at another box at a later time by picking an intermediate time, taking the probability of transit

these boxes to the designated final box in the remaining time, and adding the results together. So that if we have the fixed transitional probabilities for a unit time we can calculate transitions for all multi les of that unit time (and we can then go to continuous time limits in familiar ways).

decreasing property. As usual, the assumption consists in the claim that "past history is irrelevant" in fi urin out where the ensemble will 0 next. The re ion of box i occupied at a time by points in the original ensemble will be a

hood that a point will be in box j at the later time because it got there rom ox i. How e points got into ox i, an w at e exact spec' cation of the sub-re ion of box i occu ied b the oints is are taken to be irrelevant. It is assumed that the same portion of points in the irregular

underlying deterministic evolution of the phase points.

program for the non-equilibrium theory semble feature and a kinetic equation that describes the time-asymmetric evo ution 0 at eature to Its equi I flum con Ition are not t e on y such a proaches. But the three techni ues we have surve ed are re resentative enough to do justice to the larger variety of techniques. In the

216

Physics and chance

explanation.

e and that display time-asymmetric behavior and approach to equilibrium depend only on this sub-component of the full ensemble distribution. A recise demonstration of this is called for however and is sometimes elusive - especially in the most general coarse-graining approach to the p

p

like to obtain, and we must ask which of these results are actually obtainable from the approach in question. We would like to show that in some sense the non-e uilibrium ensemble a roaches an e uilibrium ensemble. But the sense here is fraught with subteties. What exactly is

are characterized by particular finite times, the so-called relaxation times. Can these e erived rom the approach in question? The approach to e uilibrium described b the thermod namic laws is a monotonic a proach to equilibrium. Can monotonicity be derived, or can the approach

t

1

kinetic theory approach, that the transition probabilities from mode to mo e are constant In t e master equatlon approac ,an t at e ox to box transition robabilities are constant and derivable from the uniform distribution over a box in the coarse-graining approach. To what extent

Describing non-equilibrium

217

derivation obtainable by utilizing the structure of the dynamical laws themselves? To what extent must we also invoke some special initial ensemble distribution to 'usti the introduction of the osit? If we do need to introduce such a special initial ensemble state, how can we

to justify the rerandomization posit as to use the underlying dYnamics and perhaps an initial constraint upon ensembles, to be able to bypass the rerandomization osit and derive direct! the results obtained throu h its introduction? I

mechanics: How can we possibly reconcile the results obtained by any o ese met 0 s wi t eir seeming emonstration 0 a un amenta time-as mme in the behavior of s stems with the basic time-reversal invariance of the underlying dynamical laws? Do the procedures for

v. Further readings Some general remarks on non-equilibrium theory and its varieties are

enlightening manner. e aS1C e ements 0 t e lnetlc t eory approac t roug t e ~~'-'~l" ~ hierarchy are outlined in O. Penrose (1979), Section 3.2. Bogolyubov's original work is in Bogolyubov (1962). ]ancel (1969) covers several ap-

Physics and chance

218

.

~.

. .

..

/

.

-,

lne maSLer equadon approaCH goes oaCK LO l'aUl1 \..l>,L;O), p. JU. u. Penrose (1979) summarizes more recent work in Section 3.3. Jancel (1963) covers this approach in Part II, Chapter V, Section VI. Kreuzer (1981) . .1. in rt the 10 A very clear early version of the approach using coarse-graining and c

1 •

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r

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v au.

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10' Lne eXpIUr;..t problem of their consistency with the underlying dynamics. O. Penrose (1979) outlines this approach briefly in Section 3.4. Jance! (1969) covers this abstract and Qeneral aooroach in Part II Chaoter V Sections I-IV. U'

•

7

In this chapter we take up the problems outlined at the end of the last p

non-equilibrium statistical mechanics? How are we to show their consisten with the under! in d namics? Althou h most of the im ortant answers to these questions are covered in this chapter, one approach, large," is reserved for Chapter 8. plex, it might be useful to outline here the route we shall follow through the issues. Part I deals with two important preliminaries from experimental science and from the use of computers to model the evolution of of times in different contexts. Part II outlines the programs that have

discussed. Then the "mainstream" approach, which relies heavily on the " of the mainstream approach are critically examined. Lastly, in Part IV, the remaining un amenta pro ems are out ine . Here, teo ject is to survey the results obtained in Parts II and III once more, asking whether investigation began.

1. The spin-echo experiments A series of experiments dealing with the array of spinning, and therefore if not of great importance in the ongoing discovery process of physics,

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Physics and chance

e is 0 impo ance as a es origin of irreversibility. The nuclei are first aligned with their axes of rotation parallel, by radio-frequency pulse then flips all of the axes ninety degrees so that u p r. i p e x gin 0 ro a e in the plane perpendicular to the imposed magnetic field due to the phenomenon known as precession. Variations in the perfection of the c stal lead to local inhomo eneities in the internal rna netic field ex erienced by the different nuclei, however, so they precess at differing

formly relative to angular measure. They are still perpendicular to the imposed strong magnetic field, but they point in every direction in the lane e endicular to that field. At a time A t seconds after the nuclei had their axes pointing in the

the nuclei that had moved furthest in a given direction from the original common direction of pointing 0 all the nuclei are now furthest "behind" in the" recession race." The nuclei continue to recess at their va in rates. At a time 2At after the initial time, the nuclei now once more have

scopically as a pulse using standard nuclear resonance techniques. The experiment can e repeate 0 taining a series 0 ran omizing an derandomizin motions of the directions of the axes of rotation of the nuclei. Gradually the pulses that indicate reordering of the spin axes

spin interactions among the nuclei, and to diffusion, which allows nuclei to mlgrate aroun ln t e regl0ns 0 oca magnetic in omogeneity. When these ex eriments were first erformed, man commentators emphasized the fact that the nuclear spins behaved as isolated indivi-

ginal experiments the spin-spin interaction of the nuclei was one of the actors estroYlng t e repro UCl 1 lty 0 e macroscoplca y 0 serva e coherence. More recent! , however, s in-echo ex erlments have been performed where spin-spin interaction is taken into account. In effect,

Rationalizing non-equilibrium theory

221

of spinning nuclei whose spins are all lined up in the same direction in a plane

..

..

have the nuclei in question. In the second row, a period of time ~ t has elapsed. f e in ha ces ed t different rates around th im .r ' magnetic field, so that at this point the spins, although still in the same plane, now oint in all directions "at random." In the third row the s ins have been fli ed b a radio-fre uen ulse. The s ins furthest ahead in the recession race are now furthest behind. The result is shown in the last row. The time is now twice that ela sed from the first row to the second row. The s ins have now all caught up with one another, leading to them all once again pointing in the same direction. From the third row to the last row there is the appearance of an equilibrium condition (randomized spins) spontaneously evolving into a nonequilibrium condition (spins all aligned).

a high to a low entropy state even if kept isolated from the external enVIronment. uttlng t ese lstonca acts toge er, t e expenmenta lStS who erformed the modified version of the spin-echo experiment 'ust described speak of their process for reversing the dynamical evolution at " which the points representing an ensemble are confined to a small region o e sort not usua y etefffilna e y t e amI lar In s 0 macroscopIC control. In at least this special case it is as if we could prepare a gas in such a way that an ensemble of gases so prepared would initially be

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Physics and chance

r u pr vide an important test case for any foundational account of the origin of irreversibility.

2. Computer modeling of dynamical systems puters to model the dynamic evolution of complex systems. An early discovery made by this method dealt a blow to a simple but too naive h othesis about the ori in of randomization in com lex s stems. At one time it was thought that even a small amount of dynamic interaction

guarantee randomization and approach to equilibrium. An early computer experiment by E. Fermi and others modeled the interaction of a number of sim Ie harmonic oscillators cou led to one another b a small anharmonic term. It was expected that any energy distribution over the

over time showed a non-equilibrium and periodic behavior, energy shifting ac an ort among t e osci ators in a regu ar pattern. More recent ex eriments have been able to deal with s stems of a ve large number of components. Dynamical equations are modeled by

vals. Many randomly chosen initial conditions are tried, giving a pattern o ow t e evo utIon 0 t e system e aves qua ItatIve y gIven t e constitution of the s stem and its initial state. Care must be taken that an appearance of randomization is not introduced as an artifact through the

the ability of the computer's power to handle. But checks on this effect ent t at any appearance 0 can e pe orme ,an one can e co randomness in the end result reflects the d namics of the s stem and not the computational limits.

from an initial state crosses the sub-space, one notes the point of intersectIon, an t en pots on t e su -space a 0 t e su sequent mtersection points. A system following a simple closed tra'ectory will intersect one of the sub-spaces at a finite number of points each time, and so will

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pe .i. r closes up - will intersect the sub-space in a series of points all of which lie along a one-dimensional CUlVe. If enough intersections are plotted, ortrait be ins to look like a continuous one-dimensional line in the sub-space. A system that is chaotic in its trajectory will intersect the sub-

sub-set of it. If one continues long enough, the points of intersection of a single trajectory that is chaotic appear to fill a two-dimensional region in the sub-s ace. Such a diagram can provide only part of the insight we need. We still

appearance of a chaotic trajectory will not be enough to let us know what "kind" or "degree" of chaos is actually present. Without the use of com uters but sim I usin his brilliant anal .cal powers, Poincare was able to show that the phase portraits for even

more astonishingly complex. When one magnifies the regions of apparent sta i ity, one iscovers t at on a more magn e sca e It conSIsts 0 a attern of se arate stable re ions in a "sea" of instabili the attern on this scale duplicating the pattern first seen on the original scale, and so

interactions of the component systems

trajectories. For the simplest systems, regions of instability can be "trapped" etween regIons 0 sta Ity, m t e sense t at a trajectory starte at a point in one of the regions will wander chaotically over that region but will never get into the others. For systems of many degrees of freedom,

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riven y e ,in in e pe ur ing in erac ion will generally decrease the size of the stable regions, allowing the unstable portion of the phase portion to dominate, Again, for a given relative m r 'nc m by going to more and more degrees of freedom (more and more comup u e avin many degrees of freedom opens the opportunity for many resonances and many regions of resonance overlap. The a earance of these dominatin re ions of unstable tra'ectories is very suggestive for certain approaches to the grounding of the approach

tion of a single dynamical system that will be sufficient to establish certain kinds of "spreading" behavior for an ensemble of such systems, ll. Rationalizing three approaches to the kinetic

uation

• In the BBGKY a roach to non-e uilibrium the ensemble surro ate for Boltzmann's j-function - that is, It - was shown to obey the Boltzmann 2

1

I'

an expansion in terms of density and "throwaway" higher order ou rigor ue to e a e avior of hi her order terms in the densi ex ansion. The first ste the more crucial one, was a typical modern variant of the Posit of Molecular Chaos,

underlying micro-dynamical laws, A rigorous erivation 0 t e Bo tzmann equation vided b O. Lanford. It utilizes onl the underl in d namics Ius an assumption about the initial ensemble, an assumption therefore at least

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ieaie i i scale of size determined by the size of the box confining the gas. Next, the original proof was given for a "hard-sphere" gas - that is, no intern 'I m lecul s meet and then erfect non-inte enatrabili although the results were later extended to more realistic potentials. inii differing such initial conditions. The basic idea is to impose a probability measure on the phase space, One starts with a particular value of the one article distribution function of Boltzmann and chooses a robabili measure so that high probability is assigned to the set of phase points, tmO'

set of phase points at some later time, t - t ' , it will then be highly probable that the phase points into which the initial phase points have evolved determined b the exact d amical evolution are now "near It-/ where It=t' is the I into which It-o has evolved if the evolution of the like behavior" that he derives, and not, as i~ some understandings of the 'netic equation, e avior 0 t e ensem e "on e average." We s a uzzle over this in Cha ter 7 III 7, In one version of the theory the initial probability distribution is, intui1

that would introduce inevitable correlation. Here, the constraint on an initi ensem e is, interesting y, time-symmetric. An ensem e at is i e an ensemble that satisfies the constraint exce t that each s stem in the new ensemble is the "time reverse" of a system in the original ensemble,

of the initial constraint on ensembles does not hold at any time after t - 1 1t 0 S at t - . otlce at t e constra1nt 1S a constra1nt on t e initial ensemble onl , and is, hence, not sub'ect itself to the threat of inconsistency with the dYnamic equations.

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ini i ensem e, u e eorem is more genera in s owing a any probability measure that appropriately concentrates points of the initial ensemble sufficiently tightly around those that generate the right h=o will

,

i .. 1 P P Yn is constraint is "weaker" than the one described, but more complicated to formulate. The paradox noted here is avoided by the fact that this constraint is not time-s mmetric. That is if an ensemble satisfies the constr .nt the ensemble of time-reversed systems generally will not satisfy the

of the ensemble from its special initial state only for a very short time period, approximately one-fifth of the time needed, on average, for molecules to transit between collisions! As Lanford sa s "the theorem to be stated says only that the Boltzmann equation holds for times no larger

restriction functions in the theorem indicates that the theorem can be extended, or at least that the result should remain true for arbitrarily large times. He also notes however that the theorem relies cruciall on the fact that the limit of zero density has been taken for a gas, and he is

approach is of dubious rigor - that is, because of the bad behavior of t e ig er or er terms in t e expansion. We shall return in several laces to the issue of the foundational role that these results can justifiably play. It will be useful, though, to make

that the ensembles that are such that a high probability is assigned to Inlt1a states were t e appropnate to a state IS near peo are suc at in both future and ast it is hi hI robable (within the limits of the bounds imposed by the theorem) that the phase points will be such that

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to equilibrium. Of course, we can avoid this apparent paradox by using the version of the theorem in which the initial constraint is not time-s mmetric so that an ensemble that satisfies it is not matched by a time-reversed en-

constraint will then make anti-Boltzmannian behavior highly probable. The question will be, why assume one 0 these probability measures is the a ro riate one to use in describin the world? Wh is it a ro riate to assume that initial states showing Boltzmannian behavior are highly

by its success in the world, the question is rather why does that measure that is correct wor as we as it oes w ereas its opposite ai s entire y. The roblem here is the familiar one of 'usti in a choice of a robability distribution over a set of systems satisfying some macroscopic

that there was a unique, absolutely continuous measure that remained invariant in time. Here, at is enie us ecause we are ea ing wit t e non-e uilibrium situation. We can of course "transcendentall " 'ust' one of the non-time-symmetric measures as opposed to its mirror image

doesn't fix uniquely one such probability measure as is the case (modulo sets 0 measure zero In t e equ 1 num sltuatlon. ore lffiportant y, 'ust' .n the choice of a measure in this wa oes no distance at all to offering an explanation as to why such measures work. In the case

characterize the initial states of systems, not their final states. But, once agaln, at al s to provl e an exp anatlon as to w y t IS proce ure works. The Lanford results certainly go at least some way in answering some

gas, in the limit sense of zero density, and the small size of the molecules, an an mlla constraln upon an ensem e 0 enve, Wl In severe Ime limits, the conclusion that evolution of the ensemble will be such that it will be highly probable that systems will behave in a Boltzmannian way.

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equation given by these results may run into a curious conflict with alternative justificatory schemes, in particular with one that we shall discuss in the coarse- rained a roach that is called mixin . We shall return to this point in Chapter 7,111,7. 2. The generalized master equation

From the liouville equation to the generalized master equation. The aim of the eneralized master e uation a roach is to transform the exact equation governing the dynamical evolution of an ensemble - the

a way that one can see transparently what conditions must be met for a master equation to 0 . The rocedure starts with the derivation of the Liouville e uation, Here, one simply writes a formal equation, which on the left denotes the

that governs the evolutions of the individual systems, such that L acting on p generates t e time evo ution 0 p, The next ste is to decom ose into two arts, It is this rocedure that generates the reduced description - that is, the function giving only

equation type, or at least to be intimately associated with the reduced escnptlon t at oes so 0 ey t e master equatlon. e tnc , ongma y due to S. Naka'ima and R. Zwanzi , is to introduce the notion of a projection, This is an operator that takes the full distribution function

equation that gives the dynamics of the probability distribution over one vana e, en t e prOjectIon wor s y averagIng t e IStrl ution unction over all the remaining "hidden variables" on the phase-space surfaces with fixed values for the variable we are interested in. Thus, if we

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the value of p over all the possible values of position and momentum for the oscillators consistent with that energy distribution. The values of Pp are sufficient to allow us to determine the robabili distribution over the designated variable, and the evolution of Pp is sufficient if we know

One can then decompose the Liouville equation into two coupled dynamical equations, one or P and one or Q. One can ormally solve these two e uations. Each solution contains two terms one de endin upon the initial ensemble distribution and one depending on an integral

One usually works by assuming that one starts at time zero with the uncoup e systems eac in equi ibrium, an t at t e coup ing perturbation is introduced at time zero. For exam Ie radiation in a cavi has been allowed to reach equilibrium. Then the dust mote that allows

The first term on the right-hand side of the generalized master equahon 1S 1 e t e master equation 1n t at 1t 1nvo ves 0 y testate 0 t e ensemble at time t. But the second term contains an inte ral over all states of the ensemble from time 0 to time t, so that the full information

the history of its evolution. In this way the generalized master equation 1S not ar OV1an, as 1S t e master equation or as are t e ot er 1netIc equations we wish to rationalize.

of the generalized master equation is in fact identically zero, leaving the pro ema 1C memory erm on e ng - an Sl e. e a1m 1S 0 S ow that in appropriate circumstances this term can be converted into one that refers only to the state of the ensemble at a given time, and not to

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is

a rich body

diagrams for the ensemble. The procedure becomes extraordinarily complex for systems with realistic interactions, but various posits and moves to the thermod namic limit combine to show that under certain appropriate circumstances, one can once again give long-time approxi-

that refers only to the state of the ensemble at the single time from which its evolution is being continued. It is clear that the osits needed to allow us to demonstrate the master equation solutions to be good approximations to the solutions of the

earlier than the initial time, we would be able to derive the conclusion t at t e anti-t ermo ynamlc ana ogue 0 t e master equation suita y a roximates the eneralized master e uation for ast times a conclusion we certainly don't wish to derive.

tion. Those systems that fail to show approach to equilibrium - such as t e coup e OSCl ators 0 t e erml computer expenment, or systems that satis the constraints necessa for eneral stabili theorems like the KAM Theorem to apply - will not have their generalized master

master equation is usable and when it is inappropriate, the technical 1 cu ties are severe. Sub-dynamics. An interesting variation on the generalized master

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Liouville dynamical equation and a Markovian kinetic equation that is not time-symmetric. What conditions ou ht to be im osed when we look for a com onent of the full distribution function that is to obey a kinetic equation? (1)

determined by the operators P and Q of the previous sections, for in that case it is Pp that obeys a kinetic equation. (2) The operator that selects the kinetic com onent when a lied to the kinetic com onent should leave it unchanged, because that component already obeys the condi-

by the ensemble. That is, as time goes on, the ensemble changes and so does its kinetic component, ut t e operation t at se ects the kinetic com onent out of the full distribution function ou ht itself to not de end upon time. (4) The component selected out by the operator should obey

A very interesting result claimed is that conditions (1) and (3) imply con 'tions 2 an at is, i t ere is a time-invariant way 0 ecomosin the distribution function that reduces to the P ro'ection in the limit of zero coupling, then the component picked out by that decom-

according to the general time-symmetric kinetic equation and a timeasymmetnc master equation. The formal roblem then becomes twofold. First, one must find eneral conditions on the nature of the system (of its Hamiltonian or Liouville

obey the formal condition derived. a su - ynamlc component 0 t e IStrl utlon unction can e sown to exist, there are still a number of interesting conceptual problems to be explored. One is the question of whether we can show that the sub-

whose time-asymmetric evolution we wish to derive from our statistical mec anlca mo e. 0 er IS t e ac a suc a su - ynamlcs eXlS s, there will be another component of the full distribution function, also projectible from it by an operator that is time-invariant and that reduces

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In the last chapter we examined a general schematism for describing the a roach to e uilibrium that worked b dividin the hase s ace u into a large number of "boxes," each small relative to the total size of the

for calculating macroscopic quantities, and that is to be shown to obey a time-asymmetric kinetic equation despite the time-symmetry of the e uation ovemin the evolution of the ensemble descri tion as a whole was the description that told us at any time what portion of a box was

abilities could be calculated by assuming the first designated box filled wit p ase points an ca cu ating w at proportion 0 t em wou ,un er the d namic evolution move into the second desi nated box over the time interval. The argument presupposed that the same proportion of

box were filled. In this way the past history of the evolution of the ensem e cou e Ignore in t e manner c aractenstic 0 lnetic equations. Here we shall investi ate what features of the d namics could ossibly be expected to lead to an evolution correctly described by making

constitution of a system and the underlying dynamical laws. We shall see, not surpnSlng y, t at un amenta Issues 0 t e nature 0 lnlti ensembles will also, at some sta e, arise and re uire examination. And we shall once again reflect upon the difficulties that arise when one tries

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scopic thermodynamic features of systems. namical instabili . What intuitivel must the tra'ectories of s sterns look like in order that ensemble evolution be ergodic? Ergodicity is

available phase space. This amounts to a strong restriction on the partitioning of the phase space into pieces that remain closed under evolution. An such ro er iece of the hase s ace must be of measure zero or measure one. So ergodicity is the insistence that each trajectory Cex" Is that enough, intuitively, to obtain a coarse-grained description of an evolution that matches our idea of approach to equilibrium from a none uilibrium condition? No it is not. Er odic behavior is com atible with an initial ensemble that remains "coherent" as the individual points in it

a kind of regular periodicity. It is true that in certain senses discussed in C apter 5 it wi e 'g y pro a e t at at a ran om time t e systems will be found near to e uilibrium. But this is com atible with a hi h probability that even into the distant future, if we select times appropri-

proportion of the entire phase space originally taken up by the initially e regIon. To get this result, what we will need, again intuitively, is that small regions of the phase space contain points arbitrarily near each other but

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whose trajectories, like

.

.

.

t~

and

t~,

diverge from t at an exponential rate in the

order, be found to be higWy irregular in shape and dispersed throughout the entire newl available hase-s ace. So what we ex ect to suffice to demonstrate approach to equilibrium in our coarse-grained sense is a

for this new kind of evolutionary "chaos" of the ensemble, the role played ear ier by metric indecomposab' ity in demonstrating ergodicity. A articularl im ortant formal condition of d namical instabili is for a system to be a "C system" (sometimes called a U system, Y system, or

two sub-spaces of the whole phase space. In one of these sub-spaces, trajectories ar itrari y c ose to t e given trajectory iverge away rom it ex onentiall fast in the future time direction. In the other sub-s ace the arbitrarily-near trajectories all diverge away from the given trajectory

us to derive spreading results for initially small and regularly shaped ensem es. See Figure 7-2. There are then three roblems. One is to ive formal definitions of the notion of ensembles spreading out over the available phase space in

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mechanics in any simple-minded way. This can be seen by reflecting once again on the fact that with classical systems we can have an ensemble whose members are finite s stems and that is not almost eriodic in its behavior, even if the individual systems are, by Poincare's result, almost

, systems are, even if it contains an infinite number of member systems. This w' I certainly bock any hope of proving a forma monotonic spreadin -out result of the kind we are lookin for. Of course one can then move to the thermodynamic limit of a system with an infinite number of

the classical case, is a non-trivial mathematical exercise. Varieties of chaotic evolution. As it turns out there is a hierarch of notions that correspond to our intuitive notions of an ensemble

cases, the concepts can be ranked in order of their strength, one category eing stronger t an anot er i meeting Its con ition lffip 1es meeting t e other condition but if there are s stems that meet the weaker condition but not the stronger. In some cases, it isn't yet known if two concepts

Fix attention upon a single coarse-grained box in a coarse-graining of the p ase space. serve t e proportIon 0 at ox e at any tIme y oints that were initiall confined to the re ion of the initial ensemble. Starting at any time, look at the infinite time average of that proportion

goes to infinity, becomes equal to the proportion of the allowed phase space occup1e y t e set t at c aractenzes t e 1rutla ensem e, t e s stem is weak mixing. Of course, all these references to "proportion of the coarse-grained box" and "proportion of the available phase space

space. One can show that weak mixing is stronger than ergodicity. prova y stronger con 1Ion s IS a 0 mlX1ng. lX1ng IS 0 wea mixing much as the strong law of large numbers is to the weak law. Once again, we give a coarse-graining and focus attention on a specific

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Fi - . A mixin ensemble. A and B are two non-zero sized re ions of points in the phase space r. B is kept constant. We track the evolution of systems whose initial micro-states are in the A re ion. The result is a series of T A re ions as time oes on. For a s stem to be mixin the A re ion evolves into a T(A) region "in the long run" (that is, in the limit as time goes to infinity) that is evenl distributed over the hase s ace in the coarse- rained sense. For this to be the case, it must be that in the infinite time limit, the proportion of any B region occupied by points that evolved from the A region is equal to the proportion of the original phase space originally occupied by points in the A region.

limit as time goes to infinity, this proportion approaches the relative proportion 0 t e avai a e p ase space occupie y t e origina region, then the s stem is mixin . What this sa s is that if we focus on a fixed measurable region of the phase space and look at the regions into which

lS t e set lnto w lC lS trans orme y ynamlc evo utlon at time Tand is the standard measure. (See Fi ure 7-3.) It can be shown that mixing is stricdy stronger than weak mixing, but

phase points move along their trajectories. o VlOUS genera lzatlon 0 m1Xlng lS to COnsl er m1Xlng to e n order for each finite n. For exam Ie, second-order mixin would be defined by stating that for any three measurable sets A, B,

c:

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mixing, but not mixing to every finite order. A very powerful and interesting condition of randomization for ensemble evolution is for a s stem to be a K s stem. The formal definition is a bit abstract. Very crudely, a system is a K system if there is a coarse" the coarse-graining at a time. Look at the sets generated by what happens to its boxes at a later time. Ta e a the sets one can generate by includin intersections of the ori inal sets with the new sets. Kee the process up into the infinite time limit. In the limit, one gets by this

A consequence of being a K system, one that is actually equivalent to its orma e nition, is more intuitive. Consi er a coarse-graining 0 t e hase s ace P (for" artition of the hase s ace"). Consider observations made from past time infinity to the present at fixed finite time intervals.

sense, note, for the fact that given a coarse-grained history the probabilIty IS one t at t e trajectory WI next appear In a speC! c ox 0 t e artition is com atible with a set of measure zero of such tra'ectories having that coarse-grained history but appearing with its members next

own - that is, independently of any information about the past. It can be s own a e sys ems are ose sys ems a are suc a or every partition and every time interval for measurement, the dynamic evolution is such that the resulting process has the 0-1 property. In this sense,

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s

e p ase space 0 e sys em 1S coarse-grame , discrete time intervals, the trajectory of a system is looked at. At time 0, the

.

.

.

.

0,

-1,

'

each of an infinite number of past times, which box of the coarse-graining the

..

...

, found in some specific box - say box i-at the next observation? If the system

.

, ....

°to begin with,

.

systems mixing to every finite order that are not K systems. At one tune 1t was e 1eve t at e1ng a K system was as ran om1z1ng as an ensemble evolution could be, but this is now known to be false. A strictly stronger randomizing character for a dynamical evolution is for

sequence of totally probabilistically independent trials, This is so despite t e act t at 1n t e m1cro-sense, eac trajectory 1S u y etefffilne y 1tS initial oint. A basic definition of a Bernoulli system is this. If the dynamical evo-

at fixed time intervals from the infinite past to the infinite future, For each trajectory t ere 1S t en a ou y 1 n1 e sequence 0 num ers t a 1n 1cates which box the tra'ectory was in at a specific time. There will then be a probability distribution over those sequences that determines in the

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over it will be "isomorphic" to the sequences and probability distribution over them that we would get by (1) tossing a die, each of whose tosses was totall inde endent of the others and 2 that had as man faces as there are boxes in the partition, with (3) a specific, fixed, probability on

past to the infinite future. To see how strong a randomizing condition this is, we should look at the notion of a eneratin artition. This is a artition that is sufficient! fine that the doubly infinite sequence of numbers that tells which box of

zero. In other words, the sequence of coarse-grained observations (doubly infinite to be sure w' istinguis one trajectory wit that sequence rom almost all others. If the s stem is Bernoulli then the robabili that a trajectory will be in a box of even such a generating partition at a given

there is to know about the past coarse-grained history of the trajectory, even re ative to a partition ne enoug to e generating, wi not a ow us to "im rove our odds" in bettin on which box the tra'ecto will be in at the next measurement time. One could hardly ask for more

It can be proven, with difficulty, that the Bernoulli condition is actually stronger an e system con 1t10n, t at 1S t at t ere are systems t at are not Bernoulli. It is eas to show that eve s stem that is Bernoulli meets all the other chaos conditions as well.

that satisfies an additional mathematical constraint that is usually fulfilled, w e a sys em, an ence mlXlng 1n a n1 e or ers, m1X1ng, wea mixing, and ergodic. Next, one shows that certain idealized geometric flows are C systems.

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Physics and chance

v rya u u e 0 indeed, that certain idealized physical systems are C systems as well. Two or more hard spheres in a rectangular box, a single particle in a box with own fixed convex sides and other s stems as well c n systems. With even more effort it is possible to go beyond this and show

that display instability in their evolutions, the feature of having such instability is itself "structurally stable." What this means is that a system havin the features necessa to enerate the kind of tra'ecto instabili we have been discussing is itself insensitive to infinitesimal changes

bility property as soon as the smallest change is made in some parameter characterizing the structure of the system. Rather, a range of system arameters will all lead to the s stem bein a C-s stem and havin the needed trajectory instability.

stract dynamical theory we have just seen are dramatic and important. Just ow t ey are to e app ie to give us t e answers to t e oun ationa uestions in statistical mechanics that we are seekin is however far from clear. These are issues to which we shall return in later parts of this

before moving on to deal with a few other important technical and orma resu ts t at must a so e put In pace e ore getting to t e extended hiloso hical discussion of these issues. The uestions itemized here differ markedly in the depth to which one must go in answering

we would like to obtain, at least in the ensemble sense. Can this difficulty e overcome. (2) In cases where finite time results are obtainable, a different problem arises. When we have shown a system to be a Bernoulli system,

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which such partition? Will its "boxes" correspond in any way to the kinds of macroscopic measurements we can perform? The results obtainable from d amics alone are uite abstract onI uaranteein that some such partition exists, and leaving us in the dark about just what the

to tie the results obtained here to experimental practice and, hence, to what we can macroscopically measure and to what features of systems we believe on the basis of ex erience show thermod namic characteristics. The power of these methods is a result of their great abstractness,

(4) We should notice that these results are all infected with the "except for a set 0 measure zero" pro em encountere in equ' i rium statistica mechanics and its ·ustification. The results on moon for exam Ie tell us that the size of the overlap region of T(A) and B goes to the product of

system is, perforce, ergodic, we know that the only such invariant measure at is a so ute y contInuous Wlt t e stan ar mlcro-canonica measure is the micro-canonical measure. But "absolutel continuous" means "assigns measure (probability) zero to sets assigned measure zero " of "situations with probability zero in the standard measure can be 19nore POSlt at requlres senous cntica examlnatIon. (5) The results discussed here are inde endent of the size of the systems constituting the ensemble. A system of two hard spheres in a

about limits when time goes to the infinite future in the proofs of weakan strong-moong, are resu s a ou e lml s as lme goes lnto e infinite past. Parallel to the results about lack of probabilistic determinism from past coarse-grained history for K systems are results about lack of "

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Physics and chance

ou is pas y x c for Bernoulli systems is an independence result about the probability of a system's being in a box now from conditionalizing on the future box ,.,.ro·...•....... ,.,t-i n f . to . H w e w n symmetric results to obtain the time-asymmetric results we seek for

ant question of all: What role is to be played in the justification of nonequilibrium statistical mechanics by the initial ensembles from which ensembles evolve? The results 'ust discussed make no reference to the nature of this initial ensemble. Intuitively, then, they cannot by them-

about the initial ensemble to distinguish the results that help to justify statistical mechanics from their useless time-symmetric analogue. Even roblem it for ettin about the roblem of resolvin the time-s mme seems, intuitively, that if the initial ensemble is chosen oddly enough it

How must initial ensembles be restricted to avoid this problem? What physica reason just' es the imposition 0 such a restriction? (8) Finall we must at some oint ex lore the limits of the results discussed. We know that stability results of the KAM type will provide

tally that many complex systems do fail to show the monotonic approach to equi i rium pre icte in statistica mec anics. at are t e imits to randomizin behavior? And on which side of those limits do realistic systems (with, say, realistic intermolecular potentials instead of the sin-

4. Representations obtained by non-unitary transfonnations ue to

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at least the coarse-grained sense. In operator theory, unitary operators playa role that generalizes the notion of a chan e of coordinates b rotation for ordina vectors. A set of quantities transformed in common by some unitary operator essen-

mechanics we are dealing with here treats the probability distribution as a generalized vector to be operated on by operators. Next a eneralization of the notion of a unita 0 erator is introduced the so-called star-unitary operators. A probability distribution transformed

still normalized (total probability adds up to one) and it is still nonnegative everywhere - t at is, no "pro abi ities ess t an zero" are introduced. Furthermore this new 0 erator has an inverse - that is it associates with each of the original probability distribution functions, p, a unique +

+

The importance of the new operator, A+, and the new probability istri ution unction, p = p, is t at p wi , . t e origina ynamics is at least as randomizin as to be a K s stem obe a new d namical equation of evolution p+(t) = ~p+(O). This equation has the form of a

, new distribution S = -Jp+lnp+ dJl, and one can show that this quantity sows monotonIC Increase Into t e uture un er e ynam1c evo utlon. It mi ht come as a su rise that such a transformation is ossible. The new probability distribution seems to contain all of the information present

entropy constant, by Liouville's Theorem, the new representation seems to s ow a ne-gra1ne entrop1c asymmetry In tlme. ne can un erstan better what is oing on here if one remembers the idea of a coarsegraining that is generating. The doubly infinite series of the box occupation

description in terms of the generating partition is equivalent to the original escnptlon In erms 0 a con 1nuous pro a 11 IS n u Ion. e a system can show asymmetry in time in terms of a coarse-grained entropy defined by the new partition even though its original probability distri-

244

Physics and chance

distribution from which the original one can be reconstructed, and that represents the randomization of the original by obeying a strictly Markovian kinetic e uation. The new re resentation reveals the s f ding of the initial ensemble in terms of its own fine-grained entropy.

operator A-. This operator transforms p into a new representation p-, and this new representation can be shown to obey its own kinetic equation. If one defines the fine- rained entro corres ondin to this new re resentation, one discovers that it obeys an "Anti-HTheorem" - that is, the

into the problem of initial ensembles, and we shall later critically examine how the originators of this technique use special initial ensembles to "break the time-s mmet "we outline in Cha ter 7 III 6.

(and, hence, randomizing) kinds of motion can exist. This is true at the macroscoplC eve as we. anetary motlon IS sta e. 0 IS t emotion 0 fluids at sufficientl low velocities. But fluid motion that is stable at low velocities can show a marked and very sharp transition to instability as

The transition occurs at a certain critical velocity determined by the nature e plpe. ter t e translt1on, t e traJeco e Ul an t e geometry 0 tory of even a very small piece of the fluid "breaks up" so that the iece in a very short time has its parts widely and chaotically separated from ?

Rationalizing non-equilibrium theory

245

were found that were very suggestive. They were simple equations containing a parameter. At one value of the parameter, a stable periodic solution of the e uations existed. As the arameter had its value varied at a certain point, the stable solution became unstable and a new stable

appeared. And so on. Interestingly, the amount of variation of the parameter necessary to induce the transition decreased at each stage by a constant factor "scalin " . The set of transition oints of arameter values approached a limit point. Once the parameter reached this limit point, no

More directly related to the problem of turbulence was the discovery of "strange attractors." I a ynamic system can issipate energy to the outside - sa throu h friction or radiative loss - then all the tra'ectories from a region may converge on a limited, self-contained sub-set of the

a ball rolling in a bowl with a central lowest point. The ball is started at any point in t e ow an oses its tota energy t roug riction wit e surface of the bowl each ball comin eventuall to rest at the sin Ie lowest point.

attractor can exist in the sense of a steady-state trajectory to which all traJectones In a regIon eventua y converge. en no matter ow e s stem is started, eventuall a state is reached where the s stems all follow along the same one-dimensional trajectory in phase space.

along the trajectory, no matter how small. Then there exist points in that regIon at WI Iverge rom one anot er a ong e trajectory very qUIC y as time oes on. So that althou h the s stems all conver e to motion along this single trajectory, the trajectory itself has a wildly unstable

the points of intersection form "wild" sets and two systems that pass t roug t e IntersectIon space at pOInts c ose toget er In t e IntersectIon set will find their future intersections far apart. Whether or not such "period doubling" instability transitions and tran-

246

Physics and chance ID. Interpretations of irreversibllity

With the resources provided by the last two sections, we can now proand nature of time-asymmetry in the world. I will begin with three "non- ymm ry in P ysica

facts about genuinely isolated systems governed by time-symmetric micro-dynamical laws. One major approach - that which invokes cosmolo as a art of the ex lanation of time-as et - will be reserved until the next chapter.

1. Time-asymmetric dynamical laws From the earliest days of reflection upon the paradoxes of irreversibility, the su estion has been forthcomin that the source of time-as e is to be found in some underlYing asymmetry of the fundamental dy-

Sometimes the source is sought in some specific body of laws of interaction - or examp e, e ectro Ynamics, aws that are alleged to be timeas metric. Sometimes it is su ested that beneath the com ositional structure with which our standard theories deal lies some additional micro-

set of laws is time-asymmetric is itself not a completely straightforward matter, as interesting issues in e nature 0 c aracterization 0 states p ay a role. (2) Even if a set of time-as mmetric fundamental laws is found that is not by itself enough to solve our foundational problem in statistical

with which we are familiar from phenomenological thermodYnamics and wit w ic statistica mec anics ea s. To see the oint of (1) we need onI reflect on a familiar exam Ie used to show that an apparent asymmetry of a physical law is not really

appears that the familiar "right-hand rule" for determining the direction In w IC t e WIre WI move sows us at e appropnate mIrror lffiage of the ex eriment is a situation that cannot, as a matter of h sical law, occur. But an appropriate understanding of the nature of the bar magnet

Rationalizing non-equilibrium theory

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reflection of the situation in the mirror xy, as in Fig. a2, seems to show a situation v' ~J( ~red in For in the situation oictured, the wire would suffer forces moving it out of the plane. But Figs. bl and b2 show the fallacy. For the oermanent masmet is szenerated bv internal current looos as illustrated in Fisz. bI. The reflected situation, as in Fig. b2. has the current loop reversed. and with it the polarity of the permanent magnet. So the mirror image of the situation in Fig. bl is that of Fig. b2, once again the situation predicted by the electromagnetic laws. H

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" reflection" for electromagnetism becomes an illusion. In any alleged demonstration ot the time-asymmetry ot an underlYlng law, precaution must be taken to assure one's self that one is not being similarly misled. (See Figure 7-5.) a.1U

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a system started in state T(SCt)) , the "time reversed" final state of the ongmal system, Will, governed by the same dynamlcal laws, evolve In time t to a state, T(S(O)), the time reverse of the initial state of the original system. But what is the time reverse of a given state, sct)? In-

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248

Physics and chance

xis ence of "hidden' motions (velocities or spins of sub-micro-components, for example) can mislead us into what appropriate T(SCt)) for a given S t is the one we ou ht to choose. To see the importance of (2) we need only reflect upon the one

Experiment seems to show this interaction to be slightly non-timesymmetric. The experiment actually tests the non-symmetry of the interaction under the combined 0 erations of mirror ima in and re lacement of particles by their anti-particles. The inference to non-symmetry under

system must be invariant under the combined operations of exchange of anti-particle for particle, mirror imaging, and time reversal. So if it isn't invariant under the combined 0 eration of the first two it cannot be invariant under the third either.

of thermodYnamic irreversibility. An initia y more promising attempt to oun e t ermo ynamlc lrreversibili on irreversible fundamental laws relies u on a ve eneral feature of wave generation and propagation as we experience it in the

to a stone at the bottom of the pond in such a way as to propel it out o t e water. Iml ar y, we see acce erate c arge partlc es generatmg out oin waves. These waves ca ener and "dam "the acceleration of the particle. We don't see electromagnetic waves converging on charged

The nature and source of this asymmetry was the topic of an extremely Interestlng e ate In etween msteln an tz. tz too e position that the existence of "outgoing" radiation and the absence of "incoming" radiation of the appropriately sYmmetrical kinds revealed an

Rationalizing non-equilibrium theory

249

of any fundamental time-asymmetry of the fundamental laws. From the Einstein point of view, the absence of the appropriate kind of symmetric conver in radiation was not a fact ordained b the laws of electromagnetism, but rather the absence of a situation permitted by the laws

ecules in a box, permitted by the mechanical laws of molecular interaction but never (or with great rarity, as predicted by the statistical theory) observed. A full understanding of electromagnetism, and of the difficult problem

dynamics, questions remain. Indeed, there is even a current of thought t at suggests t at t e now-o 0 ox approac 0 treating t e electrorna netic field as a constituent of the world is a mistake su estin that the field theoretic approach be replaced by a subtle and ingenious "action-

" magnetism are symmetric in the time-reversal sense. The origin of the wave asymmetry note ere, t en, is to e soug t in t e irection sugested b Einstein as a constraint u on initial conditions. From this point of view, the natural assumption for the behavior of

Over an infinitely extended time span, one would find the system almost a 0 t e hme ln a con lhon c ose to equll flum, Wlt energy elng exchan ed "randoml " between radiation field and char ed articles. Infrequently, one would find conditions far from equilibrium, say with

on charged particles and succeeded by situations in which the energy once more lsslpate lnto t e ra lahon e y means 0 outgolng radiation from the now accelerated articles. The reater the diver ence of the situation from equilibrium, the less often it would occur. The

once more the question of the origin of the actual asymmetric situation we n ln t e wor . Under certain circumstances, one can give a nice formal rendering of the condition that must be the case in order that we experience, as we

250

Physics and chance

correlation among radiation fields at spatially separated points is found only when such correlation can be traced back to a "common cause" in the radiation havin its common source at a oint char e. Such condition parallels the kind of condition we might impose on the correlation

versals. But what we would like is an understanding of why such a condition correctly describes the world of our experience. One can ima ine a ro ram to t and account for all the irreversibili in the world in terms of radiation irreversibility. Such an account would

irreversibility in terms of the electromagnetic nature of interaction among molecules. Alternatively, and this is the more common suggestion. one could t to offer an account of radiation irreversibili in terms of the asymmetry of the molecular behavior of the radiating and absorbing

versibility. There we shall look briefly at the problem of the radiation asymmetry in t e context were out oun ra iation can "go 0 to i ni "instead of bein absorbed b "nearb" char es. We shall see one ingenious account to explain radiation irreversibility in this cosmological ." Although the possibility of finding the origin of asymmetry in the timee un er yIng aws is y no means a c ose reversa non-invanance 0 issue I shall assume, alon with the orthodox ma'ori that that route is not the one to pursue. So we will continue our exploration of the search

2. Interventionist approac es From quite early days in the discussion of the foundations of the statistlca t eory, resort was rna e to t e act t at t e actua systems 0 lnterest with which we deal in the world are not, as the are in our microcanonical idealizations, genuinely energetically isolated from the exter-

Rationalizing non-equilibrium theory

251

(as they can be well shielded for in practice), there is no possibility whatever of shielding the system from its gravitational interaction with the outside world. We ought then, according to this stance, to treat systems not as genu-

intervention? First, we might use it to explain away the "paradox" of the conservation through time of fine-grained entropy. If the system is truly isolated the deterministic nature of the evolutiona d namics assures us that no "information" about the initial molecular state is ever truly lost. 1

probability for finding a molecule on the right-hand side of the box. After the partition is remove , 1 eventua Y takes on a form representing a uniform robabili for findin the molecule an here in the box. But the initial restriction on molecular position is now contained in the com-

that if the ensemble were time-reversed it would revert to its initial form representative 0 t e origina y co ne gas. But if we allow for even small intervention into the motion of the molecules by effects from the outside, the sensitivity of the correlational

the external environment and break the conservation of fine-grained entropy. Here, one usua y re 1es upon t e acts a out t e 1nsta 11ty 0 tra' ectories used to derive mixin e results to ar ue that the initiall macroscopically determinable information about the system does in fact

make the ensemble representing, say, gas that was originally in the leftan Sl e 0 t e ox, an t at representing gas t at was 1n t e ng t- an side, indistin ishable from one another. But it is not simply the non-conservation of fine-grained entropy for

we started with an ensemble of systems initially coherent in the sense of representing a constra1nt on t e mem ers 0 t e ensem e at as een removed, the continual causal intervention of the outside world would so modify the systems at a later time that no Loschmidt-imagined reversal

252

Physics and chance

spread-out ensemble into a genuine equilibrium ensemble, an ensemble spread out over the now available phase space in a genuinely fine-grained wa . Finally, we might like to go further and derive the details of the ap-

ment in the form of some randomized "forcing" superimposed on the otherwise deterministically fixed evolutionary motion of the ensemble, and derive from these two com onents an evolution of calculated from the now forced evolution of the ensemble, a behavior consonant with a

ensemble, as is done in the Lanford derivation, for example, but is suppose to app y genera y to any initia ensem e subject to the constraints im osed b the macrosco icall construed initial nonequilibrium situation.

appropriate. The time taken by a system to reach internal equilibrium, or examp e, . t e system is reasona y we insu ate rom its surroundin s is far shorter than the time taken for the s stem to come to equilibrium with the external environment with which it is so weakly

intervention will "destroy the memory of the initial state" of the system - t at 1S, flve 1tS ensem e representahon to t e equ11 flum ensem e - in a much shorter time frame than it would take for the s stem to come to equilibrium with the external perturbing environment.

tion of the test system, with all of the effects we normally subsume under t e not10n 0 en rop1C 1ncrease. econ, can t e 1nvocat10n 0 OUtS1 e intervention resolve the fundamental problem of the non-equilibrium theory, the origin of asymmetry in time?

Rationalizing non-equilibrium theory

253

to bolster their case. After all, they argue, the results of the spin-echo experiments show us that without the outside intervention, in the case where the test s stem is enuinel ener eticall isolated over a eriod of time in a manner sufficient to block "randomization" of the micro-states

vealed to us by our ability to reconstruct the original macroscopic order out of apparent macroscopic disorder by the "Loschmidt reversal" of the s in fli in. But from another point of view, and I think a persuasive one, the spin-

stages of the spin-echo process there is an "entropy increase" of the familiar sort. T at is, there is a issipation 0 macroscopic or er (the internal rna netization due to the ali nment of the s ins) into macroscopic disorder or uniformity (with the spins uniformly distributed in all

expect an asymmetric increase of the kind of entropy familiar to us from t ermo ynamics? 0 e sure, tel ormation a out t e ongina or er 0 the s stem in uestion can't vanish from the s stem as a whole without something like outside intervention to allow it to dissipate into the outside

has spread itself out into correlations among the micro-components of t e system Wlt out tru y lsappeanng a toge er. It is also im ortant to remark that the interventionist ex lanation of entropic increase requires as one of its components just the reliance

increase relies on. For it is only this that guarantees the spreading of the lfiltia o~ er mto t e corre atlons so extraor man y sensltive to pertur atlon from the outside. Essentiall ,unless the evolution of the ensemble under its internal dynamic was from a coherent to a greatly fibrillated one,

254

Physics and chance

increase, even if the perturbation from the outside never occurs. n r p ncr r, n w ig s m iv . r we start with an ensemble representing some intrinsically ordered system, random external perturbation of the trajectories certainly will result in a future ensemble that re resents a s stem that has lost some of its internal order. But here we must focus on an argument of a type that will often

to random perturbation from the outside in their earlier history. By an argument that exactly parallels the one used by the interventionist to show that ensemble entro is increased into the future b random influence from the outside, we can argue that the entropy of the ensemble

false conclusion that it was highly probable that the systems were earlier in a more disor ered state. But, 0 course, what we ought to infer is that at the earlier time the s stems were more ordered still althou h statistical mechanics won't tell us what that order was because many diverse ordered

argument from parity of reasoning. It would be to argue that the intervention rom t e outsi e is itse time- irecte . Because intervention is causation and because causation is from ast to future the intervention can only modify the ensemble toward the future direction of time. But

lawlike correlation of states, this sounds more like an a priori restrictive instruction on w en to use statistica mec anics an w en not to. is is like O. Penrose's" rind Ie of causali "concernin ensembles which says that "the phase-space density at any time is completely determined

account of why the aSYmmetry holds in the world? nce agaln, muc more cou e sal to e en aspects 0 t e lnterventionist a roach, but I will assume for the most art (with a return to these questions in Chapter 8) that we want to account for the asymmetry

Rationalizing non-equilibrium theory

255

3. Jaynes' subjective probability approach We earlier explored Jaynes' proposal to found the rationalization of the standard robabili distributions used in e uilibrium statistical mechanics upon an interpretation of probability as subjective degree of

distribution chosen. I argued earlier that the unpacking of this rationale, especially in the justification for choosing the standard measure over the hase-s ace as the one relative to which robabili was to be distributed uniformly, implicitly rested upon the use of ergodic results

equilibrium case, and it is to these proposals that we now tum. One proposal is to offer a general rule for the choice of initial non-equilibrium ensembles and to rationalize that choice. The other ro osal is to found the derivation of the Second Law on certain elementary considerations of

non-time-reversal invariant laws, external perturbation, coarse-graining, or to t e su e an powe resu ts 0 ynamics suc as mixing. In e uilibrium statistical mechanics a nes' rescri tion was to choose the probability distribution that maximized entropy (now thought of

to characterize the space of possibilities. The suggestion in the nonequi i rium case is to a i e y t is same genera ru e in etermining t e initial ensembles whose evolution is to characterize the evolution of a system prepared in a non-equilibrium condition.

rationalizable as the prescription for the equilibrium case. The rationale can pro a y e motivate as we or t ose cases 0 non-equii num in which the s stem is at least close to e uilibrium as a whole. The other common general case, where the system is describable as being locally

measure over the phase-space" rule for determining the initial ensemble pro a 1 ity istn utlon can e constructe. ut t e ratlona e or it previousl available, in which the prescri tion was backed up b the er odic result that such a probability distribution was uniquely temporally invari-

256

Physics and chance

In any case, JaYnes suggests his "maximize entropy" rule as a general

short" derivation of the Second Law

proof follows from the Jaynes method for dealing with non-equilibrium situations. First there is the in'unction that these statistical methods are to be applied only when the situation is one of "reproducible results," That is,

initial state but dynamically determined evolution does in fact follow a lawlike (or at least statistically lawlike) course, Next one starts at a time with some known values of macrosco ic parameters, 0t (or probability distributions over them). One determines ,

0

OtCto) = fpCto)0 1 and -fpCto)logpCto) is maximal, where 0t is the phase

unction appropriate to 01' To ca cu ate the time evo ution 0 physica arameters 0· one identifies the value of a arameter at a time as the phase average, relative to the evolved initial ensemble, of the appropri-

Rationalizing non-equilibrium theory

257 1

1

the identification of experimental entropy with Gibbs entropy of the maximal entropy ensemble relative to the new values of the macroscopic constraints. Then we ar e as follows: seCto ) = SGCto ) by the rule for experimental entropy. G

1

entropy at t 1 calculated using the ensemble pCt1 ) that dynamically evolved from that initially set up at to, pCto )' Now define S t as the Gibbs entro calculated b usin t . It

1

1

is obtained as the phase average over the ensemble of the appropriate phase nc~on is met. S (t ) = S Ct ) b the rule for identi in ex erimental entro with Gibbs entropy calculated by the maximum entropy distribution relative

Do we have here the explanation of time-asymmetry we have been 00 'ng or? It wou seem not. First, 0 course, t e erivation comletel i ores the search for the ex lanation of the details of the nonequilibrium process. Features of the evolution such as the relaxation

bypassed entirely. As several authors have pointed out, the "proof" even al s to emonstrate t e monotonIC Increase 0 entropy we wou 1 e to derive. It shows onl that entro ies at later times are reater or e ual to the initial entropy, not that entropy at one later time is greater than or

entropy at the time in which the initial ensemble is constructed. The cntiClsm ere IS mteresting y reffilruscent 0 one 0 e e ests cntiClsms of Gibbs' proof of the increase of coarse-grained entro . There the point was that although one could show that later coarse-grained entropies

argument showed that the change in coarse-grained entropy had to be mono onlC. Even more disturbing is an argument due to Jaynes himself. He points out that the very same reasoning applied in the demonstration suggests

made. All one need do is repeat the proof with t1 taken as referring to

Physics and chance

258 o

n -

grained entropy that follows from the determinism of the dynamic evolution in the past direction. a nes' res onse to this is to ar e that the various "rule " that nate the proof are rules that apply only to experimentally reproducible

the initial ensemble, and the rule of taking experimental entropy to be fine-grained entropy calculated by a new ensemble that is maximum entro ic for the new arameter value. But rocesses in one time direction are experimentally reproducible. The same non-equilibrium state always

distinct initial non-equilibrium states all lead to the same final equilibrium. This is certainly true. But it is just that fact that there is this parallelism in time of s stems - that distinct s stems show ex erimental re roducibility in one time direction and not the other and that it is the same time

Jaynes' argument as anything but question-begging. T ere is an important a itiona aspect 0 Jaynes' approac wort ursuin at this oint. The demonstration of the Second Law works b assigning to the later state of the system the entropy appropriate to an

evolved macroscopic parameters. From the point of view of Jaynes' o jective Bayesianism t is is puzz ing, ecause t e genera prescription it ives for ensemble construction is to maximize entro sub'ect to all of the constraints we know. What justifies this indifference to the known

to the results of the

of the historical origin of the system as (1) having evolved from a nonequl 1 flum con lt1on, ept ltS co erence In t e sense 0 not aVlng dissi ated it to the outside environment, and (3) then been Loschmidtreversed, we would get the correct results. In at least this case, then,

Rationalizing non-equilibrium theory

259

ization trick of the kind used traditionally to derive the kinetic equations will be successful. But we want to know why such a procedure will be successful when it is. And we want some indication of the conditions under which it is and is not appropriate in constructing our ensembles

values in choosing our constraints. And to this question the objectivist Bayesian (or, as Jaynes calls it, "subjective") approach provides no answer. This is the appropriate place to make a few remarks about the introduc-

related to those of Gibbs have recurred later - for example, in some aspects of the approac to irreversi i ity 0 S. Watana e. An t ey have been hinted at b E. Schrodin er. Gibbs had suggested that the source of irreversibility was to be found

ibility fraught with difficulty. One problem is that past events are as requen y a matter 0 1 erence to us as are t ose 0 t e uture. 0 e sure, if we saw a as evolve from the left-hand side of a box to fill the box we would be, from a subjectivist perspective, unreasonable in our

that the past state was one far from equilibrium. But such an argument wou stl lrec us 0 1 er e un nown past state 0 a system oun in equilibrium to be one of equilibrium and of a system found out of equilibrium to be, if its past was otherwise unknown to us, closer to

260

Physics and chance

that claim, of course, is the posit of irreversibility in statistical mechanics .. i

i

is just the fact of parallel reproducibility of systems into the future that we wanted to understand. In Cha ter 10 III we will examine briefl the claim that the ve asymmetry of our knowledge of past and future, the fact that we can

alization, rests upon the entropic asymmetry of the world. The claim, although higWy plausible to many, is one that remains problematic and difficult to establish. But in an case it is hard to see how the entro ic asymmetry itself can be thought to depend upon any relativization of

knowledge of the world. Whatever subjective elements enter into the concepts 0 statistica mec anics t roug suc notions 0 coarse-graining or the relative nature of entro it is hard to fill in an sub'ectivist theo in such a way as to convince us that the parallelism and asymmetry of

events.

its fundamental problems irreversibility in fundamentally asymmetric physical laws governing the micro-components e aVlor, In contmua outsi e mtervention into e evolution of the s stem in uestion from the outside, or in the structure of probabilistic inference understood from a subjectivist or logical theory

We wish to obtain from our account a variety of results of increasing nc ness. lrst, we WlS to s ow ow, m some sense, an approac to e uilibrium arises, and we wish to understand wh this rocess is as mmetric in time. We would like to go beyond this, however, and under-

Rationalizing non-equilibrium theory

261

comp . i n action among these components, the vast number of degrees of freedom in the system generated by its vast number of micro-constituents, and the existence of the s stem in our anti-interactionist idealization as enuinely energetically isolated from the world (at least in the case where we

evolution we describe will be not that of an individual system but that of an ensemble of systems. Or, less picturesquely, the evolution we describe will be that of a robabili distribution over micro-states of systems compatible with the macro-constraints defining the systems of

thermodynamic in kind. We s a a so nee a way 0 un erstan ing ow to "rea "t e inetic e uation that re uires somethin over and above the correlation of ensemble defined quantities with thermodynamic variables. Here I have in

system but, rather, the "concentration curve," the curve that plots the successlon m hme 0 overw e mmg most pro a e lnterme late states. Because, as we have noted, other attem ts at rationalization, such as the Lanford derivation of the Boltzmann equation, read the kinetic equation

lution, even in the statistical sense, is non-trivial. ma y, t ere lS one more crucla lngre len we expect 0 n ln any orthodox attempt to ground statistical mechanics. We presuppose the underlying determinism of the dynamical laws governing the micro-

262

Physics and chance

would have to be shown in any foundational rationale of the theory or u a ug w n n i p cip s reran omi a ion, we are entitled to posits about initial ensembles. The evolution of an ensemble is the joint product of its initial structure and the deterministic d namics of evolution of the individual s stems makin u the initial ensemble. We fully expect that it is in the nature and constitution of this

So the appropriate nature of such initial ensembles, the study of how their choice governs the results extractable from the picture, and most im ortant of all the crucial uestions about wh certain initial ensembles are the appropriate ones to choose and others are appropriate ones to

statistical mechanics.

Foundations ofStatistical Physics, published posthumously in 1950. Krylov e ieve t at e cou s ow t at in a certain important sense, nei er classical nor uantum mechanics rovided an ade uate foundation for statistical mechanics. His intended work was to begin with a detailed

ensemble spreading and fibrillation that later became formalized in the nollon 0 a mlXmg ensem e. e emp aSize e ina equacy 0 ergo iCity b itself to round the non-e uilibrium ortion of statistical mechanics and he outlined in a thoughtful way the notion of mixing as it had

Rationalizing non-equilibrium theory

263 e expresse

to characterize those exact conditions on the constitution of a system sufficient to prove that it is indeed a C system. It is a richer notion he intends however in the sense that lov is interested in discovering the foundational elements necessary to derive

would certainly not be satisfied with a demonstration that an ensemble was mixing in the mere "limit as time goes to infinity" sense. lov's most im ortant critical contribution is his em hasis on the importance of initial ensembles. Suppose we can indeed show a dy-

evolution of a system will have the appropriate finite relaxation time, muc ess t e appropriate exact evo ution 0 our statistica corre ates 0 macrosco ic arameters unless our statistical descri tion includes an appropriate constraint on the initial ensemble we choose to represent the

, will be of the anti-thermodynamic type. (See Figure 7-6.) at must appropnate lrutla ensem es e 1 e ln or er t at t elr evolution ro erl describe thermod namic behavior? The must have fairly simple shape, for allowing any degree of fibrillated complexity will

a simply shaped initial ensemble if it is too small will, while spreading out lnto a nate s ape ln t e ture tlme lrectlon, 0 ltS sprea lng too slowl to re resent the actual short relaxation time we ex erience in dealing with macro-systems in the world. Furthermore, within the

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b)

robabili within the ensemble boundaries could once more allow the design of an ensemble either anti-thermodynamic or too slow in its

of sufficiently careful procedures in preparing systems could pin them own to t eir exact point micro-state. Eac system wou en ave its evolution determined into the future in a com letel lawlike wa that specified a unique trajectory. This, Krylov believes, would be incompat-

nings. So no foundational approach based solely on classical mechanics or quantum mec anlCS can ope to groun elt er e n amenta robabilistic nature of the world or the restriction to certain ossible initial ensembles as representing the evolution of systems. And both of

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than a reflection of underlying laws e wor a ini ia con i i ns in 1 ppr pria always in statistically characterizing a system to view it as subsumed under a uniform probability distribution relative to whatever macroscopic ndi ions are known to hold of the s stem. Krylov has two major objections to this view. The first is that it would

relaxation times, and the inevitable uniformity of kinetic behavior to mere happenstance, something he holds it impossible to believe. Second, he thinks the osition fundamentall incom atible with what we know about nature.

world that whatever ensemble boundaries one picks, it is appropriate to view a system as subsumed under a un' orm istribution within that bounda . But consider A and B two intersectin but distinct re ions on the energy surface. The probability density in A is the same as that in

over the energy surface - that is, the system must be subsumable under e equi i rium pro a i ity istri ution, w atever its macroscopic constraints. But of course s s ems are usuall not in e uilibrium. Consider, also, the following argument. Suppose the probability dis-

the conservation of Gibbs fine-grained entropy over time. So if the "real ensem e was unl orm at one time, It cou not e a moment ater. e rind Ie that we ou ht to take it to be the case that s stems 'ust ha en to be so distributed in the world so that we ought always to subsume

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replace the usual classical or quantum dynamical basis, but more often i yn'c vo u i n 0 an ini ia a er a, i r a 'n ensemble once it is set up. The additional lawlike principle governs the appropriate structure for initial ensembles. The basic idea is orrowed b analo from the Heisen er ian interpretation of the Uncertainty Relations to quantum mechanics. In the early

system that "jolted" the measured system out of its exact micro-state in the very act of trying to determine what that state was - seemed a lausible readin of uantum uncertain . This readin onl later fell out of favor in the light of "measurement interference" on systems consisting

Heisenberg's "interference" interpretation of uncertainty also became less plausible in the light of "no hidden variables" proofs. Krylov does not utilize uantum uncertain to found the restriction on initial ensembles. Rather the idea is that the instability of trajectories and the ineliminable

then watched, will be so "crudely" prepared that only the appropriate, stan ar , initia ensem es wi statistica y represent t e systems so enerated. It is, then, the interaction with the system from the outside at the single

ably interfering perturbation of the system by the mechanism that sets it up in erst pace t at guarantees t at t e appropriate statistica escri tion of the s stem will be a collection of initial states sufficientl large, sufficiently simple in shape, and with a uniform probability distri-

coarse-grained spreading out in the appropriate finite time. Such ensem es are t en appropnate to t e macroscopiC parameters given at e be innin of the evolution of the s stem. The are not a ro riate later on, or for arbitrary regions of phase space taken in general, as they

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W1 no rea y e e u orm one re a ive 0 macroscopic parame ers as they evolve, but of course we will be able to treat it as such to a degree, because it will rapidly coarse-grainedly achieve such uniformity because r r n mm I i 'n hi m' 'n th ization posit of the kinetic equations finds its justification. a sec 0 a pr m w' w u count. Exactly how does this initial interference lead to initial ensembles of just the kind we need? Here, a detailed treatment of preparation in all of its ises and of the effect of re arer on re ared s stem would have to be carried out. Could this difficult program really be carried out in

detailed program. These remarks will at least cast some critical doubt on whether Krylov has pointed out the correct route to travel to find the ori ins of kinetic behavior and tem oral as mmet , One problem arises when we once again refer to the results of the

whose appropriate representative initial ensemble would be one leading to anti- ermo ynamic e avior. But t e spin-ec 0 experiment sows us that we can at least in certain exce tional cases re are initial ensembles that have just the fibrillated initial form evolving back to a simpler

doing an ordinary preparation of a highly isolated system, and then re ecting its time evo ution at a ater stage. ut t at t is can e one at all indicates the need on lov's art to show us in detail 'ust what it is about ordinary preparations that makes it impossible for them to result

upon the important use of the notion of preparation that Krylov utilizes, It is e preparat10n 0 a system at e eg1nrung 0 t e time peno over which its evolution is observed that so disturbs the s stem that the onl possible statistical representation of it is in terms of a uniform probabil-

Imagine a system constituted at one time by some process that leaves 1t or a s ort w 1e 1S0 ate rom 1tS surroun 1ng envrronment. teen of that eriod, somethin is done that reinte rates the s stem back into the energetic whole of the surroundings. If at the earlier time the system

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But suppose we have a system that is far from equilibrium at the time n np ic, wrongly, that it was to be found closer to equilibrium at an earlier time? If not, why not? I doubt that the intuitive response that Krylov would offer could be founded on somethin s ecial about the articular nature of the creating ("preparing") and annihilating ("destroying") processes.

"preparation" appropriate to representing the system by the Krylovian type of initial ensemble, because the reintegration doesn't "cause" the state of the s stem to come into existence at that time the wa the act of genuine "preparation" does. It is the intuitive idea that causation works

get time-asymmetry into his account only by relying on a temporally asymmetric notion 0 causation as istinguis ing preparations 0 systems from their destructions. This claim can be buttressed by making the following observation.

to understand how Krylov would be able to generate the essential tempora para e tsm 0 e entropic evo ution 0 systems at e nee s. For if the re aration and destruction acts are to be characterized in terms of some intrinsic difference of their nature that does not in itself imply a

tion is on the time side of its associated destruction it is hard to see why t e time irection rom act 0 preparation to act 0 estruction s ou alwa s be the same time direction. And it is hard to see why that direction would always be the time direction we take as from past to future.

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w u n su in the necessity for just the appropriate initial ensembles that when combined with the instability of trajectories generated by the dynamics, would allow us to extract the full finite time and detailed kinetic descri tion we want. And it would require an explanation of how this account will get

6. Prigogine's invocation of singular distributions for 'ni i e

legitimacy of the rerandomizing posits used to generate the kinetic equations. They also offer an exp anation 0 the origin of the time-symmetry breakin that leads us from the reversible underl in micro-d namics to irreversible thermodYnamic descriptions.

that allow for the proof of such things as mixing, K systemhood, or Bemou i system 00 or e ensem e. Rat er . erent treatments 0 the ori ins of kinetic behavior must be offered in the two cases. For the systems for which it can be shown that there remain regions

the conditions sufficient to show enough "chaotic" trajectory behavior in t e rema1n1ng p ase-space reg10ns OutS1 e t e s t a e ton or t e behavior in those re ions to be describable in a chaotic manner. Generally, the program hopes to show that in the limit of a system of a very

tori shrink to insignificance in the phase space. n ese cases, we ea W1t systems t at can e escn e y a Hamiltonian that corres onds to one obtained b neglecting all interaction among the micro-components but with a small perturbing inter-

those conditions where the orderly behavior of the trajectories in the reg10ns 0 s a 11 prove 0 eXlS y e eorem canno e extended into the general phase-space region outside those tori. Here, the idea is that the multiple overlapping resonance regions of the dynamics

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a any eXls 1ng a i iona cons an s 0 mo ion over an a ove e energy fail to determine "smooth" sub-manifolds in the phase space. It will still be true, then, that arbitrarily close to any initial condition leading n

gent trajectories. And we will not be able to partition the region outside nn . This can be the case even if the region of chaos fails to meet the strong measure-theoretic conditions, such as metric indecomposability and its stren thenin s needed to rove such stron chaos results as moon being a K system, or being a Bernoulli system. y

position of the ensemble in the manner described earlier as the program of sub-dynamics. That is, the condition is to be sufficient to show us that there is a uni ue decom osition of the ensemble into two arts. The decomposition must commute with the time evolution so that the evo-

to zero, one component just becomes the dynamic description of the evo ution 0 e ensem e 0 in epen ent micro-systems. As escri e earlier if such a sub-d amical decom osition of the ensemble evolution is possible, it can be shown that one of the components of the

time evolution at a time dependent only upon its state at the time, to satis a inetic equation type 0 Ynamics. e macroscopic eatures 0 the s stem described b thermod amics will all then be calculable from the behavior of this kinetic component of the ensemble alone.

" then link together the "geometric" nature of the phase-space flow su Clen y c aot1c e aV10r 0 traJectones OUtS1 e t e reg10ns 0 traJecto stabili roved existent b the KAM Theorem - with the a ro riate dYnamical behavior of the ensemble as described by its differential

, lution decoupled from the equation of motion of the other "correlational" component. ee ess to say, prov1ng at t e con 1t10n nee e 1S sat1sfied, in all but the simplest idealized cases, is a matter of great difficulty. In the cases where the idealization leads to a provable measure-

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y

showing that the ensemble and its evolution can be transformed by means of non-unitary transformations into a new representation that reveals the otherwise concealed kinetic ro erties of the evolution. The demonstration that such a transformation is possible has mixing as a

the original ensemble into a newly described one. The new ensemble description will generate well-behaved pro a ilities and will reproduce the same hase avera es for all hase functions roduced b the ori inal ensemble. Furthermore, the original ensemble can be uniquely recon-

Whereas the original ensemble representation will have its time evolution escri e y t e time-reversi e Liouvi e equation, t e new re resentation will have its evolution described b a time-as mmetric Markovian kinetic-type equation. And one will be able to show that in

equilibrium ensemble. In fact, one can do more. A "time" operator can e evise t at wi c aracterize t e "age" 0 an ensem e, essentia yin terms of its deviation from the "full a ed" ensemble of e uilibrium. Normally, ensembles will not have some definite age (they will not be " " " operator obeys a commutation operation with the Liouville operator, the operator generating time-rate 0 c ange 0 t e lnltia ensem e representation. This is reminiscent of the commutation relations amon the operators corresponding to "complementary" observables in quantum

ensemble in these two equivalent ways is perfectly understandable. The new represen a lon oesn pu any lme-asymme ln 0 e ensem e evolution that wasn't there already. It merely brings to the surface a time-asymmetry latent in the original ensemble description. From the

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region in such a way as to exclude there being in the region delimited by our measurements phase points whose future trajectories diverge wildly from one another - that is to sa in those re ions of hase s ace outside the KAM regions of stability where they exist, or in the general phase

be a false idealization of the physical reality. This is an issue to which we shall return when we discuss the alleged reduction of thermodynamics to statistical mechanics in Cha ter III. For now we need onl mention it, because the further development of the Prigogine view - the invocation

of individual systems. Bot approac es to t e pro em 0 e erivation 0 inetic e avior from d namical behavior we have 'ust outlined leave the roblem of time-asymmetry at least partly unsolved. Corresponding to the derived

master equation. Whereas the former has monotonically equilibrium approac g so utions in ture hme, e atter s so utions at montoruc y diver e from e uilibrium from ast times. Corres ondin to the A+ transformation that takes the ensemble to a new representation in the

in the context where the idealization allows for the clearer and more powe notion 0 c aotic evo ution 0 t e traJectones in t e measuretheoretic sense - that is, where the ensemble is taken to represent K systems.

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time in preference to the other? These initial ensembles will, in some appropriate sense, approach equilibrium in one time direction but not the other. It is claimed that no initial ensemble of non-zero measure will serve to break the time symmetry for us. We are dealing with K systems.

the limit as t~ + 00 and as t~ - 00. And for each such ensemble that approaches equilibrium, so wi t e ensem e t at is compose of a and onI those s stems that are time reverses of the s stems in the ori inal ensemble.

and "contracting" "fibers." These are ensembles of measure zero. Because t e evo ution 0 t e ensem e is measure preserving, t ey transform onl into sub-ensembles of measure zero in either time direction. Yet there is an appropriate sense in which such a fiber can "approach the

will not approach equilibrium in the future time direction. The sense of approac is, aS1ca y, 1S: ereas t e trans ormat10n 0 e 1n1t1a zero measure ensemble is not the A+ transformation of the e uilibrium ensemble, the ensembles into which the original zero measure ensemble

equilibrium ensemble. In this sense, these fibers "dilate" in one time 1rect10n ut not t e ot er. t e1r hme-reverse ensem es s ow t e reverse time-asymmet with respect to this sense of approach to e uilibrium. Figure 7-7, using the famous Baker's Transformation, a particu-

Furthermore, one can define an "entropy" for sub-ensembles. With this entropy measure, 1t ecomes POSS1 e 0 s ow a 1n1ha slngu ar ensembles that evolve toward equilibrium in the forward time direction have finite entropy, whereas the entropy, so defined, of initial singular

One can then argue that this shows that preparing a system in such a way a 1 eva ves owar equ11 num or, more exac y, prepanng e relevant collection of systems) is possible. But preparing a collection of systems showing anti-thermodynamic behavior, because it would require

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-

A

x

-

y

y

represents "Baker's Transfonnation." A square region is stretched to double its

. , .. Then the right-hand side is cut off and placed on top of the left-hand half. The . .... . ..

.

..

.

"

simplest having radical randomizing properties such as being a K system and a

..

..

,

.

,

again. It is a "dilating fiber" of the transfonnation. A vertical line, as in Fig. C, . . ." , ing fiber." Even though both the horizontal and vertical lines are of measure zero,

..

.

....

that, in a sense, the transfonnation of x will approach the equilibrium ensemble

.

...

.

t..

Next the formal devices of sin ular initial ensembles of the a ro riate kinds and the new definition of entropy are connected to our physical

randomly moving particles, corresponds to a dilating singular fiber in the er wou e t e tlme p ase space. orrespon lng to a contractlng reverse of this. We can re are the former e of s stem (and collections of them) but not systems of the latter sort. So this distinction be-

that asymmetrically distinguish the two directions of time, but represents ln a reasona e way our p YSlca POSSl llles an lffiPOSSl llles 0 preparing ensembles of systems in initial states in the world. It is this distinction that selects out the possible thermodynamic from the impossible

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problems. But like the other accounts we have looked at, it has within it aspects worthy of some skepticism. As usual, the derivation of kinetic behavior either in the form of a master e uation or in the form of the Markovian equation for the A+ transformed representation of the ensemble, ........'.J.oJ ..Jl....

J.J.J.'V......, ....J..:>,

can be inserted into the schematism that will do the job of demonstrating to us, first, that the kinetic behavior is attributable to those elements of the ensemble needed to derive the thermod namic uantities' full irreversible and monotonic behavior that we wish to demonstrate, and,

relaxation time. Next, t ere are severa questions connecte sin lar sub-ensembles as the a ro riate characterizers of h sicall preparable initial ensembles. First, are they really necessary? Although it

time limit, their behavior over any specified finite time interval can be very erent 1n ee . orne W1 monoton1ca y approac equ11 rium in one or both time directions. Some will monotonicall deviate from it again in one or both time directions. Most will show non-monotonic

without resort to the use of singular ensembles by focusing on those non-zero measure su -ensem es t at s ow a monoton1C approac to e uilibrium in the finite time periods characteristic of the experimentall observed approach to equilibrium of real systems? After all, as has been

And there are reasons for thinking the singular sub-ensembles inappropna e or escn 1ng mos p YS1ca Sl ua 10ns. ngog1ne, 0 course, realized that our system preparation will not give us exactly measure zero ensembles. But he suggests that it is ensembles close to those of

almost parallelized beam that dissipates into random thermal motion) m1g e appropna e y represen e 1n suc a way. u e more usua case - for example, a system in equilibrium relative to some imposed constraints whose constraints are then changed and that then evolves to

case, surely, is one of non-zero measure. It is one of those ensembles t at owever symmetnca y it e aves in t e positive an negative 1 nite time limits, evolves toward equilibrium for a finite time subsequent to re aration. This time will be the relaxation time ex erimentall observed for the system.

preparation en 0 t e nite time evo ution 0 a system a ways t e same temporal end relative to the measurement or observation end? Why is the direction from preparation to observation the same for all systems? It is invocation of singular initial ensembles of the dilating kind as sYffiffietry-

zero as initial ensembles gains us any ground in understanding the physical

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7. Conflicting rationalizations One further issue deserves our attention before we move on to review the results that we have been surve in and to discuss their fundamental import. We have seen a number of approaches to solving the two fun-

poral asymmetry. It is important to point out that some of these approaches to rationalizing the approach to equi ibrium can conflict in wa s that mi ht not be immediatel a arent. The conflicts I will focus on here lie at the very heart of the rationalization of rerandomization

that rely upon the generalizations of ensemble chaos that go beyond ergo icity. The Lanford a roach to rationalizin the Boltzmann e uation tells us that if we go to the Boltzmann-Grad limit, there will be an initial con-

tions evolve in the manner described by the Boltzmann equation. As or rna es pe ec y c ear, t IS IS an approac at sows not t at the Boltzmann e uation will hold "on the avera e" but that it ives "an accurate description of the time development of 'almost all' initial

short time, the evolution of "almost all" systems subject to the initial constraints. As was pointed out earlier in the survey of the historical develo ment of these issues, there is a radical incompatibility of such a claim of the

look for the mathematical reconciliation of the results in this direction.

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these results to less idealized systems. The important point here is this, as was made clear by J. Lebowitz. The "Boltzmann hierarch "from which the Lanford results are obtained "does not have underlying it any flow in the phase space." Lebowitz continues

festation of the "loss of information" resulting from the use of the Boltzmann-Grad limit - a loss necessary to make dissipative macroscopic laws consistent with reversible microsco ic d namics." In other words the going to the limit of nd 3 ~ 0 while n goes to infinity was not a mere

this limiting process that the mathematical consistency of these results wit t e Recurrence T eorem an t e mixing-type t eorems is . en. For in this limit the hase-s ace flow that enerates recurrence and mixing ceases to represent the evolution of probability.

equation in the Boltzmann-Grad limit approach to seeking the origins of 1netlc e aV10r 1S ra 1ca y at 0 s W1t t ose attempts to n t e ong1n of this behavior in such features as the mixin nature of the ensemble. The latter approach continues a tradition with its roots in the Ehrenfests'

ing through entropy values that are most probable at each time. This approac 1S a so cons1stent W1t 1 s 1 ea 0 see 1ng t e approac to e uilibrium in a coarse-grained notion of entropy. However, the former approach, Lanford's, returns to the idea of the kinetic equation as offer-

equilibrium in the manner described by the Boltzmann equation. er aps some u er c ar ca 10n can e 0 a1ne ere y re ec 1ng on the idealizations needed to obtain the various formal results under discussion. The Poincare Recurrence Theorem requires, as the interven-

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depend upon ergodicity, mixing, and their strengthenings apply only to those idealized systems (like hard spheres in a box) to which the KAM Theorem doesn't a l. A ain this is somethin not likel to be true of any realistic system.

, the Lanford result holds only for a system of an infinite number of degrees of freedom and vanishing density. From the mixing point of view the onl role la ed b the lar e number of de rees of freedom of systems is that of allowing us, in the Khinchin manner, to identify

the method of phase averages. But for the rigorous derivation of the Bo tzmann equation, t e imiting situation 0 i nite num er 0 partic es and zero densi is absolutel crucial to the derivation. But the crucial question for us is this: Which idealization is the appro-

the world? It seems clear that even at the level of attempting to understan e reran omlzing aspects 0 t e success netlc equations, muc less in t in to 0 be ond this and understand the ori ins of timeasymmetry, we still are not at the point where a decisive answer to the

non-equilibrium behavior It will be worthwhile to conclude this chapter on approaches to ration-

alizing the non-equilibrium theory of statistical mechanics by once again

point of view of asking to what extent the various results can be utilized In prOVl lng an exp ana ory accoun 0 e non-equll num e aVlor of systems that fits the models of explanation, in particular statistical explanation, outlined in Chapter 4. Do the various formal and technical

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equilibrium systems behave as they do? It is a rich body of experience that cries out for explanation. Systems not in e uilibrium can often still be described macrosco icall in terms of a small number of parameters that obey a lawlike dynamics. These are

that with rare exceptions at least, the systems inevitably follow a path from their initial non-equilibrium configuration to the equilibrium state described b classical thermod namics as a ro riate to the macrosco ic constraints on the system. This approach to equilibrium is characterized

macroscopic parameters summed up in the hydrodynamic equations of evolution. W Y 0 systems be ave t at way? Convinced b the theoretical ar uments directed a ainst Boltzmann's first understanding of the kinetic equation of approach to equilibrium

equilibrium must, in some sense or other, be a statistical or probabilistic account. But w at t at comes own to remains open to e ate. We a so resu ose althou h this once a ain is a resu osition of some ambiguity that can be unpacked in quite different ways, the subsumability

to equilibrium would be, then, the one in which individual systems are construe as aVlng Inltla mIcro-states t at etermillistlca y evo ve Into future micro-states. Probability would appear as some distribution over these micro-states at one time, leading to a distribution over evolutionary

evolving in its deterministic way, that would then be related to the o serve y ro ynamlc evo u Ion 0 a macroscopIC ea ure 0 e Individual systems. Again, which statistically calculated quantity would be related to a macroscopic feature, and how this relation would be con-

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of the place of statistics in the explanatory pattern that differ from this standard one in fundamental ways. In particular, there will be attempts to show that the introduction of statistics into d namics rests u on features of the world that when properly understood, require us to reinterpret

"tychistic" statistical state and the evolutionary equations of statistical mechanics as somehow or another describing the evolution from robabilistic individual state to robabilistic individual state for an articular system. We must not think of the equation of evolution, then, as

probabilistic notions in the account of its individual behavior. T e view 0 t e nature 0 statistica exp anation in statistica mec anics one holds will of course be uite different de endin on the view taken about the ultimate nature of non-equilibrium evolution. If one believes

genuine

1. Probabilities as features of collections of systems The most commonly held variants of the mainstream interpretation of

From this perspective, probability can characterize only a feature of some e macroscoplC aws 0 evo uensem e or co ectlon 0 suc systems. tion from non-equilibrium to equilibrium then, to be obtained by some association of the parameters in them with something constructed out

"representative" behavior of the members of the ensemble. lven a sys em ln a non-equll flum s e, macroscoplca y c arac erized, the natural way to understand this model of statistical mechanical explanation is to characterize it in terms of a probability distribution over

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n x v u i n 0 eac individual system from its initial micro-state. This seems inevitable given the idea of each individual system as having its particular evolution causally ex lainable its micro-state at one time and its d nam'c 1 then, we would have a model of probabilistic explanation as characteriz-

The aim of all such models will be to derive the characterization of systems in terms of the macroscopic parameters and the detailed laws overnin their evolution from allowable osits. Then one will to physically account for those posits. By "allowable posits" I mean to

crying out for physical explanation, "rerandomizing" posits are not to be countenanced. This is because the dynamical evolution of the system is fixed b the d amical evolution of its members and that is a iven from the underlying dynamics.

met. We would then expect members in non-equi i rium to isp ay quite i whether the initial condition for the s trajectories near those that govern the

of a collection of systems started erent e aviors epen ing upon stem was in the re ion of stable system with the small interaction

of making such claims rigorous is mathematically quite intractable. Even in e imit 0 an 1 nite num er 0 egrees 0 ree om, or examp e, it is not at all clear that the stable re ions can be ex ected to have "measure zero." Add to this the problem that even if the regions of stability

naturalness of the usual measure available to us in equilibrium theory, w ere a measure was e on y ime-invanan measure t a gave zero probability to sets of measure zero in the standard measure. We want a physical reason for thinking that it would be likely that in a collection of

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the chaotic region. We would still be far from having the full derivation that we would like of the non-equilibrium behavior we are trying to understand. We sim 1 don't et know enou h about the kind of "chaos" in the chaotic region to be able to derive from it anything like the " ergodicity can be assumed to hold. There are highly interesting suggestive results, such as Prigogine's claim that the conditions necessary for such a chaotic re ion to exist are those necessa for the existence of a "sub-dynamics" that would extract a time-invariant component of the 1 •

that in its chaotic region, with a "natural" distribution of initial conditions presuppose , an evo ution 0 components 0 t e statistica istri ution over the states of s stems evolved from that initial distribution would be properly kinetic. And we would want a complementary demonstration

sufficient to generate by association all of the macroscopic parameters at appear in t e macroscop1C equat10ns 0 evo ution. ere again, t e vast number of de rees of freedom would be invoked in a Khinchintype argument to assure us that the probability distributions for the

with impunity, for example, from average to overwhelmingly most proba e va ues 0 t e macroscop1C quantities. Even if we had all of this, we would still have to contend with the "touchstone" problem of statistical mechanics, the origin of the tempo-

even in the chaotic region to get the thermodynamic behavior of systerns, 1S actua y oun 1n e wor. t e 1S n u lon 1S suc a w en applied to final states of systems instead of initial states its application would lead us to expect anti-thermodynamic behavior of systems at the

condition, or, better, a distribution that when applied to states generates an asymme 1n pre 1C 10n an re ro 1C lon oc 1ng any app 1ca lon 0 it to final states that would lead us to expect anti-thermodynamic behavior, we know that there will be another such distribution that would lead

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y, w y n n no me more "symmetric" distribution. The real issue here is the parallel time behavior of systems and not the fact that equilibrium is approached in the "forward" dir cti n f time taken s iven. If we move away from the realistic case of systems describable by the

interactions (like "hard spheres in a box") where the measure theoretic generalizations of ergodicity hold (mixing, being a K system, being a Bernoulli s stem we obtain as com ensation for our reater idealization, rigorous mathematical results and a firmer grasp on just what "chaos"

(being a K system), a new representation of the probability distribution governing the ensemble can be obtained, by a star-unitary transformation a re resentation that shows manifest! kinetic behavior. The transformed ensemble obeys a Markovian equation of evolution of its face.

idealization of systems as having vast numbers of degrees of freedom is, in one sense, not a necessary con ition or t e exp anatory sc erne to a I. The vast number of micro-com onents of realistic s stems does playa role, however, in once again allowing us to infer that the reduced

to the behavior of associated macroscopic quantities, the large numbers o egrees 0 ree om p ay a ro e. The results of chaos that can be obtained are used to rovide the rationalization of Gibbs' posited "spreading in the coarse-grained sense"

Markovian behavior of the coarse-grained ensemble description. This is nee e to go eyon a emonstratlon 0 entroplc lncrease to a emonstration that the route of evolution followed is that posited b the standard kinetic equation. Another virtue of this program is that at least some "

"

by the dynamics or the nature of the system, is vitiated. Some results will o 0 any coarse-gralnlng 1 e mlXmg . ers 1 e elng a ernou 1 system) will prove the existence of a coarse-graining having the appropriate behavior.

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clearly delineated by the Ehrenfests. The hydrodynamical equations are derived from the evolution of reduced probability distributions, the macrosco ic arameters bein functions of these. The kinetic e uations governing evolution of the reduced probability distributions are inter"

"

of most probable states of the systems in the collection. Thus, the monotonic approach to equilibrium is reconciled with the Recurrence Theorem. When however we ask how the results obtained b means of the method that generalizes the Ergodic Theorem are to be associated with

us with rigorous mathematical results, hinders an easy understanding of its app ica i ity to rea p enomena 0 rea systems. Some 0 t e resu ts tell us what ha ens "in the limit as time oes to infini ." But we wish to understand what happens "in the short run" - in particular, in time

numbers of degrees of freedom is used but where the loss of accuracy incurre in ea ing wit rea systems aving vast num ers 0 egrees 0 freedom but not infinite numbers of them can be assessed the method provides us with no way of judging how long a time is "long enough"

the theorem. n ot er cases were resu ts are rame 1n terms 0 n1te times, ot er related roblems of in the results to the observed henomena complicate the understanding of the place of the results in our explanatory

coarse-grained cells necessary to derive kinetic behavior for coarse-grained 1stn ut10n nctIons an monoton1C 1ncrease 0 1 slan coarse-grame entropy, the problem is that the method guarantees us that an appropriate coarse-graining will exist but doesn't help us to understand, for

behavior we want for the macroscopic observables associated with the appropna e coarse-gra1ne s a 1S 1ca quan 11es. 1S one 1ng 0 s ow that because of the instability of the dynamics a probability distribution over an ensemble of systems has an appropriate representation that shows

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pro a 1 lty IS n u Ion. namical behavior of the ensemble of this kind with real relaxation times and the real hydrodynamic behavior of local pressures, temperathermodynamic equations of approach to equilibrium.

approach to the explanation of non-equilibrium phenomena. But of course

nature. As Krylov emphasized, the initial ensemble would have to have w" nnw u u v n spread in the manner appropriate to the observed relaxation time of realistic systems. Here, all the problems about the physical ground of the a ro riateness of initial ensembles re resentin s stems in the world reappear once more. What guarantees that systems could not be such-

fibrillated as to have the ensemble condense in a reasonable finite time, even if guaranteed to spread out once again as time goes to infinity? What uarantees that the a ro riate robabili distribution to ick within the boundaries will be uniform with respect to the usual phase-space

to us in equilibrium theory, what can explain to us physically the place played y the usual p ase-space measure in our theory, a question we must answer even if we acce t the uniform robabili osit (relative to that measure) as given to us from some principle of indifference whose

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u

swers are not yet fully satisfying. Once again, the problem of the asymmetry of systems in time points out with transparent clarity the fundamental o en roblem here. Bein derived from the time-s mmetric underl in dynamics, all the results of the generalizations of ergodic theory are

indeterministic in the future time sense (K systems) are probabilistically indeterministic in the negative time direction as well. Bernoullian coarserained behavior is also time-s mmetric. So would be their conse uences such as Markovian time behavior for the transformed ensemble. If a

representation obeying an equation Markovian into the past. Why, then, are systems asymmetric in their temporal behavior? A representation of this time-as mme in our theo can onl come about in this ers ective, from an appropriate choice of initial ensemble to represent the

eralizes beyond ergodicity. It is shown that under the conditions posited, e one-partlc e istn uhon nction 0 eys t e o tzmann equation, an one can then use the usual association of uantities defined b means of this function and its appropriate short time behavior as governed by the

regard to evolution is a return to the idea of the theory as generating an overw e mlng y pro a e course 0 evo utlon, an not as generahng a concentration curve throu h states overwhelmin I robable at s ecified times. This, of course, makes the approach conceptually at odds with

As usual, idealization is a crucial component of the explanatory account. s we note , t e resu ts 0 on y or very s ort hmes a er e initial time, but there is some hope that they remain true, if not rigorousI provable, for longer, perhaps realistic, times. In this approach, the large

density goes to zero and when size of particles relative to size of contain1ng ox a so goes to zero. t 1S unportant to remem er t at t 1S 1 ea 1zation is unlike the usual one of going to the thermodynamic limit of an infinite number of de rees of freedom. In most cases we do have a grasp of the degree of inaccuracy entailed by our idealization when we

such a way as to be properly represented by such an ensemble, but not

at time zero, do not propagate forward in the time and that the time-

2. Probabilities as features of states of individual systems As an alternative to the view of statistical explanation in statistical me-

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statistical explanations as those appropriate to such systems. The general idea is that systems have no "finer" states than those associated to them b the robabili distributions attached to macrosco ic conditions b the statistical mechanical theory. The postulation of pointlike (in phase-

to be a false idealization. Along with this revisionist ontology for the states of systems comes, of course, a new understanding of what we are doin when we use statistical mechanics to rovide us with an explanation of the dynamics of a system.

characterized by the probabilities of transitions latent in it to other states of the system. Dynamical evolution is the transition of the system from state to state re lated b the irreducible robabilities of transition latent in the states themselves, and otherwise undetermined by some deeper

explanatory scheme on us. It is the absence of those deterministic features, ea ing us to a statistica account as t e eepest possi e exp anatory account of the d namics as it is in itself. Analogy is frequently made with the explanatory scheme of quantum

state, by the rule of associating "characteristic states" to measured values IS quantum state en evo ves In a pe ect y etermlno 0 serva es. istic wa accordin to the Schrodin er e uation in a manner fixed b the energy function for the system in question. It is in using this evolved

and explanation comes in. For the quantum states determine only proba IIt1es over e POSSI e outcomes 0 er measurements 0 0 servable uantities. An im ortant feature of the scheme is the so-called Projection Postulate, which specifies that if a complete measurement has

.

.

system to attribute to the system for new predictive purposes is the c aractenstlc state etermlne y e va ues 0 e measure 0 servabIes. This rule, or its generalization, Luder's Rule, that applies when the measurement is incomplete, posits that future statistical inferences about

to that state being irrelevant for future statistical predictive purposes.

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reality of systems become endemic. Here, we wish to present only this fragment of the quantum mechanical scheme to see how closely statistical mechanics can be made to resemble it. The analogy to the "observables" of quantum mechanics would be,

complete set of observables, In statistical mechanics, this notion of "completeness" of a description would be given to us, presumably, by considerations of er odici . A s stem that is er odic has a limited set of macroscopic conditions that can be imposed on it in a time-invariant way

global constants of motion would lead to metric decomposability and, ence, to a vio ation 0 ergo icity. T is is important or us ere, or otherwise one could ima ine macrosco ic conditions of ever reater complexity until the specification of all of them did constrain the system

distributions of statistical mechanics are the finest representation of the actua state 0 t e in ivi ua system imagina e. Fixin a s stem in a macrosco ic state would then determine transition probabilities to other macroscopic states, These would be determined

ensemble evolution), would determine the much less likely transitions we cou expect m e orm 0 uctuatlona p enomena. 0 e attrl ution of a robabili distribution to the individual s stem of as in the left-hand side of the box would tell us both to expect the gas to be in

further from equilibrium macroscopic condition as a possibility and would te us Wlt w at pro a llty to expect suc a rare occurrence. There is even, in statistical mechanics, some thing like the Pro'ection Postulate of quantum mechanics. This is the "rerandomization" posit used

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in macroscopic dynamical equations of evolution. These macroscopic equations have the further evolution of a system depend in their nature onIon the s stem's macrosco ic condition at a iven time ast histo being irrelevant. In the statistical mechanical analogue, the rerandomization

, probability distribution to the new "observed quantity," and throwing away any information about how that quantity arose. Of course, the analo is onl that. In uantum mechanics the ro'ection occurs onl when the system is "observed," a very problematic notion with which

strated star-unitary transformation to a new ensemble representation obeying a Markovian equation or y some 0 er means is taken as Ie itimate even when the s stem is "unobserved" in an reasonable sense. Despite these analogies, the disanalogies of probability distributions in

These are the so-called "No Hidden Variables" proofs. In statlstlca mec anlcs, we ave, lnstea, Ul t lnto t e very rna ematics from which the robabili distributions are derived the underlying phase-space with its pointlike representatives of exact micro-states.

theory misrepresent the physical reality - the claim, for example, that e systems, x or ecause we cannot, ecause 0 t e mst~ llty 0 ourselves the exact oint state of a s stem that would uarantee some determinate future trajectory for it, the positing of such deterministic

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e spin-ec 0 experunen an re a e p enomena ere, e resu s 0 become crucial. Granted, such phenomena are rare in the world and that the preparation of the peculiar reversed states is a trick requiring an fact that the initial non-equilibrium macroscopic state can be reobtained p in i x i r - yn e sys m exis e a a me. Or at least, it shows that if a region of dispersion around a point state was the physical realization of the system, it was a dispersion much smaller than the s read-out robabili distribution that statistical m chanics would attribute to a system having the macroscopic character

invoked by statistical mechanics as representing the individual states of such systems seems pretty implausible. We will return to this question once a ain in Cha ter . Improbable as a rendition of the ontology of the situation, and hence

lawlike nature of the results of thermodynamics does seem to require of us, as Krylov emphasized, some understanding of just what the "in rinci Ie" limits on re aration of collections of s stems must be. Some kind of "irreducibility" of the statistical account does seem to be implied

noted earlier, there are good reasons for thinking that such singular states are neit er necessary or t e t eory nor su cient or a comp ete theo . The fundamental roblems as alwa s are the re roduction in the theory of actual finite time behavior and the explanation of the

probability distributions to break finite time-symmetry, even if all nonsmgu ar ensem es e ave on a par as tune goes to p us an minus infini . And even if we do invoke sin lar ensembles, we still need an account of why those need be invoked that give the right asymmetry in

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'n

tion" required to prepare the anti-thermodynamic singular ensembles begs the question in the definition of entropy chosen.

3. Initial conditions and symmetry-breaking of rationalizing the kinetic equations of statistical mechanics. Exactly how dYnamics and idealizations are to function in a derivation of the approriate statistical-mechanical ensemble evolution that is to re resent the approach to equilibrium is still a matter of grave controversy.

a posited initial probability distribution is essential to eliminate the objectionable reran omization posit use in ear ier erivations 0 t e inetic e uations. Even when rerandomization is obtained from d namics alone initial probability distributions are still crucial in obtaining the short-term

again, it is important to emphasize that it is not the fact that entropy increases in t e ture time irection t at is at issue ere. at mig t e obtained ultimatel b some version of Boltzmann's osit that our ve concept of the future (as opposed to past) time direction is itself grounded

parallelism presupposed by the Boltzmann argument, that can only be o tame y a POS1t a out appropnate ffiltla ensem es t at 1S not matc e b an corres ondin osit about "final" ensembles. Or, more accurate1 we need a posit that systems all are appropriately represented by one of

systems. e ave seen severa approac es to ratlona 1zlng 1S POS1t 0 tlmeas mmetric bounda conditions for ensemble evolution. One roup of approaches relies upon a notion of preparation. It is in how systems are

appropriate initial ensembles (covering a large enough region of phase space, e1ng oun e 1n a su C1ent y uncomp ex or u n ate way, having a uniform probability distribution over this largish, simple region) is to be found. As we have seen, this proposal suffers from being, for the

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seriously the radical distinction in the conceptual basis associated with the underlying dynamics between laws and initial conditions. This aproach enerall resu oses that initial conditions are totall unrestricted by any lawlike constraints, and argues that the time-asYmmetric feature

by Reichenbach, Griinbaum, and others. But, following Krylov, we have seen that this account, or, perhaps, this counsel of despair, is afflicted with man difficulties. Most im ortant is its failure to round the a arent "lawlike" status of the posited aSYmmetry. No one is claiming, of

ence of fluctuational phenomena for individual systems. But the status of t e statistica surrogate at rep aces at aw in our groun ing 0 it in statistical mechanics seems too universal too ervasive and too inviolable to be simply dismissed as a grand cosmic accidental congery of

of the view as usually formulated. If it is taken to be a fact about all macroscopica y c aracteriza e systems at e "rea ensem e" s a uniform robabili distribution a ro riate to it we soon obtain the conclusion that all systems ought to be inferred to be in equilibrium. For

has remained isolated from the world, the posit of uniformity of proba 11ty over t e p ase space a owe y 1ts ater macroscop1C con 1hon is sim 1 inconsistent with that osit a lied to it at the initial time. This result is, of course, just one more application of the invariance of fine-

of time itself that accounts for the aSYmmetry of systems. The trouble ere 1S t at t 1S approac as one c ear meanmg an one vague one. e clear meanin is the osit that there are fundamental laws of nature that are time-asymmetric, and that this lawlike time-asymmetry underlies the

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haps) - in the theory of elementary particle interactions - seems to be grossly inadequate to account for the familiar pervasive asymmetries we are in to account for. The va uer inte retation of the a roach argues that something about the asymmetry of "time itself" expresses itself

is that it is far from clear what such an asymmetry of time itself is construed to be. It is even more unclear how it would be supposed to ovem the initial conditions of s stems. The roblem here is that the underlying dynamical theory seems to tell us that only the laws and

of the system from the outside, for any other factor. If there were some such "asymmetry in time itse ," ow wou it fin room to pay a ro e in the d namical evolution of s stems? But there is one other approach to examine in some detail. Here, the " " approach to time asymmetry in statistical mechanics, and to it we tum in C apter 8.

Early investigations into the generalized master equation and the technlques 0 su - ynamlcs are a aJ1ffia an wanzlg an (1964). Later work can be found in Pri ogine and Grecos (1972). Jancel (1969), Part II, Chapter VI, Section V, summarizes the approach. A lengthy

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tion to the technical aspects. Devaney (1986) is a superb treatment at the

especially Chapters 5 and 6. A brief discussion is Sklar (1974), Chapter

(1983),

Krylov's critique of the traditional approaches as well as his "preparation" analysis of initial ensembles is in Krylov (1979). For two critiques, see Sklar 1 8 and Batterman 1 O. Prigogine's introduction of singular initial conditions can be found in

An alternative rationale for non-equilibrium theory not discussed in this book is found in Hollinger and Zenzen (1985).

8

I. The invocation of cosmolo cal considerations

The late nineteenth century saw two attempts in science to invoke the previously been thought to be purely "local" in their nature and hence One of these novel approaches to explanation was Mach's introduction of the overall structure of the universe as acornponent of his roposed solution to the problem of absolute acceleration in Newtonian non-inertial motion, a physical difference revealed by the presence in the the case of inertial motion, by positing that genuine accelerations were accelerations relative to "space itself." Later, related theories posit the inertial frames of space-time, in the case of neo-Newtonianism or the structure of space-time, in the case of general relativity, as the reference found the postulation of "space itself" methodologically illegitimate, and sou t or an exp anation 0 t e asymmetry etween inertia an noninertial motion in some theory that would involve only the relative motions Mach found his proposed explanation of the inertial effects in the

generated were higWy independent of the distance between the relatively

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with that determined visually by its relative motion with respect to the "fixed stars." It will not be to our oint here of course to follow out the Ion and fascinating career Mach's speculations have had in the space-time

a foundational scientific difficulty by introducing cosmological facts as surprisingly relevant to local phenomena. Much more relevant for our ur oses is the fascinatin if briefl sketched, set of proposals made by Boltzmann and noted by us in

existence of the equilibrium state - the attractor state to which all systems converge an t at is itse sta e over time - y c aiming that state is the "overwhelmin I most robable" combination state for a s stem the state obtained by far more permutations of the micro-components than

improbable on his terms? His suggestion was t at t e universe as a woe ts in equii flum. Given the universe's vastness in s ace and its vast duration in time the theory of fluctuations would imply that we ought to expect large spatio-

ingly dominate the non-equilibrium, why do we find ourselves in a portion o t e universe ar rom equii flum. ou n t it e muc more pro a e that we would exist in an e uilibrium re ion? No. Here Boltzmann invokes a "selection by the observer" argument. This kind of argument is

regions, for an observer must be a fairly complex, stable, macroscopic system. uc systems cou on y e sustaine over time y energy ows enerated b the existence of local non-e uilibrium. Sometimes it is said that Boltzmann's argument "explains why our

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n

evitably be found by us to be the case in our vicinity. But it is not, of course, some substitute for a causal explanation of why this region of non-e uilibrium exists. The statistical-causal ex lanation for the re ions of non-equilibrium existing, and, hence for the possibility of observers

vastness of the system that leads to high probability for regions of great size to obtain entropies far below maximal values for extended periods of time. It is sometimes objected to Boltzmann that because a much smaller

his argument as trying to do more that it intends. We can show that the conditional probability of our existing in a local universe such as the one we ex erience is much hi her than the conditional robabili of our existing in an equilibrium region of the universe, for that latter probability

than the probability of a region like this. And the probability that sentient eings wi exist in more mo est uctuationa regions t an ours is on y somewhat smaller than the robabili of sentient bein s in a re ion like ours. So it is reasonable to suppose that the probability, given that a

entropy of our region, is substantially higher than the probability, given sentience, t at e region is e ours. But the Boltzmannian had a reasonable re 1 . The universe is filled with fluctuational regions. Many have less divergence from equilibrium

kind that is more common than other even more extended and lower entropy uctuatlona regions. ur actua eXistence as y is argument been placed in a statistical pattern that is compatible with the alleged actual probabilities of situations in the world. And this is one version,

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the future direction is grounded in entropic phenomena, a claim we will discuss in Chapter 10. The ma'or contem ora ob'ection to Boltzmann's account is its a arent failure to do justice to the observational facts. As far as our observa-

direction of entropic increase of systems toward what we intuitively take to be the future time direction that we encounter in our local world seems to hold throu hout the universe. uite a different icture of the overall structure of the cosmos now has overwhelming acceptance from

increased with their deviation from the equilibrium state. But, as we shall see in Section 8,11,2, some e ements 0 a Bo tzmann-' e picture 0 rea ear in some s eculative cosmolo ies still under consideration.

2. Big Bang cosmologies Relying on a combination of important observational data and systematic eorizing, a view 0 e overa structure 0 t e universe or, as we s a see in Section 811,2, in some accounts 'ust of our "com onent" of it) has become orthodox in contemporary cosmology. The picture of the

Observationally, we discover that the distant galaxies seem to have t e 19 t we receive rom em re -s te in an amount at m lcates a veloci of recession of them from our ala . Such an effect is 'ust what would be expected from a uniform expansion of the matter of the

time, we see what would appear to be convergence of all the matter to a smg e pOint in a hme peno 0 some tens 0 1 ions 0 years. t certainl looks as if all the matter of the universe is expanding outward from an initial singular pointlike concentration in what we call the past

Cosmology and irreversibility

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"cosmological principle" assumes that each point in a spacelike slice of the universe is (in a smoothed-out sense) like any other, and that all s atial directions are alike the erfect cosmolo ical rinci Ie assumes that the universe is structurally alike at every moment of time as well. ,

,

........u ' V ..... j;;, ......

such simple worlds as rotating universes would be incompatible with it. The perfect cosmological principle is the foundation on which steadystate models of the universe were built and if we can trust the data of the Hubble red-shift and the standard interpretation of it, that cosmological

count of the universe's dynamics as well. General relativity, like Newtonian gravitationa eory e ore it, pictures gravity as a tota y attractive orce. Because all the matter in the universe is bein attracted to all the other matter by the gravitational attraction, a stationary model of the cosmos

cosmological models within general relativity in the form of the cosmo ogica constant term 0 t e gravitationa e equations. But e was deli hted to be able to dro the term when the d amic model became observationally preferred. Crudely, the various dynamical mod-

down. If thrown fast enough, it might "go off to infinity" with its velocity never s owmg to zero. r it mig t, exceptiona y, e at Just t e transition between these two ossibilities if its kinetic and otential ener ies 'ust balanced at the moment of its release.

theoretical t1xploration ought to begin. Assume that one can partition space-time into spaces at a time, so t at events can e a e e y a global "cosmic time." Assume that the spaces at each time are locally isotropic. Assume that they are homogeneous as well. Then the worlds

+1, -1, or 0; and by a scalar parameter that varies with time, R(t). The

case correspon 0 a universe a as c ose ni e vo ume spaces at a time. These are three-spheres of constant curvature. These universes are the analogues of the ball that returns to earth. After a

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universe at a time, going from zero at the initial singularity, reaching a maximum and returning to zero in the k = +1 case, and going off to infini in the other two cases. ust how R will va with time de ends on how the universe is constructed - whether, for example, it is filled

by quantum field theory, and so on, The existence of an initial singularity in these models - of an initial state in which the universe was concentrated in a oint where all enuine physical quantities became divergent, including the scalar curvature

choice of an idealized perfect symmetry. More recent work has shown, owever, t at any wor 0 eying genera re ativity at as a converence like that of the universe of the Robertson-Walker models backward in time, must converge to a singularity, whether that universe is sym-

incorrect in the very small anyway, and that a proper treatment of the 1nitia singu anty must e in terms 0 a quantum state 0 space-time wit its manifold of evolutiona ossibilities overned onl b robabili distributions. We will have a little more to say about such matters as this

as homogeneous and isotropic in the spaces at those times and as exist1ng ill t erma equ11 num. orne 0 e consequences 0 a POS1t 0 1S kind have been marvelously confirmed by contemporary observational data. The relative abundance of the elements that would be formed in

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that matter becomes mostly neutral and radiation becomes decoupled from matter, the universe should be filled isotropically with radiation. As the universe ex ands lowerin the fre uen of this radiation and increasing its wavelength, this radiation should now be apparent to us as

ard model would lead us to expect that the observable matter of the universe at present should, if we use a large enough scale, be found to be homo eneousl and isotro icall distributed in s ace. Althou h there are now some doubts about this due to the discovery of non-uniformity

empty space devoid of matter, there is some reason to think that the expecte un' ormity in t e arge can sti e accepte . How would thermod amics and statistical mechanics a I to the description of such a model universe? To that we turn in the next

this can be handled by dealing with finite portions of the universe of arger an arger Size, eac contalnmg itS sma er pre ecessor in e se uence, and takin the behavior of the whole as the limit of the behavior of the finite sub-systems. Or it can be handled by using one of the

numbers of degrees of freedom. More problematic, as we shall see in ectlon , , , is ta mg account 0 entropy resl ent not in matter ut in the vitational field - that is, space-time itself. Why does the universe as a whole have its entropy increase in the forward time direction?

ordinary thermal processes, expansion at a finite non-zero rate (say, of a mo ecu ar gas m a con alner oun e in pa y a mova e piS on is accompanied by an entropic increase. Couldn't the situation be the same with the universe as a whole? First, it is important to notice a crucial

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us an a universe e pure y wi ra ia ion e nonzero rate with no entropic increase whatever. If the universe has both dust and radiation, it is true that non-quasi'n

may respond to the expansion differentially in such a way that a joint x

v v

kind of heat flows and other relaxation processes that normally result in entropic increase of a system. Could this then be the ground of the universe's entro ic increase in the sense that an ex andin universe must have its entropy increase?

increased in the expansion - basically, that the usual Second Law behavior holds - tells us that entropy will increase in the recontraction stage of cosmic evolution as well 'ust as the entro of a molecular as increases under non-quasi-static compression. Because we are, as usual, assuming

expansion and recontraction, would be a universe that had an expansion wi entropic ecrease 0 owe y a recontraction a so wit entropic decrease. If such a universe is lawlike allowed then ex ansion b itself cannot be the origin of entropic increase.

that at the moment of termination of expansion and initiation of recontraction t e entropy egan to ecrease instea 0 increase. Now we know that such astoundin correlations can be induced in some ways. They show up in the spin-echo experiment, for example. But there

component of the system is acted upon directly by the reflecting radio pu se. e a sence 0 suc a mec anlsm ln t e case COnsl ere ere is clear. How do the micro-constituents of the universe "know" that the turning point of expansion-contraction has occurred? How could such

Cosmology and irreversibility

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plausible to associate a strict necessary connection between expansion and entropic increase, the possibility of a model of this kind can't be i iv I dismissed. As we shall see in Sections 8 II 1 and 2 an "improbable" condition must be imposed somewhere in order to get the

the kind we shall explore in Sections 8,11,1 and 2. These latter depend on an initial constraint on the system characterizable in a simple, local, macrosco ic wa . Low entro is" ut in" to the s stem much as we are accustomed to putting it into temporarily isolated sub-systems of the

trived n-particle correlation function (where n may in fact be infinitely large) at all space-time points simultaneously, that makes the posit of a be ins to decrease at the moment of cosmic universe whose entro expansion tum-around so implausible. But nothing in physical law makes

, stars will still radiate heat to outer space, cream will still stir into coffee an not out, an , in genera, entropic increase wi continue in t e time direction we now call the future time direction and not in the antiparallel direction posited by the account just discussed. If expansion by

A brief look at the irreversible behavior of radiation in the cosmolo ical context is in order at this point. Once again the claim may be made that

coherent inbound radiation spontaneously inducing an acceleration on a receIvmg partic e, In Icates a un amenta aw I e Irreversl Iity In t e nature of radiation. We have seen how, in the case of radiation confined to a bounded environment, the solution accepted by most is that the

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more complicated, but we shall ignore the refinements needed. "' "' r n cc 'c g orev r, never being absorbed. But we know not to expect such coherent radiation coming in from infinity. It is sometimes argued that the existence of such inbound coherent radiation can be dismissed on dee er rounds than the mere resort to an asymmetry of initial conditions. But it is hard to see

coherent radiation coming in from past-forever to accelerate a receiving particle, are on a par. Of course, we can make the former situation occur b a local action of causin a article to accelerate in a situation where some of the radiation it emits may be unabsorbed for all time. But we

spacelike domain. But this is just the familiar fact about initial conditions in our world that we experience in ordinary statistical mechanics. Barring s in-echo-like tricks we can induce future delicate correlations in article motions by a local, macroscopic manipulation of the system at an

from macroscopic equilibrium to macroscopic non-equilibrium. It isn't c ear w y t e greater spatia scope 0 t e con itions nee e to set up inbound coherent radiation either from time minus infini or over a finite spatial region from a finite past time, differs from the familiar

tion or the receiver of it) will do the job of selecting outbound coherent ra iatlon as a owe an moun as or 1 en. se -consistent so ution in which onI outbound coherent radiation a ears follows from the universe being constituted as a "perfect absorber" of the outbound co-

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it receives the radiation. An interesting variant is obtained if one drops radiation as a substantial field existin in its own ri ht as is done in "action-at-a-distance" electrodynamics introduced to handle some of the problems of field theory that

with relativistic constraints one needs the action of particle 1 on particle 2 to be "retarded" by what, in the standard theory, is the propagation time of the wave from the one to the other. The Wheeler-Fe nman version of this theory posits both retarded and advanced action of each

interaction is explained away in terms of an asymmetry in the initial conditions 0 t e "a sor er," t e u 0 t e universe at arge with which the test article is interactin . Once a ain one can obtain for example, damping of the accelerated motion of a charged particle. This

and retarded interactions of the particle with the rest of the particles in t e universe. But one can a so 0 tain tea sence 0 anti- amping t at would s ontaneousl ut unaccelerated char ed articles into accelerated motion. Ingenious improvements in the arguments allow for self-

symmetric retarded-plus-advanced direct interaction view even if, from t e e pOlnt 0 Vlew, t e unlverse IS transparent an some ra latlon is allowed to "esca e to infini ." Let us assume, then, that the source of temporal asymmetry on the

asymmetry is to be accounted for along with all the others by a general asymmetry 0 t e type amllar to us rom statlstlca nlec anlCS. nce again, we need to find some appropriate ground for this asymmetry in the cosmos as a whole. D. Conditions at the initial singularity

1. Initial low entropy A direction in which to explore when seeking the explanation of the

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is correc , ce e course 0 e ac a we can, curren cosmo 0 the universe back to a localized initial state, the initial singularity and the "small" universe developing immediately from it. Can we find a way of c rizin n' I th rl of reasoning familiar from the thermodynamics and statistical mechanics

Ordinary systems show entropic increase when they are started in lowentropy initial conditions. The gas is confined on the left-hand side of the box and the artition is removed. Given the usual assum tion with which we are now so familiar, that the micro-state of the gas at that initial

more non-equilibrium condition, we expect immediate entropic increase of the gas. That is, we expect the gas to flow toward a condition of more uniform densi . Makin the same ar ment at each further time invokin the rerandomization posit), we expect a monotonic increase in the system's

At first glance, the usual description of the cosmos just after the Big ang is unprom1sing rom t 1S perspective. For matter 1S usua y escribed as bein in a thermal e uilibrium state in the earl sta es of the universe. Perhaps the expansion combined with a multiplicity of phases

another source of low entropy is far more important. The early universe 1S spat1a y omogeneous an 1sotrop1c. ut suc a un1 orm 1stn utIon in s ace-time is actuall a low-entro state when ravi is taken into account. The purely attractive force of gravity is such that such a uniform

, clumping takes place (into superclusters of galaxies, into galaxies, into stars , m w at or er 1t c umps 19 or sma er structures rst, w at t e fluctuational structure of the early universe is that generates the clumping by a positive feedback effect in which slightly clumped matter gravita-

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a low-entropy smooth space-time state to a clumpy distribution of matter, and, hence, of local space-time curvature, takes place. It is this clumping that rovides the low entro necessa to ather matter ori inall in thermal equilibrium, into the radically uneven distribution we experience

The evolution of a universe that expanded and then recontracted would follow a pattern of entropic increase in both p ses of its evolution from this ers ective. In the ex ansiona hase matter moves from its initial thermalized state to one of thermal non-equilibrium, the transition being

radiate into cold space, adding more entropy to the universe in the process. T e processes 0 oca c umping 0 owe y t erma ra iation continue even after the ex ansion has reached its maximum and the universe has begun to recontract. In its final stages, just prior to the "Big

entropy by a vast amount, the entropic increase of the universe over the two-p se process. lone is un appy In attri uting an entropy to t e two sin larities themselves one can deal of course with the se uence of nearby non-singular universes just after the Big Bang and just before

evolution being time-reversal invariant. The time-reverse picture of the unIverse IS 0 one startmg m a 19 ang t at lffiffie late y goes to a new universe of extraordinaril hi h entro . As it evolves, once a ain expanding and contracting to a Big Crunch, entropy steadily decreases.

fect uniformity as a limit as it nears its final singularity. In this picture, as In t e stan ar vIew 0 empora asymme ry In n1te systems, It IS t e special nature of the initial condition, the low entropy of the Big Bang, that determines the actual entropic course of events, not some time-

2. Accountin

or the initial low-entro

state

Suppose the pervasive entropic increase of the universe is then attributed to the special nature of the Big Bang - that is, its being characterized as

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initial state has the features it has. wp n n. a, W entropy is identified in the statistical mechanical picture with "improbability." Why should such an improbable initial state exist? The idea here is to ima ine the collection of "all ossible initial states of the universe " all possible initial conditions for space-time and matter in a universe U .., ......~Jl.....

1"t

W

would give the class of worlds like ours that have such "unusual" initial states in small measure. Why, then, should such an improbable initial state have occurred in our world? This "improbable" condition of the initial state is often associated with

domains of causal influence, causal propagation being assumed to be imited y t e spee 0 ig t as maxima y ast causal signal, even when we trace the re ions backward in time toward the initial sin lari . Although distant galaxies get closer together backward in time, the amount

assuming that the rate of expansion we now experience can be extrapoate ac ar in time in t e stan ar manner a t e way to t e Big Ban we find that the universe has a structure in which ortions of the world remain forever causally unconnectible no matter how far back in

positions at early enough times? If the regions were in causal connection, some causa process m1g t e 1nvo e to re ax non-un orm port1ons into the eneral uniformi , but if the re ions are immune to causal contact as far back as we go, how does one region "know" the condi?

Other "improbabilities" are noted as well. If we try to determine whether our un1verse 1S c ose ,open an W1t negative curvature, or open ut at the bounding condition of having its s aces at a time flat, we discover that it is at least fairly near to flatness. Scaling conditions on the features

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just as it is improbable that a ball tossed up in the air with a "random" impulse will have exactly the kinetic energy needed to achieve escape veloci no more or no less. What accounts for this su risin 1 improbable condition holding in the world? " " with pitfalls for the unwary. The exploration of the sort of questions confronting us here hardly only began with the discovery of the Big Ban and its somewhat su risin features thou h. Consider the most obvious question of all about the initial state of the

ing from the invocation of a deity as "uncaused cause" of the universe to specu ations that t e universe mig tea "quantum uctuation" out 0 "nothin "to such amusin observations as that because there are so many ways a universe can be, but only one way in which nothing can

ing that any coherent question can be asked here, much less answered. I we ta e it t t t e on y exp anation we can give or t e existence 0 somethin is a causal ex lanation accountin for its existence b reference to the existence and nature of other ordinary objects and their features

ordinary world, then we might argue that the search for the explanation o t e eX1stence 0 everyt ng 1S Slmp y an 1 eg1t1mate extens10n 0 t e ve notion of what can be ex lained. Kant, for example, in a persuasive series of arguments in the Critique

theory in which causal reasoning is applicable to the world only because causa structure 1S 1mpose on t e wor y t e m1n 0 t e su Ject to whom the world is given as a world of experience, the general structure of his account will appeal to many who reject his views of causation as " thing has a cause, we are driven inexorably to ask the ultimate cause of eve ng. u suc a eman 1S u u a e. or causes are a ways 1n the world" demanding for themselves further causal explanations. The very idea of the "First Cause" of everything, whether called "God" or

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Paralleling arguments to the existenc;;e of God as uncaused first cause

the order of the world as well- so-called Teleological Arguments. Usually these were based on the existence of the order of lawlike regularity in the world. 1m ressed with the s stem and order of the world summarized by the laws of nature, it was asked how such an order could exist

watchmaker? Here it is the "improbability," a priori, of there being any lawlike order at all to the cosmos that is taken as requiring an explanation for the order that we do find and in the traditional version as implying an orderer.

it is frequently taken as a given. It is now the existence in the early universe 0 t e "impro a e" con itions suc as t e ow entropy 0 a uniform s ace-time or the near flatness of the s ace-time that are taken to require an explanatory account. Here it is argued that even given the

from an a priori point of view. Hence, some explanation is required as to w Y t is con ition, an not one 0 tear more "pro a e" ones we would ex ect actuall obtains. But of course there have been many who, in the context of critiques

phenomena in the world to a transcendent explanation of the basic p enomenon 0 t e wor 1tse t e c a1m 1S t at 1t 1S 1mpro a e initial states that cry out for ex lanation, the critic will fre uently den the very propriety of attributing "probabilities" to these initial or overall

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talk about the probability of a universe is, from this point of view, incoherent. For there is no "real collection" of systems from which this universe is "chosen" that can serve as the needed reference class relative to which a probability can be grounded. As Mach said, in a somewhat " "

, the universe's order as some may be, others have offered "explanations" of why the improbable initia state 0 the universe is as we have found it. Some have resorted althou h usuall in a sufficientl evasive wa as to avoid direct criticism, to positing a "benevolent" creator in the familiar

effect that the improbable initial state is a necessary condition for the u timate existence in t e universe 0 sentient i e, t ere y imp ying t at the Desi ner was desi nin with us in mind as Its(?) ultimate aim. Of course, why that ultimate aim might not have been the existence of

ingenious and worth noting. One approac is to argue at re atlve to certaln spec e constraints, such as the standard laws of nature holdin and the total ener of the universe being like that of this world (or things of that sort, there being

over the class of possible universes obeying these constraints, that there e a great entropy terence etween one en 0 e unlverse an t e other. This ar ment takes its most natural form in the case of a universe that has expansion and then recontraction from and to singularities, but

entropy will be greatly different at the singularity end and at the limit as tlme goes to 1 nlty In t e non-smgu ar en . The ex lanation as to why entropy is low in the past direction and high in the future is then accounted for in the manner of Boltzmann's

chosen there will be as many universes with high entropy at the initial slngu an an ow a e na as e 0 er way aroun ,or e ormer are just the time-reverses of the latter, and the collection of time-reverses of an ensemble of systems (even whole universes) is of the same natural

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the philosophers put it) taking the notion of what is future time in one "'........ ~....,y n ses in a given time direction and in the other half it increases in the other time direction. But in each world, observers will say that entropy increases in the future time direction. This Boltzmann "way out" will of course have its critics. One objection

entropic difference, or the degree of low entropy at one singularity end, that we find in this world. If we take it that the Big Bang really has the " erfect" s ace-time smoothness credited it in man familiar models can that really be said to be "probable" in a standard measure over the class

Some solutions look to taking the class of imagined universes just discusse an viewing em as a "equa y rea ." Or the proposal might even 0 further and allow for a broader class of existin universes including those with constraints in such form as total mass, and so on, that

of this approach involves quantum mechanics, and the curious attempt e "Many to reso ve its so-ca e "measurement pro em" t at is ca e Worlds" inte retation of uantum theo . In this understandin of uantum mechanics, an understanding that is fraught with conceptual prob-

manifold of alternative universes. Could it not be that most of these unlverses ave a more pro a e nature, ln some natura pro a 1 ity measure, than ours, and that the universe we find ourselves in with its puzzling low overall entropy, is merely an exceptional "branch" or " Here the analogy with Boltzmann's "way out" is clear. Although he poslte t at our oca regl0n 0 t e unlverse lS a rare ow-entropy overall e uilibrium, now our fluctuation in a vast sea of hi h-entro entire universe is taken to be a rare exceptional case in a sea of co-

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the universe we find ourselves in is of low entropy, it is argued, for only in such universes does the structure of the universe allow for the evolution of sentient bein s. Nor should we wonder about the articular time direction entropic increase takes in our particular universe, for, once

much more probable universe would be enough to sustain sentience, and that the probability that the universe would be as exceptional as it is even conditional to the existence of sentience is low often a ears as an objection to this account. Needless to say, the whole model of a

understanding of the world. One variant 0 e approac just escri e is particu ar y interesting here. We have remarked that the initial sin ulari 's uniformi is uzzling given the causal unconnectibility of regions of the space-time for all

of these puzzles simultaneously is found in the so-called "inflationary" cosmo ogles. Here, eatures 0 quantum e t eory. are lnvo e to give a characterization of a universe that, startin from a sin ulari ,has a period of slow expansion (or even stasis) in which a vast store of latent

"jolt" to the universe. This occurs in a manner analogous to the fast crysta lzatlon t at occurs w en a supercoo e UI na y as Its tranSItion to a solid that has "held off" become tri ered, sa b the introduction of a seed crystal. When the phase transition is finally completed,

in which causal contact could have occurred throughout the spatially e space-time an wou a so account or t e near separate regIons 0 flatness of the currently observed universe without assuming the almost perfect flatness of the universe near the singularity of the standard

". of which constituted its own observationally closed universe. Once again, one cou ry 0 an e e ow-en ropy pro em or our wor y some combination of Boltzmannian arguments of the kind noted. Once again, the familiar objections would hold.

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this initial irregularity is smoothed out over the course of time by a ao i an e n n is appearance of uniformity, the absence of chaos, is due to the usual process by which irregularities are smoothed out in thermalization. But as R. Penrose has em hasized with reat clari this would lead us to expect a large entropy of the matter fields. The entropy of matter

tum field theory, by the ratio of the number of photons to the number of baryons in the universe, a ratio estimated at being something like a billion to one. Sometimes this is said to be alar e value so erha s the appearance of low entropy for the initial space-time is misleading. Per-

small given the entropy contained in space-time itself. In whatever way we ook at t e cosmo ogica situation, we must admit that we have entro increase in the evolution of the world. When we consider how high the space-time entropy could be in the Big Crunch at

compared with the vastly higher entropy at the Big Crunch, and that t e existent present entropy 0 matter cannot e enoug to a ow us to et awa with a osit of initiall hi h entro ic s ace-time. Basicall the problem is that if we wish to account for the evolution of entropy

of finite systems, we must posit for the universe the same kind of initial ow entropy in t e orm 0 a macroscopiC constraint 1 e e gas a bein on the left-hand side of the box). And we must osit a hi h probability micro-state underlying the macro-state to avoid anomalous

"improbable" uniformity of the spatial structure at early times. ina y, one cou ry 0 0 Via e e appearance 0 ow pro a 11 for the initial condition by positing constraints in the form of laws of nature such that, relative to these constraints, the condition no longer

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of lawlike constraints and hence make the condition we actually do find a probable or even inevitable one. If this law is to permit the much hi her s atial entro ex ected at the Bi Crunch of a universe that cycles back toward a singularity, though, it must have a time-asymmetry

Just such a proposal has been made by R. Penrose. He suggests a new law of nature governing space-time singularities. The particular one he su ests is that Bi Ban - e sin ularities be overned b the condition that their conformal space-time structure, described by the Weyl tensor

problematic time-reversals of the familiar black holes predicted by genera re atlvlty. n ee , enrose suspects at ey are lmpOSSl e. 0 ow methodolo icall reasonable is it to invoke such a "law?" Or does the posit simply come down to a posit about the initial structure of space-

as given and simply invokes a time-reversal non-invariant law. In these respects It IS rat er a enla 0 t e genera program or n lng ways out of the statistical mechanical paradoxes that Boltzmann suggested in his replies to his critics and that we have seen repeated in metamorphized

The profound questions on the very possibility of explanations of what are POSI e as cosmo oglca or aSlC con 1 Ions 0 e unIverse a ground all normal explanations is one crying out for deeper exploration. This need is made crucial by the recent tendency of practicing science

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reasoning in non-equilibrium theory as a component of a time-symmetric theory to probabilistic reasoning as explaining the origin of the notorious thermod namic as mmetric behavior of s stems in the world?

m. Branch sterns

In whatever wa we are to account for the observed obal entro ic increase of the universe we observe, we must accept that such an in-

want to account for in statistical mechanics, the Second Law behavior o "sma ," in ivi ua systems? Can we get e asymmetric erma time behavior of s stems that has so far eluded us out of the iven as mmetric time behavior of the universe? That the introduction of this global asym-

the laws. This symmetry could only be broken in the earlier accounts by "putting in a istinctive pro a i ity istri ution over micro-states 0 e s stem at the "initial" state and never at the "final" state of the s stem. Those who hope to find the key to irreversibility in cosmology often

in splendid energetic isolation from past time infinity to future time inn1ty. ereas conS1 eration 0 suc permanen y 180 ate systems may be useful for some u oses - studies of lobal stabili of tra'ectories or investigations into the Recurrence Theorem, for example - perhaps the

energetically isolating a piece of the energetically interacting universal woe rom 1ts surroun 1ngs, an en ett1ng e system eXlst m energetic isolation for some finite eriod of time. During this time, the s stem's thermal behavior is watched. At the end of this period, the system's

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we use the facts about cosmological entropy increase and the fact that systems studied in thermodynamics are branch systems to account for the irreversibili whose ex lanation has reviousl eluded us? Now, one might try to prove one of three different things using cos-

intuitive future direction of time. Next, one might try to show that the overwhelming probability is of entropic increase in branch systems in that direction of time in which the entro of the universe as a whole is increasing. Finally, one might only try to show that it is overwhelmingly

the intuitive future time direction or the direction of time in which the entropy of the global system increases or not. Those who seek a demonstration of the latter two results usuall t to extract the former as well by an application of Boltzmann's "argument by definition" - that is, by

whelming probability entropy of systems increases. We start wi teo owing acts: 1 e systems wit w ic we ea in thermod namics are branch s stems. The live in ener etic isolation from the bulk of the universe only for finite periods of time, being in

verse, or at least of our region of it, is extraordinarily low (note that this IS true ot at e egmnIng 0 a ranc system s eXIstence an at ItS end). (3) The entro of our local re ion of the universe is increasin in one time direction. The hope is that we can use these facts to derive one

we now turn.

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the direction of entropic change of the main system from which the in opposite directions_ of time in some spatial portions of the universe at a given time, or at different times even in the same place, and that such a difference of direction in time of increase of entro of the main stem will result in counterdirected thermal behavior in those regions of the y

Reichenbach says, with justice, that Boltzmann just assumed that the branch systems would (in the probabilistic sense) have their entropy increase in time arallel to each other and arallel to that of the main system. But, he claims, again with justification, that to infer this requires

Reichenbach is mainly concerned with the claim, to be discussed in this book in C apter 10, t at our intuitive notion 0 the past- uture distinction can be ounded in the distinction between the two directions of time that in which entropy decreases and that in which it increases. So if

will serve this ultimate Reichenbachian purpose. This is the direction in w .c t e ranc systems ave t eir entropies increase para e to one another. But of course it leaves unex lained wh the s stems behave probabilisitically, in parallel to one another and to the main system,

other assumptions and to this extent "explained." Assumption 1 is just at e unlverse now as ow entropy an lS sltuate on a "s ope 0 the entro curve - on a lace where its entro is hi her in one direction of time and lower in the other. That is, it isn't currently in a local

of their existence, but energetically isolated from it for the duration of elr nlte lves. Assum tions 3 and 4 are more im ortant. Assum tion 4 is that, of the branch systems, the vast majority have one low-entropy endpoint and

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seems unjustified as an assumption. But one can just read Assumption 4 as delimiting those branch systems with which we will be concerned, the ones that suffer a serious entro chan e durin their brief lifes an. The aim is of course to get their changes parallel to one another and to

Assumption 3 is the crucial posit, the one that provides the core of the premises from which the otherwise question-begging Assumption 5 is to be derived. Assum tion is that "the lattice of branch s stems is a lattice of mixture." What does that mean?

its value at small, equal time intervals as the systems evolve from their starting point. Keep trac , or examp e, 0 t eir Bo tzmann indivi ua stem entro values. Make a s uare arra of the values obtained: n

2n

of a at one time in a row to some other value of a at some other time 1n t at row 1S 1n epen ent 0 tea va ues at any com 1natlon 0 t1mes in an combination of other rows. The rows are robabilisticall independent of one another.

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ra er an our previous re erence c ass, w ic was a co umn J pairs in a single row one column apart), a has the value a* in column j + 1 given that it had value a** in column j. Lattice invariance is the ability of a** being immediately followed by a* in a row. Intuitively, it S1 ar n na 11 wi . r w . Y These two conditions define what it is for the lattice to be a lattice of mixture. From them, one can prove, among other things, that no matter how the values of the a's are fixed and distributed amon the left-hand most column of the array, the distribution of a values in the right-hand

over any of the rows. If, then, we started with ergodic systems, almost all of which were at their initial states as represented by a values in we would find that the the left-hand column in states of low entro distribution of entropy values in the final states, as represented by the

terns at high entropy values. From this, Reichenbach argues, we can see that Assumption 5, which ex resses the arallelism in the robabilistic sense of entro chan e of the branch systems with each other and with the direction of change

entropy ends and that the array of their entropy values constitutes a attice 0 mixture. Is e correct? The basic ar ment oes like this: Even if we don't have an reconceived notion of which direction of time is the future direction, we can

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to the joint assertion that individual systems are probabilistically independent of one another, and that the transition probabilities in the ensemble ou ht to du licate those in an individual s stem? What could be more natural? And because the probabilities in the individual system,

how could we have begged the question in favor of parallelism by making the innocent assumption of mixing for the array? To see that the assum tion is not as innocent as it first a ears consider the following "paradox." Once again, consider a vast collection of

direction from its systems 2 end as in every other system. Assume that ha these systems ave tel en at ow entropy. W at oes e missing osit tell us to infer about them? Answer: That the are almost all at hi h entropy at their 2 ends. Now let the other half of the systems have low

collection, now a collection of systems half of which have low and half o w ic ave g entropy at t elr en s, w at oes mixlng ea us to infer? Answer: That almost all have hi h entro at their 2 ends (and similarly with 1 and 2 interchanged).

entropy at 2 with their states lined up starting with the 2 end! The leftan co umn 0 e array Wl conslst 0 a ow-entropy states. !Xlng will then lead us to infer that the right-hand column will consist of nearly all high-entropy states. But now the two groups of systems have their

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same time direction from the right-hand states that a "bias" toward par-

patible with that macro-condition at the initial state of a system, we get predictions confirmed in experience. But if we use the same probabilistic reasonin to a "final " but still low-entro state of a s stem we et the wrong results. Yes, that is true. But what we wanted to know is why the

is, just imposes parallelism' on the systems without explaining it. But, to be fair to Reichenbach, he never really claimed he was explainin it. Yet the a earance that an ex lanation has been iven has I believe, misled many readers. Worse yet, the appearance is that one

On page 138, Reichenbach says this:

He goes on to argue that this probability "wouldn't help us very much" ecause It re ers to very ong time peno s In t e story 0 e unlverse, and that the best reason for thinkin 2-5 true is em irical confirmation of them.

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increase, and they parallel in this the direction of entropic increase of the universe as a whole. But, the question is, does the direction of entropic f f he niverse as a whole e. lain the direction of entro ic increase of the parallel evolving branch systems?

tion 5 posits both parallelism of branch systems and their parallelism to the main system. As we have seen, the parallelism of branch systems of Assum tion is obtained b Reichenbach onI b use of Assum tion (that the array is a lattic of mixture). And, as we have seen, it is obtained

parallelism, then a kind of argument that the evolution of the main system will be in the direction 0 the par el branc systems might simply be that if we assume that it is hi hI robable that an iven s stem will have its entropic increase in the same direction as that of the bulk of

other systems, which is equivalent to their paralleling it, because parallel irection 0 entropic increase is a symmetric re ation. But that hardl constitutes what we thou ht we were oin to be given. What we thought we were going to get was an explanation as to

increase of the main system. What we get is an argument for parallelism at comes own to posItIng It, an a reverse- Irection I erence to t e arallelism of branch and main s stems. One can see what is going on here if one looks at a simplified version

entropy. If we find a branch system in low entropy, it is much more ley to ave een separate rom t e maIn system t an to ave s ontaneousl fluctuated to that low-entro state from a hi her-entro state while isolated. Now, consider a system in a low-entropy initial state,

condition is given in the usual way, we ought to infer with high proba Ilty t at ater states 0 t e sys em WI ave g er entropy. e wou also be impelled to infer the same thing about earlier states of the system if it had any, but having been just created, there are no earlier states of

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seen already that the entropy of these new isolated systems will almost certainly at IS, It Increase, In epen ent 0 our conventIOn a out t e Irectlon 0 tIme. may be asserted confidently that almost all branch systems will show parallel entropy change. It is the asymmetry regarding the formation of branch systems which brings about the parallel increase in all (nearly) branch system entropies. . . . If the branch s stems are re arded as not existin in the ast then the entro of the overwhelming majority of these systems will increase with time. It is through branch s stems that the customa intuitive notion that entro increases with time is derived.

He continues: With these considerations an entropy law can be formulated which is valid even in the case of a totally permanently isolated system. For if there is an ensemble of such s stems which on a articular occasion are in randoml selected states of entropy S«SMAX then from the forgoing arguments it is clearly overwhelmin I robable that all these s stems will be in molecular chaos at an entro minimum, and so all will increase with both greater and lesser t, Le. in both

, occasion (and too widely separated from the first) called S~ and S~, then the

IC ever way It IS C osen to Increase, or 0 c ange In a para e Irectlon as seen. I A > A t en B > S B, ut i time is measured the other way and S~ < sl so S~ < S~ and the product is positive in both cases. This is Schrodinger's formulation of the entropy law, and it reduces to the usual form if A, say, is allowed to become the system of inference, and B becomes the rest of the world. Then if the increasing direction of time is defined as the direction in which the entro of the whole world increases [the roduct Ie ~S ~ ere S = 2 - S1. Th' e I Clausius.

W

of entropy change of the branch system is parallel to that of the outside world

system isolated for a finite time, that its low entropy is a result of its lnteractlon Wit t e main system an not an lmpro a e uctllatlon rom e uilibrium. But, as far as I can see, that is all that cosmolo b itself can really give us.

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of the main system doesn't explain the direction of entropy change of the branch systems. Rather it is just that any two systems being expected to stem ou ht to be ex ected to evolve in arallel evolve in arallel an to the main system. Then the Boltzmann trick is performed of saying that

who would take "future" to be defined, rather, by the parallel direction of entropic increase of branch systems), and we have the usual version of the Second Law. So everything hinges on the "demonstration" of parallelism. Once again,

even if we take for both A and B the 1 end to be in the same time direction from the 2 end for both systems, and we then invoke the usual h othesis of random micro-state at the 1 end taken to be of low entropy, we get the prediction that the 2 end will be of higher entropy, and

1 and 2 ends of the systems in opposite time order from each other, and wit e 1 en again 0 ow entropy, we wou sti, i we posite random micro-states at both 1 ends redict the 2 ends to be of hi her entropy than the 1 ends and we would predict oppositely time-directed

be made at the same time end for all other systems. Once again, suc a POSlt is reasona e 1 we want to escri e t e world correct! . S stems do (in the robabilistic sense) have their entropies increase in a time direction parallel to each other and to the main

To emphasize what has gone on here, consider Davies' argument to tee ect t at we are oc e rom 1 errmg a 19 -entropy past or a stem created in low entro y b the fact that it has no ast. Ve ood. But now consider a system in low entropy about to be reabsorbed into

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y. u ni new u misi as n ure i guided, of course. If it has low entropy in its final state, it almost certainly got there by coming from an even lower-entropy initial state when it n

lifetime of a branch system created and later reabsorbed. It has both a

higher-entropy future. But, of course, the second inference is reasonable and the first isn't. It is Ie itimate to osit randomness of the micro-state for low-entro initial states of systems, but not for their final or intermediate states. Let The principle, which I shall call the principle ofcausality, is simply that the phase-

Indeed, it is notions of causation that we are likely to find invoked at t is point. T e initia ow entropy 0 a branc system is exp ained by its havin been cut off from the main s stem. That is how it was "caused" to be brought into being. Because its state at that initial time then "causes"

tion, reference to a probability distribution over these initial micro-states is in or er in exp aining uture micro-states or pro a i ities 0 tern. But the final state even if of low entro ,can't be ex lained b reference to the forthcoming reabsorption. Nor can the states that led to it be

by a posit of probability over the later possible micro-states of the final ow-entropy ~tate 0 t e system. is is in ee t e rig t way to 0 statistical mechanics. But we were supposed to be avoiding bringing such obviously time-

system, the universe as a whole. I have argued here that in Reichenbach's account to get para e ism 0 time lrection 0 entroplc c nge or ranc s stems, a osit has been made that is close to that sti ulated b O. Penrose. It doesn't talk of past and future, but it does demand that

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parallelism of systems with each other, and hence with the main system as one system among many. But it is wrong to think that these arguments constitute in an wa a derivation of arallelism from some time-neutral probabilistic assumption, such as that probabilities of "space ensembles"

mixture. The probabilities in the individual system are time-symmetric. Parallelism is gotten out only by a judicious and question-begging insistence that the robabilities of the individual s stem alwa s be a lied to the same time end of systems whose evolutions are to be comparatively

with only finite lifetimes. It is also wrong to think of these arguments as in any way exp aining why the time direction in which entropy increases in the branch s stems whose entro is chan in is alwa s the same in the probabilistic sense. It is true that given the positing of parallelism for

universe does indeed provide a pool from which low-entropy systems can e "cut 0 "an 0 talne ar more eaSI y t an y creating an equilibrium s stem and waitin for a fluctuation to a low-entro state. But that is as far as it goes. Concerning parallel entropic increase, it seems,

An additional note is in order here. Sometimes it is suggested that

e ongln or asymmetry or t e oca systems can e oun In some combination of the idea that these s stems are never reall totall isolated from their surrounding environment and the cosmological facts. But this

tion to posit would be random disturbance of the system by the outside envIronment. OSIt1ng InterventIon WI ow entropy or e system at e initial time might indeed lead to a justification of the inference to higher entropy at the later time. But positing low entropy at the final stage of

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u i wor into system. Nor does the fact that the global entropy of the universe is increasing in a time-asymmetric way - in one but not the other time direction - seem to hel here. That is indeed an as mmetric fact but how could that fact be used to account for some special asymmetry of

fully determines all its subsequent states. Then we could simply posit an initial state that gives rise to parallel entropic increase of branch systems with each other and with the main s stem. But to characterize the state in that way would, of course, not be offering us an explanation of the

times and a "probable" micro-state compatible with that macro-state, and be ab e to derive the Second Law from that. But to derive the Second Law from a bald assertion that "initial conditions were such that the would lead to Second Law behavior" hardly seems of much interest.

the main system also suffers from the defects noted here. It relies upon a u ious interventionist account 0 entropic increase or not-rea yisolated branch s stems. And it osits the a ro riate non-correlation of the external world with the micro-states of the system in question in one

the world, it is hard to see how such a posit, as opposed to a posit, say, o pure ran omness or outsi e intervention app ica e in 0 time directions can be otten from an sim Ie lausible characterization of the initial cosmic state that doesn't just postulate parallelism of branch

increasing, if we partition the universe into energetically isolated pieces t at as a tota 1ty e aust t e woe un1verse, 1t can t en e 1 erre y the sheer additivi of entro for the s stems (entro of the whole being the sum of the entropy of the parts) that it must be the case that

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entropy of the universe is low and increasing in the future direction of time that explains why a specific sample of gas equilibrates in the future and not the ast time direction. IV. Further readin

A treatise on Machian ideas in dynamics is Barbour (1989). Volume 1 dealing with pre-relativistic physics has appeared, and a following volume on relativistic matters is forthcomin . Earman 1 8 is useful on Machian ideas in dynamics and space-time, especially Sections 2-1; 4-8

For Boltzmann's original cosmological ideas, see Brush (965), Volume 2, Chapter 10. A treatise on "anthro ic ar ments" in h sics and cosmolo is Barrow and Tipler (986). Chapters 1-4, 6, and 7 are most relevant to the

(971). A goo survey 0 i ationary cosmo ogies is Lin e 198 . For the basic connections between cosmolo ies with its cosmic ex ansion and thermodynamic features of the universe, Tolman (934) is a

the nature this must take is R. Penrose (979). This piece also outlines Penrose s proposa t at t 1S 1n1tla ow entropy 1S to e accounte or y a new law of nature. On "teleological" or "design" arguments in general and on their con-

The seminal exposition of the notion of branch systems and their use 1n c aractenzmg entrop1c asymmetry 1S e1C en ac . ranc s stems are defended in Davies (1974), especiall in Cha ters 3 and 4. Griinbaum (973), Part II, Chapter 8, and Mehlberg (1980) Volume 1,

Physics and chance

332 ~

.

.

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.

.

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ror a criuque or me rOle or orancn syslems, see ;)Klar

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~l~O / ).

Another account of the role of branch systems and cosmology (using interventionism) can be found in Horwich (987), Chapter 4. On the "r~l1C;:~1

.

(1979), Section 1.1.

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1. Positivist versus to unify a greater and greater range of phenomena in more and more comprehensive theoretical schemes. This unifying process often takes the form of a theory that encompasses the phenomena in one domain " phenomena handled by the reduced theory now being handled by the phenomena for which it was originally designed as well. Examp es are mani 0 . We are to t at Kep er's aws 0 p anetary motion reduced to Newtonian mechanics, that Newtonian mechanics On the other hand, we are told that Newtonian mechanics reduced to duces to the theory of electromagnetism, a theory already obtained by t e searc or t e uni ing account to w ic t e ear ier separate eories of electricity and magnetism reduced. Nowadays we are told that the r

n

to the electro-weak quantum field theory and that there are hopes that strong interactions, to the grand unified theory. And that theory, along Wit t e quantum t eory 0 gravitation, may e re uce to t e eory 0 super-symmetric strings. Ho es abound that whole disci lines can be reduced to others. Isn't there at least the hope that biology can finally be reduced to chemistry

, there is hope that an ultimate reduction of biology to physics is in view.

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ruu pcr of science concerned with unpacking just what we might be claiming when we claim that one theory reduces to another, As we shall see, the sub'ect is one whose full richness and com lexi have so far not been fully explored, One thing we shall note is that many important cases of

classical mechanics to quantum mechanics, of genetics to biochemistry, of mentalistic theory of one kind or another to physicalistic theory are all roblematic in this wa . And as we shall see in Sections II and III the alleged reduction of thermodynamics to statistical mechanics is another

account of reduction can do justice to all the special cases in mind. A ook at a mode 0 reduction once promoted by phi osophers enamored of the ositivist idea of theories as mere instrumentalist devices for generating lawlike relations among the observational data avail-

posed a model of reduction that takes one theory to be reduced to 0 e 0 servationa anot er' t e atter eory is a e to generate conse uences of the former and is su erior to the former in some wa , One way in which the reducing theory might be superior to the reduced

Someone who holds to a realist interpretation of theories will naturally e 1ssat1s e W1t 1S mo e. ure y, an 1mportant part 0 t e genu1ne co nitive content of a theo is its descri tion of a osited real realm of theoretical structure, a theory can only be said properly to be reduced to

The reduction of thermodynamics

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role of theories in science is tenable are also likely to be dissatisfied with the Kemeny-Oppenheim account of reduction. If we look at actual cases of reduction in science we find that there is alwa s a close relationshi between reduced and reducing theory at the level of theoretical struc.........."-J.L . L"""',

to find a reducing theory that could account for all the phenomenal data accounted for by the reduced, but which bore no connection whatever to it at the level of the two theories' theoretical structures. Even if both theories are then nothing but instruments for generating observational

theory with elements of the theoretical structure of the theory that does the reducing. The sim lest relation of reducibili of a kind stron er than the ositivist would be a simple derivation or deduction of the reduced theory

scheme from which the earlier generalizations can be derived and also rom w .coer genera izations covering a i erent range 0 p enomena can be derived. It is often claimed for exam Ie that Galileo's law of falling bodies and Kepler's laws of planetary motion were both shown

Newtonian physics is derivable from special relativity and from quantum mec anics as we ,an t at speCla re ativlty IS enva e rom genera relativi . As is obvious from the latter cases cited, but as is clear even in the

If Newton's theory is correct, Galileo's law is not true. Objects accelerate 1 ewton saws 0 , more e c oser t ey are to e ea s center. Ke ler's laws are not true. The lanet and sun travel around the center of mass as focus, not the center of the sun, perturbations disturb the

Newtonian physics is derivable from special relativity only by a "limiting" process t eve OClty 0 Ig t gOIng to 1 nlty, not y any SImp e envation at all. The exact details of the relationship between the space-time of special relativity and general relativity is a subtle one. The only ob-

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the narrow sense. e w of the now refuted, reduced theory are good "approximations" to laws strictly deducible from the reducing theory won't do. First, the complex of "takin a limit" e of 0 erations that characterize the interrelations of the theories in important cases - such as the limit of infinite velocity

the case of the reduction of classical to quantum physics - are far too subtle and formally complex for any simple notion of "approximation" to cover the relationshi between the theories. A ain im ortant uestions of "limitation of domain," specification of the kinds of situations in which

But there is a yet deeper problem than these. Once we decide that the meaning of a term is ed by its use in a anguage, e consequence that the meanin of a theoretical term in science is fixed b the totali of roles it plays in the network of assertions of the theory in which it

terms of the theory as well also becomes almost a foregone conclusion. But t en it ecomes pro ematic to assert t at e aws 0 t e re uce theo can in an wa be "a roximations" to those derivable from the reducing, for if the theories are in any way incompatible with one an-

to those of the other, even in an approximative sense, impossible? 's seems pretty u ious' one is comparing, say, G i eo's or Kep er's laws with Newton's. Surel it will be claimed whatever meanin is the kinematic terms referring to position, velocity, and so on mean the same "

Newtonian mechanics, or that space and time as projections of spacet1me 1n re at1v1ty eones S1mp y on t count as t e same eatures 0 the world as s ace and time did when the a eared in re-relativistic physics, are all too plausible and, not surprisingly, frequendy made.

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else asserts "not 5," they cannot be contradicting one another, for the terms in "5" and "not 5" have, by the doctrine, different meanings, and e two assertions are not the ne ations of one another. An assertion to the effect that the value of some physical parameter was slightly dif-

greater than previously thought would have to count, because the new "electron" couldn't mean the same thing as the old "electron," not as attributin a chan ed attribution to some class of h sical entities but as denying that one class existed and positing something new altogether.

want to say in the radical theory shifts and yet also to do justice to what we want to say in small changes of theory, is something we have yet to et sufficientl clear about in hiloso h of Ian ua e. As we shall see in Section 11,2 in this chapter when we discuss the relationships between

cepts function in the two theories will appear to force us to be cautious in ee in using t e same term to try an ea wi eatures 0 t e wor attributed to the world b the two theories. Exactl what "entro "as used in statistical mechanics has to do with "entropy" as used in thermo-

meaning change as a consequence of theory change that we have noted ere.

2. Concept-bridging and identification

weak positivist sense? ear y attempt to ea W1t t 1S pro em y . age suggeste t at the reduction was constituted b the postulation of "bridge laws," lawlike propositions containing the concepts of both reduced and reducing theo-

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Physics and chance

u

n

account. If the reduced theory follows only from the reducing theory and the bridge laws, why say that a reduction to the reducing theory h s 0 urred at all? Aren't the brid e laws th mselv s in ne f explanation in terms of the reducing theory alone if a genuine reduction

Can't we always derive the reduced theory from the conjunction of the reducing theory and a new bridge theory that simply posits that if the reducin theo holds then so does the reduced? One important response to this problem, proposed originally by H. II'

Star are one and the same object (also called the planet Venus), so light waves are nothing but electromagnetic waves of a particular frequency. But "The Momin Star" doesn't mean the same thin as "The Evenin Star," and it is by no means a trivial proposition established by the "

Star is the Morning Star" is not a priori or analytic, even it if is, according to some p i osop ers, a "necessary" trot . Simi ar y, Maxwe 's iscovery that li ht waves are nothin but electroma netic waves - a bit misleadin because what he actually said was that the optical aether was identical

of experience. Yet i entities, even' esta is e a posteriori, ave a . erent status when ex lanation is the issue from that held b "brid in laws." If Maxwell had simply asserted that each electromagnetic wave was accompanied

waves, there is no "lawlike correlation" of the presence of light waves an e ectromagnetic waves to e exp a1ne. ere 1S a ot to e exp aine , of course. We want to know wh , for exam Ie, electroma netic waves revealed themselves in the world in the manner we were accustomed

the complex story that involves not only the identification of light waves W1t e ectromagnettc waves, ut teem e 1ng 0 t at re uct10n 1n a eneral scientific context that includes other reductions as essential components. Another component of this overall reduction is the identifica-

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take place in a context in which some apparent features of the objects to be identified have already been stripped off these objects and put somewhere else erha s in the realm of the "sub'ective," The "sensed yellowness" of yellow light may be a puzzle for us, for example, for how ....'U''"',,L ...._,

is one in a scientific tradition that has already, for better or worse, stripped "sensed yellowness" from light, making it a "secondary quality in the mind" and leavin h sical li ht with 'ust the sorts of features s atiotemporal locality, velocity, frequency, and so on) easily attributable to

Could we reduce the psychological description of the mental phenomena 0 experience to, say, neurop ysio ogy y, at east in part, an i entification of some of the ontolo of the former with some of that of the latter theory? The hope was that, for example, "having a yellowish sen-

Some are the result of perplexities one gets into when it is events or propertIes or processes one wants to I entl WIt ot er events or ro erties or rocesses, rather than identi in one class of thin with another, But the graver difficulty is the implausibility of finding any place

Just as in the case of the simpler derivational reductions, in the alleged I entl catory cases one agaIn must co ront Issues generate out 0 e fact that it is rare indeed that the reduced theo suffers the reduction process unchanged, The very act of reducing one theory to another

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u n spo en 0 by the reducing theory? Or should we say, instead, that what we have discovered is that the elements spoken of by the earlier theory don't exist their ex lanato role bein taken over b new elements suf ciently like them to be called, misleadingly, by the same common noun

of Mendelian genetics as having been reduced to modern molecular biology or to speak of Mendelian genes as being "identifiable" as DNA molecules. Rather it is claimed the new molecular biolo rovides a correct ontology of genetics showing us what was and was not correct

constitution of matter and statistical mechanics, it is not surprising that we wi n muc to say a out just t ese issues when we discuss the details of the relationshi of thermod namics to statistical mechanics. So complex is the situation that we will be a bit surprised that people still

mean kinetic energy of molecules," as if that latter "identification" (if that is in ee w at it is were as straig orwar as a t at. One standard concern of those who deal with reduction b identification is how we could tell whether a class of entities at one level was

opposed to correlation, is forced upon us, unless we are willing to make su stantive c anges e sew ere in our eories. Once we attri ute massener to Ii ht waves and to electroma netic waves the conservation rules for that mass-energy won't leave room for distinct light waves and

correlated to them, we will have to so modify our theory to deprive the 19 t waves 0 mass-energy, 0 ot er eatures momentum, or ex-amp e as well. Once we sa that the Ii ht waves are 'ust the a ro riate electromagnetic waves, the problem is gone.

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tion, whenever we can. That is, we should identify whenever the assertion of such an identification is not blocked by some feature of the situation icall b the reduced enti havin some feature enuinel not attributable to the reducing.

scientific reduction by identification. This is the reduction of the theory of macroscopic matter to its micro-constituents by the identification of the macrosco ic entities as structured out of microsco ic entities. But whether the reduction also involves, in any simple way, the identi-

non-thermodynamic terms is a subtler matter.

domain of nature reducing to the other, but where it is correct to say that everyt .ng at appens at one eve is e or etermine y w at ha ens at the other. An account of the world at the second level then if complete, constitutes enough said about the world for all of the facts "

domains. But superveruence can 0 or many 1 erent reasons. loso hers and aestheticians who held that moral and aesthetic ro erties ("goodness," "beauty") were autonomous features of the world not

aesthetic features were, however,

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fully determined by our neural condition, but the sensory states are

the neural, at least on some traditional views of the matter. A rather different set of issues, issues more relevant for our purposes, center around those cases where a sim Ie reduction of one theo to another is blocked not by the appearance of some "irreducible" funda-

framed in terms of complex relational features of the world that fit into an "autonomous" schema not relatable to the way the world is described at the lower level in an sim Ie wa . The basic idea is that whereas the reducing level description of the world is "complete," the reduced level

izes the order of the world in terms of relational features introduced for specia purposes. T is "recategorization" of t e things and eatures of the world is such that no sim Ie "definition" of the reduced level cate ories in terms of those of the reducing level can be given or ought to be

the way we sort objects into household goods. Surely a chair is nothing e wor escri e ut a comp ex array 0 t e micro-components 0 b h sics. But an ho e of definin "chair" in the standard fundamental vocabulary of physics founders on the complex, vague nature of that

A much more interesting case might be that of evolutionary biology.

Notions i e t at 0 genotype, p enotype, species, tness, environmenta stress, natural selection and so on, form the fundamental vocabula of this theory. The theory has its own explanatory principles that func-

, defining these terms in terms of the concepts of fundamental physics seems out 0 t e question. et t e eXlstence 0 e comp ex re ationa structure of world in which evolution takes lace, and the usefulness of characterizing the world in the categories used by evolutionary biologists,

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world that somehow puts the objects of its domain outside the realm of ordinary physical explanation. It is often claimed that the art of our mentalistic discourse that deals with such notions as cognitive states (such as beliefs, desires, and so on),

complex functional structure of the world exists, treated naturally in terms of its own appropriate discourse. Just as the totality of physical facts "fixes" all the evolutiona facts or facts about furniture so too it fixes all the facts about mental cognitive contents, meanings of words, and so on.

layout some simple derivation from the laws of physics of the principles of explanation we invoke (or examp e, exp aining ehavior in terms of beliefs desires and the aim of the a ent to maximize ex ected utili ) familiar from these disciplines.

that if a physical structure becomes organized and structured to a certain egree 0 comp exity, new p enomena "emerge" not present in simp er h sical structures. Now emer ence can be (and has been) used to mean that what emerges is something genuinely ontologically novel

cepts and lawlike regularities that is said to emerge. This is an ontologically muc more mo est calm. A claim to the effect that humans (and some machines) are or anized in such a way that their workings are best described by positing states

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p enomena so ar eri e , mi prove use u w en e re a ion 0 thermodynamics to statistical mechanics, and of the two together to the remaining body of fundamental physics in which they are embedded, b' Once one has posited these autonomous concepts for the upper-level

, n for predictive and explanatory purposes, the question immediatefy arises as to the relationship between this conceptual and generalization structure at the u er level and the conce ts and laws of the lower-level theory, the alleged reducing theory. Simple derivation of the former from

new structure is somehow to be taken as incompatible with the underlying structure certainly have a problem on their hands. The less radical thesis that the u er-Ievel structure is com atible with the lower level but somehow still "independent" of it also requires some understanding of " " organization can exist. Even when it is asserted that supervenience holds, so that a at goes on in e upper eve is" xed" y the ower- eve attern of causation even if our descri tion of the u er level behavior is not derivable from the framework in which we characterize the lower-

conceptual scheme of behavior as governed by states having cognitive content an our sc erne 0 anguage as governe y semantica ru es t into our icture of a ents as biolo ical bein s in a h sical world, And our concern is with the overall place of the schema of thermodynamics

components evolving under dynamic evolution. As we shall see in Sections an , 1n t 1S case we can get c ear, to a egree sti not 0 ta1ne 1n the other two cases, on 'ust what the lace is of the thermod namic view in the dynamical (or the dynamical supplemented by probabilistic posits).

The reduction of thermodynamics

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n.Thec~eofilittmodynMWcs~dsm&~w

eones

, endy into our general description of the world than do the concepts of more ordinary physical theories. The theory is astonishingly universal. It is a licable to s stems of radicall different structural constitutions and to systems that are governed in their evolutions by radically different which this theory "cuts across" the usual hierarchy of structural reduction and dynamical genera ization that characterizes the rest of fundamenta h sics. Whereas notions such as that of the equilibrium state, temperature,

systems' evolutions strikes us as quite different in kind from the usual ynamica constraints at are t e joint pro ucts 0 t e structure 0 t e s stem inc1udin its internal and external forces and the basic d namical laws governing the response of systems to forces. The basic concepts

isolated systems, are quite unlike what we expect from ordinary dynamica conSl eratlons. If we ask what ives the conce ts of thermod namics their meanin , the natural place to look is at the roles played by the concepts in the

to the measuring devices, described in a non-thermodynamic way, that are stan ar: y use to attn ute magrutu es 0 t ermo ynamlc eatures to s stems. These are most obvious in the earl introductions of tern erature into physics by the association of degree of temperature with read-

role in the axiomatic versions of the theory. In particular, the Zeroth, lfSt, an econ aws 0 ermo ynamlcs c aractenze t e aSlC eatures of equilibrium (such as its transitivity as a relation among systems) and temperature. They also provide the ground for introducing the

Physics and chance

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to output states of a computer - that is, by their place in programming aviora inpu -ou pu eories 0 a en e avior, or examp e. si ar understanding of entropy (and perhaps temperature as well) would go some way in clarifying our understanding of these concepts. Here, the characterization of states of s stems in terms of their features th af I specifically thermodynamic (say, of a gas by its volume and pressure and

earlier, with the thermodynamic features of thermodynamic temperature and entropy introduced in a manner analogous to the functional states. When we look at the details of some of the s ecific conce ts in the next section, we shall see a little more of this analogy appear. "

highly dependent on the particular constitution of specific kinds of systerns. A typical such law would be the ideal gas law, PV- kT for certain kinds of idealized ases or the other similar e uations of state describing the interrelations of thermodynamic and non-thermodynamic variables

ing in a non-equilibrium steady state. They, again, will vary with the speci c system eing descri ed. Evidence from within thermod namics that its conce ts are not of the usual physical sort is also obtained from a little reflection on "Gibbs'

discriminable by means of the different alignment within them of the two nuc ear spins in a mo ecu e, an t e entropy 0 mixing is non-zero. So entro chan e is relative to a "level of descri tion" of the substances involved. Now it is clear that accusations of "subjectivity" for entropy and

of meanings of that notoriously ambiguous word) is quite another matter. alffis at entropy c ange 1S su jectlve ecause 1t 1S re atlve to c aracterization of the substances or to the level of fineness with which we are concerned in a given problem are as misguided as claims to the effect

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hydrogen if all we are concerned with is the degree of uniformity of hydrogen distribution, but that entropy change is non-zero if it is uniformi of distribution of h dro en with a s ecmc ali nment of nuclear spins in its molecules is perfectly understandable. In the former case,

,

,

distribution of hydrogen. In the latter case, work would need to be done from the outside to restore the original partitioned state of hydrogenwith-s ecmed-ali nment-of-nuclear-s ins. No "sub'ectivi "is involved. Nonetheless, the fact that entropy has this relation-to-detail of structure

entropy as fixed entirely by its functional place in the general thermodynamic aws an , su sequen y, y e constitutive equations is imortant. For it is in ex lorin how those enerallaws are functionin in allowing us to predict and control that we begin to grasp the meaning

being universally applicable to all manner of physical phenomena, even ose ra 1ca y erentlate rom one anot er 1n 0 er ways. 1t 1S like thermod namics in its mode of "cuttin across" the usual theoretical hierarchy of constitution and general dynamical principles of evolution.

nomena rather than "deterministic" accounts. or erst tlme, statlstlca mec an1CS 1ntro uce 1nto p YS1CS e 1 ea that the aim of a hysical theo could be not to provide an account of what must happen, but of what might happen. Phenomena were now to " " probability" and macroscopic phenomena as being reflections of what appens mos pro a y, or some 1mes on e average, a e m1croscopic level. The probabilistic theory differs from the earlier theories, as we have

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Physics and chance

both in what it takes to be its fundamental explanatory goals and in the w i io 0 e probabilistic posits along with the structural facts and dynamical facts about the system requires that we work in an idealized context. All a lication of theo to the world re uires some de ree of idealization of course. But, as we have seen, the idealizations utilized in statistical

sense of providing no insight into how deviant from the ideal a real system can be expected to be. This introduces one more factor that will com licate the anal sis of the reductive relationshi of thermod namics to statistical mechanics.

2. Connecting the concepts of the two theories Muc of the prob em concerning the reduction of thermodynamics to statistical mechanics can be localized in the ex loration of the com lex of relations that hold between the special concepts of thermodynamics

level theory. How are we to "connect" the concepts of thermodynamics wit statistica mec anics, or, on teo ject eve 0 .scourse, ow are we to associate features of the world osited b the thermod namic theory with features of the world posited by the reducing theory?

ground identifications as components of background reductions function ln ot er ways as we to prOVl e a context ln w lC we can ope to reduce thermodynamics. Some of the concepts that appear in our thermodynamic description of

The reduction of thermodynamics

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container to which the moving molecules are restricted, Radical theories to the effect that any change of assertion is a change of meaning aside, it seems re clear that we ou ht to take "volume" and other urel kinematic terms) to refer to the same physical feature of a system in both

that even such a primitive notion as that of time takes on a new meaning in the statistical mechanica context from the simp e parameter-of-change function that it la ed in thermod namics and in our ordina kinematical and dynamical descriptions of the world.

reduced and reducing level. As a consequence, we are directed unequivoca y towar e concept at t e re ucing eve at we must associate with a s ecific conce t at the reduced level. Once we acce t the First Law of Thermodynamics and accept the model of a gas as consisting

bIage, and therefore with something constituted out of its molecules' 'netic energy 0 motIon an potentIa energy 0 InteractIon, seems necessa . Or a ain once we view the as as consistin of molecules that impinge on the walls of the container and bounce off it, the association

and unit time due to molecular impact. oug a trect argument, en, ea s us rom ermo ynamlc to statistical mechanical conce t in these cases, it would still be uite misleading to baldly assert that the reduction proceeds by a simple-minded

example. There is, for a particular sample of gas at equilibrium, the actua momentum trans erre y t e mo ecu es ImpIngIng on a wa 0 the box in a short time, and there is its average value per unit area er unit of that time. On the other hand, there is the quantity calculated for

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expect to find as features of the individual systems has been the subject

.

pressure, a ramifying of concepts at the statistical mechanical level, and a situation sufficiently complex that any hope of doing justice to it b sim I sa in that a thermod namic feature is "identified" with some feature of the system at the reducing level is misguided. It should not

the "thermodynamic analogies" when he outlined how thermodynamic functional interrelations among quantities were reflected in structurally similar functional relations amon ensemble uantities. He carefull avoided making any direct claim to have found what the thermodynamic

more generalized macroscopic picture, it is the end state of spontaneous evo ution. e constitutive equations c aracterize e interre ations among the other osited characteristics of a s stem when it has the characteristic of being in equilibrium. Within statistical mechanics we could think of

this "equilibrium

the ways described in Chapter 5. Of

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having arrived there one stays there, will also now not be a statistically true generalization, but (if we can somehow rationalize non-equilibrium statistical mechanics a full "lawlike" ro osition. But once a ain the "exceptionlessness" of the "law" has been obtained by a reconstrual of

of a probability distribution constructed relative to a specified kind of system. Fluctuation from equilibrium is no longer part of the theory, but onI because e uilibrium itself is now construed as somethin that contains within its definition the fluctuational phenomena. The equilibrium

probabilities) as well. With the concept of temperature we begin to see some of the richness of conce tual varie that develo s when we move from thermod namics to statistical mechanics. As usual, we must distinguish at the reducing

, pIe, as the average kinetic energy of the molecules of a particular ideal gas at a particu ar time, rom t e concepts app ica e primari y to ensembles (tern erature as a definin constraint of a canonical ensemble or as a parameter of the micro-canonical distribution). As usual, the former

ceptual innovation. e statlstIca mec anIca surrogate or empInca temperature as c aracterizin e uivalence classes of s stems in mutual e uilibrium with each other) and absolute temperature (characterizing maximal efficiency of

mechanics that it can account for the place of temperature as fixed by stan al1 measunng Instruments as we. gam, IS IS a su t er matter than when one is dealing with pressure or volume. Here, the embedding of the thermodynamical reduction to statistical mechanics within the

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temperature values of systems. So the statistical mechanical reduction " u

i

magnitudes of its quantities to systems. This is a virtue of theories much sought for. In space-time theories, it leads to attempts to frame general relativi in terms of the motion of free articles and li ht waves these being more easily construable in their behavior within the theory than

of measurement, in contrast to the accounts of measurement of Bohr and of the idealists, for example, that attempt to characterize the measurement a aratus and the measurement rocess within the world described by quantum theory itself.

have the same temperature and be in equilibrium with one another, although radically distinct in their natures. Thus, radiation, for example, can be in e uilibrium with a as. The microscopic feature of the system associated with its temperature

with any characteristic of a system in microscopic terms (such as the average .netic energy 0 t e mo ecu es, e eature associate wi tem erature in the case of the ideal as). Now we have seen a reason why a too naive application of the notion of identificatory reduction

system sample at a time, but, rather, to some feature of a probability istri ution over systems 0 a speci e type. But even stic ing to "tem erature" as referrin to a ro e of an individual s stem does the variation in microscopic correlate from system to system make an asser-

is "identical with" one feature for one kind of system and with a different eature or some ot er n 0 system. ut is ar: y prec u es our sa in that in each case the tem erature of the s stem is identical to its relevant microscopic feature. It is sometimes argued in philosophy of

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ble. This argument is, at least in part, grounded on the assertion that the functionalist states can be realized in a variety of physical processes, just as a 10 .cal state of a ro ram can be realized in a com uter whose workings are mechanical, electronic, or optical. In the case of thermo-

feature of the system with which to identify temperature. And insofar as it is temperature as the instanced property of a particular system that we have in mind there seems nothin to block a strict assertion of the identity of that temperature for that particular system with the appropri-

micro-features of a system, we can then, as S. Yablo has suggested in the min -bo y context, t in 0 e re ation 0 temperature to t e microfeature on the model of a determinable ro e 's relation to a determinate property falling under it. That is, philosophers think of yellow, say,

tion beyond the realm dealt with at the upper level by the concept. emperature prOV1 es a goo examp e 0 1S 1n 0 concept extens10n a lied after a reductive association of conce ts has been achieved. The ordinary notion of absolute temperature has all such temperatures

of temperature with the measures of order and disorder in statistical mec ames ea s to a natura extens10n 0 tea so ute temperature conce t in that theo . A ical situation in which the extension is a lied is that of nuclear spins of atoms locked in a crystal. If they are aligned

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here where the maximal energy state is a peculiarly ordered one). The reduction has directed us in the ways we have described to fix on an a ro riate conce t at the reducin level to associate with the conce t of temperature at the reduced level. Once we fix on the formal surrogate

the original thermodynamic concept was never intended. Such a natural extension of the reducing level concept is, of course, conceptually harmless. And it is scientificall effective in doin the additional work cut out for it. This phenomenon of concept extension - where a concept

higher-level concept - is a familiar feature of the reductive process. T e concept 0 entropy is t e most pure yermo ynamic concept 0 all. Its rima meanin is fixed entirel b the role entro la s in the characterization of systems that follows out the method introduced by

its place high up in the theory and unrelated to immediate sensory qualities or primitive measurements as temperature IS re ate to t ese , It IS not su risin that in seekin the statistical mechanical correlate of thermodynamic entropy we have the least guidance from the surrounding

do we have guidance from the connections of thermal characterization to measunng Instrument construction, somet ng at prOVl es some uidance with res ect to tern erature. We need to find some feature of systems, characterized in our reducing theory, that will fill the theoretical

surrogate for entropy. This is exactly what we find - that is, a wide vanety 0 entropIes to corre ate WI t e t ermo ynamlc concept, eac functioning well for the specific purposes for which it was introduced. First, there is, as usual, entropy as a characteristic of an individual

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ideal gas, deriving the right substitute for the original simple system entropy is sometimes a difficult task, but the general conception is always the same. Given the fact that the value of this entro will de end u on the chosen partitioning of the molecular phase-space into finite "boxes,"

position-momentum partitioning he chose, even at its origin, rather than the a priori equally good position-energy partition. His choice was based u on his earlier work on the stationa solutions of his kinetic e uation and their characterization as having the maximum value of the usual

correlate of this kind for thermodynamic entropy is fixed by a multitude of consi erations, inc u ing e a itivity 0 entropy or in epen ent s stems in thermod namics and most im ortand the demand that the function be maximized for the equilibrium configuration of the micro-

Constrasting with this entropy of individual systems is entropy as a c aracteristic 0 an ensem e or 0 a pro a i ity istri ution re ative to a s stem kind. What differentiates entro from the other thermod namic concepts is the wide variety of ensemble "entropies" suggested as the

it fits nicely into the functional relation among ensemble features that 1 Suse m IS 1SCUSS10n 0 t e t ermo ynam1c ana ogles In statistlca mechanics where he found the a ro riate analo e to the thermod namic quantities. And it has the appropriate maximization properties for

constraints. e re ation 0 t s entropy to t e o tzmann1an IS e amI 1ar one 0 the relation of an ensemble feature to a feature of individual s stems. The ensemble over which the average of the "index of probability"

Gibbs associated an ensemble quantity found by averaging over all of ese sys ems WI e ermo ynam1c en ropy, 0 zmann aSSOC1a es that feature of individual systems that we know, from Khinchin type of results, to be overwhelmingly dominant (in the probabilistic sense) among

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The usual alternative exists of sticking with a concept most like the 1 1 gu ri , v ping the unexceptionless law. We expect a system held in energetic isolation from its environment to have its Boltzmann entropy fluctuate, even after lee uilibrium" has been obtained. But its Gibbs fine- rained entro will of course, remain constant, like the entropy of a system in equilibrium

fluctuational behavior. The Gibbs fine-grained entropy, by averaging over the ensemble of systems, has absorbed the fluctuations into the improbable but resent low Boltzmann entro members of the collection. But, of course, making this distinction between individual system

mechanics. We have seen how, in an attempt to characterize the nonequilibrium dynamical process, the idea of coarse-grained entropy was introduced. Ensemble fine- rained entro is rovabl invariant over time. But the coarse-grained entropy can vary with time, and thus have

Fully understanding the role of coarse-grained entropy is a complex matter. First, t ere are t e i cu ties in connecting e c nge 0 coarserained characteristics with the observed chan es in real macrosco ic features whose dynamics we wished to understand in the first place.

have the behavior one wants - in the case of entropy, monotonic increase e t eory is a towar t e equi' rium va ue. Getting t is resu tout 0 matter not full resolved. The best results hold for certain kinds of ideal systems (C systems and the like). In an idealized sense, we get the right

as to be connectible to the macroscopic variables with which we are concerne. onet e ess, 1t 1S P a1n t at t e 1nvocat1on 0 t ese coarserained notions of entro does hold some romise for roundin what might ultimately be a successful attack on the general problem of non-

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tion of the reduced theory to a broader conception allowed by the richer resources of the reducing theory. Already within thermodynamics, as we have seen the conce t of entro had been extended in natural ways from the equilibrium case to non-equilibrium cases. When the non-

integrated over the entire system. In the coarse-grained case, entropy is being generalized in a different wa . Here the entro is a Gibbsian ensemble uanti . For e uilibrium ensembles, coarse-grained and fine-grained entropies will agree, because

will generally disagree, the fine-grained remalnlng constant, and the coarse-graine ,one opes, evo ving towar e entropy appropriate for the e uilibrium state of the ensemble relative to the im osed constraints. Whatever the difficulties there are in applying coarse-grained entropy for

one wants it to behave, the process of concept extension here is clear. It is wo w i e, at t .s point, to iscuss t e issue 0 t e egitimacy 0 conce ts such as coarse- rained entro . Here the issues I wish to discuss are not those centered around the suitability of using coarse-grained

physics. The accusation of subjectivity comes from the fact that we can coarse-graln e p ase space In an In rute num er 0 1 erent ways. or an 'ven actual ensemble of s stems, in an finite time, coarse- rained entropies can be found that will vary in any way one likes: increase,

coarse-graining represent the objective increase of entropy associated Wl approac to equl 1 flum. One su estion, that of A. Griinbaum, althou h in enious, won't work. He suggests that for each coarse-graining, a majority of systems in the

each coarse-graining. But for any coarse-graining there will be ensembles w ose coarse-gralne entropy re atlve 0 at coarse-gralnlng Wl 0 whatever one likes, and, more importantly, for any specified ensemble there will plainly be coarse-grainings that make the ensembles' entropy

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Physics and chance or coarse-gralnings.

the coarse-grained entropy. Proofs that a system is a Bernoulli system, on

w

c tainly strong enough to give us monotonic increase of coarse-grained entropy relative to that coarse-graining. The much more im ortant oint is that a relativit of coarse- rained entropy value and behavior to a coarse-graining chosen is no mark

ization of the system. It is, of course, of vital importance that coarsegrained entropy does indeed behave the way it should with regard to certain "natural" artitionin s of the hase-s ace ones based on the usual breaking up of the phase-space region into cells small with regard

simple shape, and are based on cells whose definition is framed in the usua parameters suc as position and momentum. For it is t at kind of coarse- rained behavior that we can ho e to connect in the ri ht wa with the observables whose dynamical evolution we wanted to com-

the expected manner. But none of this should lead us to characterize coarse-graine entropy as in any way "su jective." ere are, to e sure interestin ob'ections to the Ie itimac of the invocation of such partition-relativized notions into the theory, and proposals to avoid that

systems, to handle the problem of arbitrariness by proving the existence o a new ensem e representatIon t at IS Mar ovian In ItS evo ution. But it is uite misleadin to exclude the coarse- rained notions a riori on the basis of some claim that they are subjective.

concept of entropy that is obtained by first making the transition to the new ensem e representatIon t at 0 eys a ar oVlan equatIon 0 evolution, and that rovabl exists when the s stem is randomizin enou h (being a K system). He then defines a new entropy from the new rep-

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program, the permitted singular states from those that are excluded and that would result in anti-thermodynamic behavior. In another context Pri 0 ine su ests a different but related extension of entropy concept. We have seen how in the spin-echo experiment

a high- to a low-entropy condition. This can be construed as a violation of the Second Law more radical than that entailed by the existence of fluctuational behavior in an statistical mechanical account. Iri ractice the entropy generated in the external environment to generate the spin

Prigogine suggests that although the intervention into the system seems to generate a spontaneous decrease in entropy subsequent to the flip, this result is mere! the conse uence of an inade uac in our usual notion of statistical mechanical entropy. The statistical mechanical entro-

tropy, for example, utilizes only the one-particle distribution function, ignoring t e corre ationa i ormation containe in a t e ig er partic e functions and contained entireI in the full n- article distribution function. In the spin flip, this is what happens: A system originally of low

Prigogine suggests that a new entropy be defined that pays attention to e corre atlona I ormatIon as we as to e one-partlc e I ormation. He seeks such a notion that would have the followin features: (1) Like the one-particle entropy, and unlike the entropy we would get using

one-particle entropy, this new entropy takes account of the information n t e a out corre atlons among t e components 0 t e system. case of normal system evolution, where information dissipates from the one to the n-particle distribution, the entropy so defined will, as usual,

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in ro uce y e osc reversa process, e en opy a es on a very low value, below even that of the original spins-parallel state. With this new entropy so defined, we can hope to describe the spindecreases the entropy of the spin system. The spontaneous evolution i WIn p n aw i av u ,0 ourse, the anomalous nature of the evolution of the system after the flip is as surprising as it ever was, and the possibility of the Loschmidtian reversal rocess as im ortant as it ever was. Once a ain we see the ossibili of saving a regularity by modifying a concept. The new notion of entropy

the old notion. And the benefit gained by moving to this novel concept will be the ability to save the old regularity from apparently refuting h sical ossibilities. Summarizing, the situation is this: When we move from thermo-

level to associate with the concepts at the reduced level. We are directed in our search for the appropriate concepts by the place played by the conce ts at the reduced level in the eneral thermod namic laws in the constituent equations, and by the standard measuring processes in-

broader scientific reduction of macroscopic systems to arrays of constituent micro-components, an y t e pace 0 t is t eory in t e genera kinematic and d namical icture. Concepts exist at the reducing level that designate properties like those

property referred to at the reduced thermodynamic level has been ident' e wi a eature re erre to at t e re ucing, statistica mec anica, level. With this association of features at the two levels the laws at the thermodynamic level become modified by the reduction into statistical

side the scope of the associated concept at the reduced level can be lffiportant. Conce ts at the reducin level can also be used to desi ate features not of individual systems but of probability distributions relative to a

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a subtle and complex one. The full understanding of the association will

dynamical feature can be extended and modified in quite distinct ways. No harm is done by such conceptual modification as long as it is always ke t clear about how the novel conce t is utilized in some ex lanato scheme, and as long as one avoids the pitfalls of confusion due to " "

m. Problematic aspects of the reduction 1. Conseroative versus radical ontological approaches of posited entities of the world and, simultaneously, in a process of decreasing t e num er 0 t ese posits. Nove t eories intro uce entities into our world icture reviousl unima ined. But theoretical novel also consists in telling us that entities and features of the world we

us that what we initially thought were two classes of entities in the world are ut one c ass. ere we once t oug t t ere were ot ig t waves and electroma netic waves we now know that there are onl electromagnetic waves. Not because light waves don't exist, of course, but

Intertheoretic reduction sometimes encompasses a rejection of portions eory, an so a 1sm1ssa 0 some 0 1tS prev10us y o e now re uce osited ontolo . And intertheoretic reduction often consists in art in the identification of a class of entities or features posited at the reduced

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liminary to the later reduction.

to statistical mechanics. We need not worry how that latter theory is to encompass "felt hotness and coldness" because these have already been stri ed off the h sical s stem and erha s out of the h sical world by the implicit metaphysics of the reduced theory.

at a time, identified with features of the system characterized by the reducing theory. The reduction is not a simple identificatory reduction, as we have seen because the terms correlated with the thermod namic terms are in many of their uses being used in a way quite distinct from

ture, and its entropy, interpreted as concrete features of the individual system, it oes seem p ausi e to say at t ey are not lee iminate " by the reduction but rather assimilated b identification to the relevant kinematic-dynamic features of the system described at the level of the

tern to serve as that identical with some thermodynamically characterized eature 0 t e system may e statistica in nature, an may require re erence to the s stem as a s stem of a s ecified kind (that is as a member of a particular ensemble). Thus knowing which mechanical feature of the

features of that ensemble as being an equilibrium ensemble. Nonetheess, t e eature 0 t e 1n 1V1 ua system 1S not 1tse a stat1stlca eature, but is definable in urel mechanical terms. Thus for an ideal as mean kinetic energy of the molecules is just the average of those purely me-

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includes all the features of neurons and their processes in its domain. It doesn't seem that the claim of the reducibility of thermodynamics to statistical mechanics can be ob'ected to in this vein. As we have seen thermodynamics has already eschewed such things as felt hotness as

features of the system construed as a structure of micro-constituents in the usual manner of identificatory reductions. Yet it fre uentl is asserted that thermod namics is not reducible to the mechanical world-view. We will examine the grounds of one such

theory. But here it might be useful to say just a little about one such toes ave an important onto ogica aspect - t e c aim 0 c aim Pri 0 ine and others a claim noted briefl in Section IV 2 of eha ter 7 that the relation of thermodynamics to mechanics requires us to radically

not identifiable with any mechanical feature that allegedly blocks the re uction 0 ermo ynamics to mec anics p us statistics. Instea , it is claimed that thermod namics and the facts about mechanics we discover in an attempt to reduce thermodynamics to the latter theory show us that

holds and blocks measure-theoretic chaos over the whole phase space, rna es t e posltmg 0 exact mlcro-states or t e system represente y oints in the hase s ace) a false idealization. If we cannot, in rinci Ie, "fix" the exact micro-state of a system, ought we not, on good positivist ?

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sys em 0 e some appropna e pro ail IS n u ion over a regIon 0 phase space. In other words, shouldn't we take ensemble representations to characterize individual system states in the manner discussed in ?

These general considerations are backed up by arguments that rely S U I 1 pro ii is ibutions that are such that they are represented as singular concentrations of all the probability on a single phase point or other set of measure zero will still be such that their non-unita transforms will evolve with time into "delocalized" probability distributions reveals formally the "false

On the basis of these considerations, the analogy is drawn with quantum mechanics, which, in nearly all accepted interpretations, denies the existence of an exact osition-momentum state for a s stem. Instead uantum mechanics attributes to the individual system the wave-function, which "

But, as we noted in Chapter 7,IV,2, this radical revision of our standard ontology is raug t wit problems. The argument rom non-precisefixabili of the ointlike micro-state to its non-existence is surel too quick. Even from a positivist perspective in which the postulation of

of concern, presupposes the existence of the exact micro-states of the course, e necessity or in ivi ua systems as its un er ying asis. this resu osition is 'ust what the radical reviser of our traditional ontology will deny.

tions whose representatives are no longer totally concentrated upon sets oug suc transo measure zero IS a so not entrre y persuaSIve. formed re resentations rna ve well ca ture interestin as ects of the dynamics of the ensemble, the fact that the original phase-point repres-

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of the Bohrian interpretation of the wave function as being a "complete" description of the state of the system, and against earlier interpretations of uncertain as the mere result of e istemic limitations on our abilities to know the values of existing but "hidden" exact parameters, such as

tern represented in the theory by the points of the phase space, is presupposed by the construction of the phase space that provides the arena over which robabili distributions are constructed and over which their dynamic evolution is followed.

amplitudes of the component waves, allowing interference effects where amp 1tu es un 1 e pro a 11t1es can cance one anot er, 1S one 0 the stron est ressures behind treatin the wave-function in uantum mechanics as something more like a concrete feature of a system than

But, of course, such interference phenomena are totally lacking in t e ensem e pro a 11tles 0 statlst1ca mec an1cs. n quantum statlstlca mechanics, such interference must be taken into account, of course, but that is at the level of the wave function attributed to an individual system,

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Physics and chance no as some us to that attitude

micro-states when the systems are simple enough to have stability of mo ion or pe r e mo ion a is s 1 in e reg10ns 0 s a 11 , would also seem to incline one to view the exact micro-states as real even when instability makes it impossible for us to determine which ex ct micro- te a s stem 0 sesses But th nt I reply to this that the postulation of the exact micro-states in the stable ~.L~>".L.LLJU

states, all of whose trajectories are coherent with one another for at least some future time. Here the argument would be at least analogous to those who would ar e in the uantum context that the ostulation of "classical" systems is unnecessary, things behaving classically just being " " echo-like results. Here, it would seem, the appropriate ensemble probability distribution to pick at the moment of flipping is the spread-out distribution in some a ro riate coarse- rained sense at least corresponding to the system's overt randomized nature. Yet the possibility of

all of the subtle correlational facts among the spins, exists. Otherwise, ow to exp ain tea i ity 0 t e system a er ipping to "n its way home?" Added to these considerations is the fact that denying the existence of

patible with the generation of an appropriate statistical structure at the eve 0 statistica mec anics to give us a statistica exp anatory account of the thermod namic features of henomena even if ettin that account requires some puzzling and problematic posits about the prob-

resolving the most crucially puzzling questions such as those centered aroun t e ongm 0 hme-asymmetry. et er we ta e t e pro a 11ty distributions as directl descri tive of concrete as ects of individual s sterns, as the ontological revisionist desires, or, instead, as probabilities

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equilibrium time-asymmetric results we want. It is far from clear why anything in the ontologically revisionist position helps with this fundamental roblem.

If the arguments of the last section are correct, then thermodynamics requires no ontological posits about the world not already present in the osits of the theo of the micro-com onential structure of matter and in posits of the kinematic and dynamic theories governing the behavior

with systems or features of systems posited at the reducing level. Nor does the structure 0 t e re ucing theory require any radical ontological novel in order that it do 'ustice to the world described b thermodynamics. Yet the intuition persists among many who have thought

intuitions rest? e ermo ynamic picture 0 e wor 0 structure on the world that doesn't find an eas lace to reside in the eneral dynamical world picture. In particular, the idea of equilibrium states as

of the world that require the greatest insight in constructing the reducing mlcro- eory at IS Inten e to account or em. at 0 we nee at the reducin level to do 'ustice to this thermod namic structure? Mechanics by itself, along with the facts about the constitution of the

, in one sense, do the job by itself either. What I mean here is that the arguments stemmIng rom erst cntlca responses to 0 tzmann s attem t to save the H- Theorem b makin the inter retation of it probabilistic still carry strong weight. In order to get the results we want

metrically apply our natural probabilistic assumptions to the dynamical s a e 0 a system a a tIme. e must app y t em on y to Inltla states and never to "final" states. Only thus can we get the correct thermodynamic predictions out of non-equilibrium statistical mechanics and block

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heart of arguments for the "irreducibility" of thermodynamics to any lower. ri in uc ive rea-

soning, and so immune to the demand for a physical account of why they are true, seems dubious. Except for the pure equilibrium case, where the ensemble robabili distribution can be iven its "transcenden 1" rationale as being the only stationary distribution constant in time and

matter how "natural" they appear, constitute an independent posit of the theory, not derivable from mechanical considerations alone. Even in the ure e uilibrium case the assum tion of absolute continui with res ect to the standard measure is a posit that cries out for further justification.

relaxation times for systems. One way in which claims of "irreducibility" for thermodynamics can be understood is in terms of a stron em hasis bein ut on the necessi of the basic posits about probabilities over initial conditions. The claim

most crucial features of the thermodynamic aspects of the world, the eatures i e approac to equi i rium an time-asymmetry t at appear as a su rise from the urel mechanical view oint of the world a ear at the reducing level only because a basic posit, essentially thermodynamic

straints on the world imposed by the latter theory by its fundamental autonomous aw - t e econ aw - Wlt a structura constramt on robabilities of the initial conditions of s stems characterized at the micro-level.

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a imp wi e pure y dynamical world-picture. As we have noted, although rerandomization might prove inconsistent with dynamics, imposing the distribution on t on ime ca 0 be. It i rath rem that the general context in which the mechanical world picture develops

a model of a possible world is taken to be the lawlike constraint that once an initial state is chosen states at other times be lawlike compatible with that choice. But now we seem to be sa in that the choice of initial states of the world is constrained, at least in the peculiar probabilistic

initial conditions appear in other portions of physics as well. It has been proposed that one can have a theory of special relativistic space-time Minkowski s ace-time and et tolerate the existence of su er-luminal causal signals - signals traveling faster than light in a vacuum, so-called

that would be construed as transmission of a signal "into the past" from e point 0 view 0 some ot er 0 server moving wi respect to the first. This seems to enerate the ossibili of a closed causal 100 in which the occurrence of an event causes something to happen in the past of the

posed solution is just to insist that initial states of the world that are so causa y incompati e wit t emse ves cannot exist. Here again we see the su estion that there mi ht be constraints on initial states over and above the constraints on pairs of world-states imposed by the usual laws

world models of that theory that allow for closed timelike curves - that IS, one- ImenSlona sets 0 events at are causa y connecte to one another and so arran ed that, once a ain, an event can "cause itself" at a finite time interval along the curve. Here again, the possibility of causal

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that the initial states are so distributed in the world. But, again referring

Law phenomena seem so universal and fundamental that describing them all as the result of some "mere accident" about the configuration of initial conditions which on the model of the distance from the sun the earth just happens to have) seems absurd. To go from the dynamic world

probabilistic constraint on initial conditions that is the core of the statistical mechanical approach to non-equilibrium dynamics. I think it is this all im ortant fact that lies at the heart of claims that the thermod namic structure of the world is primitive, ineliminable, and irreducible to any

with some other putative possible reductions will be enlightening here. Philosophers 0 biology 0 en express the opinion t at the ontology of enetics and evolutiona theo is exhausted b the usual biolo ical entities and features such as DNA molecules, their cellular environments,

organisms develop, the role of self-reproducing molecules in the genesis o t elf structure, 10 og1ca acts a out transm1SS10n 0 genet1c c aracter in the re roduction of or anisms, the lace of the environment in aidin or hindering the survival of an organism, and so on. This gives us at least

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"macro-level" experience of species differentiation and species dynamics in evolutionary history. From this perspective it can at least be argued that there is no need to osit an in over and above the usual h sical entities and features of the world and the usual physical laws governing

hope of accounting for the reduced-level structure in terms of that of the reducing level will founder, even if less than kind-to-kind identificatory reduction is all that is demanded. A similar case can be made for the claim that one can at least offer

cognitive states, content, intention, and so on in our "folk psychological" account 0 uman ehavior. Here, an account in terms 0 the place of or anisms in their environment the structure and evolution of the neural processing machinery of the brain, the behavioral strategies likely to lead

that results in behavior usefully describable and explainable in terms of e possession y e agent 0 e ie s, esires, an so on. Once again, there will be man ske tical that an such ar ment can reall do the job. But at least the hope for a physicalist account of the place played by

tropy, and irreversibility in our account of the world, as contrasted with e egree to w IC we can oun ese notIons on t e common notIons of the remainder of h sical theo that is in uestion. Here, the connection between the thermodynamic structure and purely

neurological structures. We know pretty clearly what we must super-add o e usua ynamlc escnp Ion 0 sys ems to e a e to generate e situation where the applicability of thermodynamic concepts in a useful descriptive and explanatory role becomes clear. It is the posits about the

(facts about instability of trajectories, for example), are what provide the c araetenstlc structure 0 e wor at rna es t e ermo ynam1c p1cture applicable. So the additional elements necessa to lace thermod namics into the general dynamic picture are clear, although, as we have seen, much

theory. It is this that is at the core of the strongly held intuition of many

of thermodynamics, is utilized in choosing the basic probabilistic posits espec1a y 1n t e non-equ11 flum case, 1t 1S essent1a to ave some thermod namic a reciation of the s stem in order to even begin to look for the correct initial probability distribution that must be posited in Frequently, we have little guide as to how to choose this probabilistic

statistical mechanical picture, it is this already existing macroscopic ac-

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at our disposal, we generally cannot hope to find the general solutions of it. Instead, the search for solutions is guided, once again, as in the Cha man-Ensko mode described in Cha ter 2 b the use of what we know, from the macroscopic thermodynamic-hydrodynamic level, about

, ing us, for example, to derive the values of transport coefficients previously inserted into the hydrodynamic equations on the basis of their ex erimentall determined values. But it is vitalI im ortant to reco nize that the very structure of the solutions found is guided by our antecedent

The apparent ineliminability at certain crucial stages of previously obtained characterizations 0 systems, c aracterizations ramed in ermod namic terms and established b em irical observation in formulatin the correct statistical mechanical model posited to underlie the thermo-

must have a subtly contrived model of reduction in mind.

in science and posits lawlike "bridges" as essential. The many types of lnte eoretlc re uctlon are lscusse ln ar 00 er is an extensive review of hiloso hical models of reduction and their problems.

On the notion of supervenience, its formal structure, and its applicaHon, see 1m , an For the issues plaguing attempts to characterize the relation of mind to brain (or mental theory to biophysical theory), see the readings selected

infinity" and "negative absolute temperatures," see Landau and Lifschutz

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or msey IScusslon 0 concepts is Ehrlich (982). A discussion of the variety of entropies encountered in statistical memany of the issues, focusing primarily on allegations of subjectivity for

non-equilibrium entropy based on a change of ensemble representation

" 1 See also Prigogine (974), Goldstein, Misra, and Courbage (981), and

(967) and (969).

10

I. The Boltzmann thesis

Let us review once more the basic components of Boltzmann's final account of the as mme of the world described b the Second Law of Thermodynamics. In his final picture of statistical mechanics, we would Excursions to states of very low entropy ought to be rare, and we should before improbable excursions from a close to equilibrium condition. How can we reconcile this account of the probabilities of micro-states in the world with what we actually find? What we find is a universe apparendy the future direction of time, but, as far as we can tell, seems to be ever In addition, this entropic asymmetry of the universe as a whole is matched by the para e ism 0 entropic increase of branch systems temporari y isolated from the main system. the world to reconcile his probability attributions with the observed facts. tiny fragment of the whole universe. Both in terms of spatial extent and at t e universe as a woe is vas y tempora uration, it is assume larger than the universe we can observe at the farthest reaches of our vaster universe is, in fact, overwhelmingly at or near equilibrium in its near equilibrium for anyone spatial region over most of that region's tempora uratlon. But, the argument continues, one will, given the standard probabilities ri fi" universe in conditions far from equilibrium for "brief" periods of time. duration of the universe constituting the standard of size. So we would

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e universe a we can 0 serve expec 0 n regions as vas as a 0 in conditions as far from equilibrium as ours is for periods of time of the order of the billions of years revealed to us by the evolutionary lifetimes Next, Boltzmann applies his version of what has come to be called the " u mo a or near equilibrium for a long period of time, a place characterized as inhabited by sentient observers will have a probability, given that condition of bein ve far from e uilibrium. For onI in a uite lar e portion of the cosmos that for quite a long period of time is very far from

by continual energy flows. When we add that these creatures arose by the process of evolution by natural selection, the need for an even vaster and Ion er lived excursion from e uilibrium becomes clear. anI such a vast fluctuation could provide the conditions under which the evolu-

librium in space and in time but that we are in a "small" region of entropic uctuation. T is region is a so ute y improba e ut hig y probab e relative to the resence in it of livin creatures althou h as usual not as probable as the much smaller region of fluctuation that would be

why is it that we find the high-entropy end to be in the future time irection an t e ow-entropy en in e past, an not teo er way around? It is at this oint of course that the claim is made that the direction of time in which entropy is increasing is "constitutive" of what

t ere is more t an one

, future direction of time the direction of time "parallel" to our past time lfectlon. creatures cou lve In e reglons 0 e unlverse more or less at e uilibrium, which is im ossible, the would have no future (or past) direction of time at all. There would, to be sure, still be two

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standing of the cosmological structure of the universe. The new orthodox account is of the entire universe in a state of disequilibrium. Starting from the Bi Ban in a state of ve low entro the universe is currend expanding. Its entropy as a whole is constantly increasing, the increase

its now concentrated energy into the now cold reaches of empty space as the stars radiate light and materia entropy increases. The assumption is then that the universe as a whole is on one of those slo es of the entropy curve, with one direction of time pointing toward the higher-

allels the direction of entropic increase for the dominant class of branch systems, a oug t e reason or t is para e ism 0 ranc systems wit each other and with the main s stem is far from clear. Once again, we note the possibility that this picture might be embedded " universes are all expanding "bubbles" in a more encompassing manifold an e space-time t at constitutes e entIre wor 0 eac u e universe. In other versions the multi Ie universes are the" arallel" worlds each of which represents one possible evolution from the initial state

life. Again, the problem of why our universe has such an "excess" of ow entropy WI anse. But, in an case, the view will be su lemented once a ain with the final Boltzmannian argument. Why is it that in the universe as we find

encounter the claim that the very constitution of what distinguishes ture rom past tIme trec Ion IS 0 e oun In e en OpIC asrmme of the universe and its branch systems. The motivations behind such a claim are clear. If the claim were true,

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s of what counts as future. Of course, the existence of a time direction of parallel entropic increase of isolated systems would remain nondefinitiona and in need of an ex lanato acc unt F rt r b naturalism would hold that all of the facts of the world ultimately require .i

of the world resides in the asymmetric temporal facts about entropic behavior of systems, then any asymmetries implicit in the notion that the future direction of time is distin ished from the ast direction would somehow or other, have to supervene on the basic physical asymmetry

what kind of claim of "definition" , "dependence" , "constitution", or "supervenience" the claim that the future-past asymmetry as a whole is "defined b ""de endent on " "constituted b "or "su ervenient u on" the entro ic asymmetry is supposed to be. It is not a trivial matter to get entirely

claim comes down to, it is hard to know what kinds of arguments can support it an w at in s can co ute it. Once one as ecome c earer concernin the nature of the claim the soundness of both the su orting and confuting arguments can be evaluated more clearly. I shall be

the "entropic theory of temporal asymmetry" than I will about what t e structure 0 suc a c aIm comes own to.

n. Asymmetry of time or asymmetries in time? a pervasive asymmetry of the behavior of physical systems in time. Both or e cosmos as a woe, an , In a pro IStIC sense or e terp.oraril isolated branch s stems, later states are hi her-entro states. The main questions in the issues of the "direction of time" problem are

there is a preliminary topic to which we first ought to devote a little attention. ug t we to say at e entropic asymmetrIes ill t e wor reveal an asymmet of "time itself" or ought we to say, rather, that they only represent an asymmetry of the world in time? Indeed, what would

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1. Symmetries of laws and symmetries of space-time Symmetries of the observational phenomena in the world play a vitally im ortant role in contem ora h sics. The observation that s ecific location in space is irrelevant to the evolutionary dynamics of a system

is irrelevant to its behavior underlies the conservation law for angular momentum. And that the specific time at which a process is allowed to evolve is irrelevant to its evolution rounds the rinci Ie of conservation of energy.

identical to those in any other led, in the context of Newtonian mechanics, to e princip e of G i ean re ativity. Generalized to encompass electrorna netic henomena and with a now necessaril modified mechanics taking the place of the Galilean-Newtonian mechanics, the uniformity of

In quantum field theory, the invariance of the behavior of systems un er t e com ine operations 0 going rom a system to its mirror ima e re lacin articles with anti- articles and examinin the evolution of the new system from the "time reverse" of what was the final state

invariance shows up in the identity of the original quantum-mechanical transltlon pro a 1 lUes Wl t e new ac ar pro a llt1es. n many cases the various transformations, taken one at a time, constitute s metries of systems as well, although, for example, invariance under mirror

symmetries, that, like symmetry under replacement of particle by antipartic e, are not lrect y connecte to space-tlme structure, suc as t e au e s mmetries of modern field theo ,are im ortant as well. Failure of symmetry can be as important as the existence of symmetry.

Second Law that is our main concern. The realization that weak quantum e processes al e 0 e symmetnca un er e exc ange 0 a system for a mirror image of that system was a crucial breakthrough in the understanding of the interaction of the elementary particles with one

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Physics and chance y ica

effects on systems making their behavior quite different from that of similarly constructed systems that were in uniform motion, was crucial to the Galilean-Newtonian mechanics and to the theo of s ac an posited to underlie it.

law that itself is such that it informs us that two systems related by some appropriate transformation of the one into the other (the first a mirror ima e of the second for exam Ie or the first in accelerated motion with respect to the second) will behave in different ways as determined by the

having that law subsumed under some higher order more general law that is itself asymmetrical in nature. But in other cases, the asymmetrical nature of the law is taken as fundamental or foundational not to be derived from some more basic law. How then can its aSYmmetrical nature

never found in any system in uniform motion. A general proposal is to lntro uce just enoug asymmetry In t e space-tIme to account or t e as mmetric henomena summed u in the as mmetric law and no more. The critical examination of Newton's notion of absolute space, and the

Once a fixed reference frame, at rest in "space itself," is posited, the notIon 0 tea so ute ve OClty 0 a system, ItS ve OClty WI respect to s ace itself, becomes well defined. Yet b Newton's own theo ,the absolute velocity of a system will have no discernible effects, all phe-

asymmetry of space-time posited by Newton seems excessive. One can, In ee , move 0 a space- lme In w c e s nc Ion IS re lne between those systems that are in uniform motion and those that are absolutely accelerated, but in which no notion of absolute rest or of

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381

but no one of the inertial frames is taken to be "at rest," with the others possessing some uniform absolute velocity. Similar refinements of the s ace-time lie behind the move in relativistic theories from the Minkowski space-time of special relativity that retains the global inertial frames of

global frames of uniform motion can be characterized. So, the general line here is to look for symmetries and for asymmetries in the foundational laws of h sics. Some of these rna have no obvious relation to space-time, such as the symmetry of processes that captures

and asymmetries we motion wit respect of the underl in s aSYffimetries among

have just noted among systems in various kinds of to one ano er, can e connecte to t e structure ace-time. When such s ace-time s mmetries and the phenomena exist and are captured by such

lying space-time as to account for the observed lawlike asymmetries and no more. Before oin on we ou ht to note however that man methodolo ical, epistemological, and metaphysical puzzles are encountered in trying

tory virtue symmetnes observed s world, is a

obtained by positing "space-time an aSYffiffietnes, an t en USing mmetries and as mmetries amon spurious virtue. For reasons of a

itself" with its various at POSit to exp ain e the henomena of the familiar reductionist or

that some of these are space-time-like in their nature having to do with e symmetnes an aSYffimetnes among systems re ate to one anot er in s ace-time-like wa s. The systems rna be related b relative osition or state of motion or mirror imaging, for example. The relationists further

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Physics and chance

really differ from one another, is something we cannot do. Instead, we isu owe ' i pui u easye 0 phenomena we have been exploring - the asymmetry summed up in the Second Law of Thermodynamics or one of its statistical surrogates - to an as mmet of s ace-time itself in somethin of the manner in which the dynamic asymmetries we have noted are so "explained." Or, instead,

from clear.

tme

summed up by the slogan that in an isolated system, entropy increases, at east in the statistical sense, in t e ture time direction. But should we view the world as one in which s stems show a remarkable as mmet in time or, more "profoundly," as one in which "time itself" is asym-

quently proved attractive. Over t e years, anum er 0 t .n ers, rom H. Me erg to P. Horwic , have ar ed that an as mmetric nature can be attributed to time itself only if the asymmetry displayed by physical systems is one that is

asymmetry of time as explanatory of the fundamental lawlike asymmetry possi e. But the continue the as mme summed u in the Second Law is now known to be a mere "de facto" asymmetry not grounded in a

sembles of systems that is represented by the differing appropriate representlng stat1st1ca ensem es or 1n1tla an na m1croscop1C states of s stems that have common initial and final macrosco ic conditions attributed to them. But if the physical asymmetry is merely an asym-

The direction of time

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planatory factor? Perhaps some of the intuitive plausibility of this argument comes from onsiderin as metries of s stems in the world that are rounded in "mere de facto initial conditions" and that are much less "grand" than the

sible stereo-isomers (mirror image molecules) that plays an organic role. Why is it that the DNA of living creatures is overwhelmingly found to twist in one s iral direction and not in the other e uall ossible direction?

elementary particles, although none of the attempts to so explain the biological asymmetry have been very convincing. Usu y, t e asymmet is credited to the "accident" that in some initial life forms the one stereo-isomer and not the other happened to play the crucial biological

duction. To be sure, this account also faces a number of puzzles, but let us assume, or e sa e 0 argument, t at it is t e correct one. Would then the existence of this molecular biolo ical as mmet be grounds for attributing some asymmetry to space under mirror imaging? " for this asymmetry of the systems without any recourse to something as at ramahc as an un er ylng asymmetnca nature to space-hme ltse. seems correct. But does a similar argument do justice to the claim that, likewise, the , " Alas, we are hard pressed to know what to say here, if for no other reason an t e act t at we rea y ave no soun account 0 t e exlanato rounds we ou ht to take as established that would ex lain the physical asymmetry. But, as N. Krylov so forcefully argued, any at-

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Physics and chance

equ11 rium a macroscopic sys ems invana y s ow - i 1S no a mere matter of fact" either. Indeed, the very distinction between what is forced by the laws of nature and what is a merely contingent initial state seems Perhaps a cosmological solution to the physical asymmetry of entropy acc u s r e u i ry r y pr pa a e initial "accident." But that doesn't seem too likely. Even the usual characterization of the cosmological picture - with the entropy of the s ace-time at the Bi Ban ve low to rovide the resetvoir of low entropy from which low-entropy branch systems are subsequently selected

of Penrose, to be explained in terms of a fundamental lawlike asymmetry. In the light of all that we have seen so far, it seems a misrepresentation to think of the entro ic as mme as "merel de facto" in an sense that would reduce it to some sense of pure contingency without

asymmetry is something that ought to lead us to attribute an asymmetry to time itself, or by H. Reichenbach and A. Griinbaum, who think of it as constitutin a sufficient basis to s eak of "time's as mmet ." Finally, we must return, once again, to the question of whether any

versal asymmetry in the distribution of initial conditions and final conditions 0 micro-states 0 systems c aracterize in simi ar ways rom a macrosco ic oint of view the initial distribution lackin the "uniformi " appropriate to a close to equilibrium state and the final distribution having

the grand asymmetry of entropic increase, isn't that enough to say that y itse it constitutes t e asymmetry 0 "time itse ?' When one reflects u on the difficulties encountered in in to osit symmetries and asymmetries of space-time as some "independent fact"

or asymmetric structures to the space-time really amounts to little more t an a conven1ent way 0 summ1ng up t e aw 1 e ynam1c symmetries and as mmetries obsetved of s stems in the world, it becomes not implausible to say that the admittedly still mysterious aSYmmetry of initial

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m. What is the structure of the Boltzmann

time itself - is not, of course, the crucial one for the Boltzmann thesis. His thesis requires, as its final step, the justification of the assertion that "what we mean b the future as 0 osed to ast time direction 'ust is that direction of time in which the entropy change is an increase."

makes us say that the future is very "unlike" the past? If we can get a little clearer on at, en we wi e a e to get a itt e c earer on just w at it mi ht be for the entro ic as mmet to unde in the more intuitive asymmetries by which we characterize the future-past distinction.

just isn't like the earlier-than relation. Getting straight on just what this mig t mean is not easy, ut it certai y inc u es t e i ea t at t e aterthan (or earlier-than) relationshi is as mmetric. It is unlike sa the relation two things bear to one another when they are "next to" one

pathological global time topologies left to the side). And, we feel, we ave some sort 0 trect awareness 0 two non-Slmu taneous events that we ex erience, an awareness that tells us which of the two events is later than the other. We need not, if the events are both given to us

about which we have only indirect information such an inferential process ffilg t e necessary. It is a little easier to come to ri s with the intuitive as mme between past and future than is summarized in the very different kind of

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Physics and chance

the original event in such a way as to allow us to infer the occurrence ve s u y uu v a so a w us s pr se to infer that the future evehts will occur. But there is some difference in the relation of present to past events and the relation of present to future events that makes the s ecial kind of correlation that enerates a "record" of a past event unavailable for future events.

And, surely, this is not a mere formal triviality in our confining the word "memory" to our knowledge of past events and refusing to apply it to our inferential knowled e of what will occur. There is somethin about the past and its relation to the present that makes it possible for there to

even a profound question as to whether or not memory is just one species of recor akin to physical records of past events, although general h sicalist considerations make us think that somethin like that must be the case, and we speculate about "memory traces" in the brain cor-

We have, as well, the intuitive idea that the direction of causal influence is a ways rom past to ture. Once again, it seems ar too s a ow to sim I claim that of events lawlike connected to one another we choose, by stipulation, to always call the earlier event the cause and the

to pin down why we hold to such intuitions with such firmness. Surely e causing - t e rna .ng or etermininge asymmetry in time 0 relation amon events is one of the crucial features about the world in time that, along with the epistemic asymmetries just noted, constitutes a

Human concerns are also radically asymmetric in time. We have a very erent attltu e towar past an uture events. ast events may e re retted or remembered with nostal ic leasure. But the cannot be planned for, anticipated, or worked for or against. The idea that the

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no use crying over spilt milk, is again one that pervades our everyday intuitive distinction between how we think and act vis-a-vis future and ast. Those metaphysically inclined find even deeper grounds for the dis" either true or false. But the future is the realm of a reality that is at best "indeterminate." This is not a denial of determinism in the sense of past events lawlike necessitatin what will follow but a denial of an kind of "determinate reality" to future events at all. There have always been

as early as the thoughts of Aristotle, is that although the past can be said to e a rea ity, t e ture, or at east t at part 0 it at is at present sti contin ent cannot be said to have an determinate reali at all. Exploring anyone of these alleged asymmetries at length would be a

2. What is the nature of the proposed entropic theory of

future from the past in their special natures are "grounded" in the entropic asymme ry 0 p YSIca sys ems In Ime. u w a oes groun e mean here? The very nature of the alleged "reduction" is sometimes, I believe, misconstrued. Only when we have become a little clearer on the intention

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Physics and chance

improper, reversed direction that is, with the initial images being of the .s argumen assumes, 0 course, a e un amen a aws canno e. of nature are time-reversal invariant and that the film is not of some process among the elementary particles where time-reversal invariance sn' hoi. But if there are entropic features of the processes involved, then one

by the spontaneous confluence of small dispersed agitations of the water in the pool is one of those processes declared so improbable as to be ruled out of court b the usual Second Law considerations. It is ar ued then, that entropic features are our guide to which events in a process

ment for accepting the entropic thesis? No, it isn't. It may very well be true that in trying to decide of a record of a rocess like a film which is the tern orall be inrun sta e and which the temporally ending stage of the process recorded that we must

those states are the earlier and which the later. It has been argued, justly I t i ,t at we etermine 0 events at are wit in our experience w ic are earlier and which later b somethin that can be called if an hin can be so-called, a totally non-inferential process of acquiring know-

and which is earlier. But we have no trouble in knowing which way the a is ro ing, an nee not see some su y associate states t at 0 have entro ic characterizations and that are simultaneous with the states of the ball in order to determine by some inference which states of the

that works by trying to convince us that all our knowledge of B phenomena IS summe up In an eVl entia aSlS t at IS tota y c aractenza e in terms of A henomena. It then uses a verificationist account of meaning to try and convince us that this evidential relationship between the "

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to assertions about the immediate contents of awareness. But such an account of reduction will not do to convince us that the phenomena of tern oral as mmet reduce to entro ic henomena. A much more persuasive case for the entropic account can be found

intuition that space, at least in our vicinity, is asymmetric in that there are distinguishable directions of up and down. The two directions in space are associated with a varie of as mmetric h sical henomena - rocks fall down, flames go upward, and so on. Indeed, some early theories of

tell which direction is down in a non-inferential way. Even locked in a darkened room we now, imme iate y an wit out i erence, w 'c wa is down. But now we understand that the up-down distinction is one grounded

that is the downward direction. The theory explains why things behave as t ey 0 wit respect to e up an own irectlons. It te s us w y, for exam Ie unsu orted rocks accelerate in the downward direction. The theory even explains how it is that we can have our "direct and

Understanding all of this, we now realize that it would be quite paroC 1 on our part to ta e t e lfect10n 1n space t at 1S own or us an declare that direction to be the downward direction everywhere. It is much more reasonable to declare at each point "down" for that point to " "down" is for us. What we mean by sameness of direction is determined y para e transport 0 1rec 10ns rom one p01n 0 ano er. , we then realize, one consequence of thinking of "down" in this way is that there may very well be places in the universe where there is no up-down

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e , wi py a e in ui ive ure-pas distinction. If the direction of time in which entropy increases is the direction of time that we label the future on the basis of those intuitive h asymmetries is totally accounted for explanatorily by the existence of

..,...,y increases as it is to say that "down" just is the direction of space that is the direction of the local gravitational force. And there would be no more oint to askin wh entro increases in the future time direction than there would be in asking why the gravitational force points in the '-J.J. ....

of the entropic account of the future-past distinction. Just as the gravitational theory of up and down provides an explanato account of how we can have non-inferential knowled e of which direction of space in our vicinity is the down direction, so we should

which earlier, given that we know that there is some time interval between em. Just as e gravitation theory 0 up and down allows for there bein laces in the world at which down varies from its direction here, so the entropic theory would allow for at least the possibility of

gravitational theory allows for regions of space in which there is no upown istinction at a ,so t e entropic t eory wou a ow or regions 0 the cosmos in which althou h there were still two directions of time it would be incorrect to refer to either of the directions as the past or as

entropy increase of small systems was in the opposite time direction rom t e time irection in w'c systems ave eir entropy increase in our vicini . And the re ions of the universe at e uilibrium for Boltzmann in his speculation the overwhelming bulk of the cosmos in space and

Surely the entropic theory of the future-past distinction thought of 1n 1S way 1S at east 1n1t1a y co erent. ut to exp ore 1tS p aus1 11ty 1t mi ht be hel ful to first look at another distinction in the world, one that, unlike the up-down distinction and its association with the direction of "

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x

2

y

The failure of invariance under mirror ima in in what are now taken to be the fundamental laws of particle interactions tells us that a system

governed by the weak interactions, the mirror image of that system will e one s oW1ng a ecay pattern spaha y arrange 1n a way t at 1S never observed to occur in nature. We could use this lawlike correlation between orientation and other, not intrinsically orientationally defined,

left-handed system or counterclockwise rotation) without even showing e person an examp e 0 e system 0 e 1n 1n queshon. e cou do this b havin the other erson construct a h sicall ossible system and then viewing, say, its rotation from direction fixed by the emission

But surely a claim that handedness is somehow "reducible" to behavior un er wea 1nteract10ns 1S a sur. e ce a1 y on etermlne w 1C of two gloves is left-handed by first examining weak interaction decay processes and then using them to fix appropriate orientations. But a

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seems manifestly absurd. Granted that there are many puzzles about the e 'g - a ys em may ai 0 a oca IS 1 io e global distinction if the space of the world is non-orientable and that the distinction is also connected with the dimension of the space into which the ob'ects are embedded there 'ust doesn't seem to be an lausibili to the claim that somehow or other the theory of weak interactions will

and one in the other all along. The theory that seems plausible is that the orientational distinction is fundamental and irreducible and that it is our erce tual awareness of the world that gives us direct awareness into the orientational structure

tion in the world, and possibly correct that without the entropic asymmetry there would be no future-past distinction of the standard kind, it 'ust seems absurd to sa that had weak interactions turned out to be invariant under mirror imaging there would be no distinction at all be-

dubious of claims to the effect that if no other phenomenon were connecte wi a given re ation, eit er y a aw 0 nature or y some ing uasi-Iawlike like the connection between entro ic increase and time order, the given relation would have to be taken as "unreal." There

nor quasi-laws, that were non-invariant under mirror imaging. Similarly, c aims to tee ect t at were t ere no suc associate p enomenon, a world with a iven structure of the relation in uestion and one that had all the relations reversed by some symmetry operation would be "one

quite misleading to make such a claim. Even if all the fundamental laws were 1nvanant un er mIrror ImagIng, ere IS no 1mme late reason to assume that a claim to the effect that the world and its entire mirror image would be the same possible world would be correct. Even claims

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p

tion, say, as the distinction between a left- and a right-oriented object seems to be, we could indeed distinguish the one from the other without ere bein an associated henomenon such as a differential deca pattern, to provide us the means for the distinction.

we seem to have in this situation is the existence of a distinction between two kinds of spatial systems, left- and right-handed ones, and the discovered existence of a lawlike correlation between some other henomena (weak interaction patterns) with handedness in the world. But the

been completely symmetric under mirror imaging, the distinction between the two kinds of handedness would remain t e same. The situation is uite different in the u -and-down case. Here we have discovered that the reason the direction we call down in our vicinity has

gravitational force at that point. Had the gravitational force been in some ot er irection, t at ot er irection wou ave een t e one t at was the down direction. And had there been no ravitational force at all in our vicinity, there would have been no up-down distinction selecting out

, distinction is proposing. At any event location, there are two directions o time. We cou ar itran y a e t em, say "p us an "mInus." Un er the circumstances of the world bein time-orientable we could extend this labeling in a consistent way so that there would be a global "plus"

these time directions as future at an event location are features existing at t at ocatIon so e y ecause ere IS, In t at regIon 0 space-tIme, an entro ic as mmet in the behavior of h sical s stems. Just as in the gravitational case, such an account will allow for one of the directions of

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w rr a ion of entropic increase with the future direction of time is to be viewed as a contingent fact about the world. If the entropic theorist is correct, it is not a contin ent fact that entro increases into th t r notion of what the future is is given by the stipulation that it is the

Before looking at a few arguments that try to make the entropic account plausible, it will be helpful to notice some other features of the ro osed account that make it somewhat different in its detailed nature from the gravitational account of the up-down distinction. Although in-

higher-entropy end state of that system will be later than the lowerentropy end state. For many isolated systems - say, a single particle acted u on b a force in a frictionless and otherwise non-dissi ative environment - there will be no entropies distinguishing states of the system. In

increase of entropy into the future is only a statistical probability and not an inevita i ity. Consider similarl the role la ed b the notion of a "common cause" in arguments about the source of our intuitive notion of the direction of

tion between such "spreading out of consequences of a common cause" Wlt t e t ermo ynamlc pnncip e 0 entropy increase w en we iscussed Einstein's claim that the as mmet of radiation (correlated outbound radiation from an accelerated charge, but no converging correlated

consequence of it. ereas t e events an ent, if we conditionalize all robabilities on the common cause event, C, probabilistic independence will be restored. That is, while

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with B. Associated with the view that earlier stages of a process are the lower entropy stages is the view that the direction of time is determined the fact that such screenin off common causes is earlier in time than the pairs of events whose correlations they screen off by their serving as

, , statistical theories as quantum mechanics cases where a common cause, in the intuitive sense, fails to screen off the probabilistic correlation between the two caused events. Other cases can be constructed where a screening off common cause can be found subsequent to the corre"ground" the temporal asymmetry can be constructed. One needs to take a somewhat su t er view of the relationship behenomtween the future- ast distinction and the aIle ed roundin enon. The basic strategy to employ was made clear by H. Reichenbach.

of two pairs of events, A and Band C and D, that D is the same time at is, we ave a notion 0 time parirection rom C as B is rom A. allelism. It is not our concern here how that tern oral relation is rounded nor how we can gain epistemic access to it. Given this parallelism, if we

, one event being a record or memory of the other or one event being the e ect 0 teo er as cause, were t e re atl0n IS In e cases In question founded u on some existin entro ic as mmet a licable to the cases in question - we can then "project" that asymmetry onto all other cases

, other as effect, even when no entropic consideration characterizes the na pOSition an events at a ,suc as In e case 0 t e Inlt1a an veloci of a oint article. Here, we could sim I decree that we take the earlier of two such lawlike connected events to be the cause and the

so building temporal order into our causal notion is that in the entropic cases, some 0 er reason eXiSts or IStlnguls Ing e cause event rom the effect event, some reason rounded in entropic considerations. The claim then continues that it is the parallelism in time of the entropic order

future) as the one in which to seek effects. Once an asymmetry of future an past as een esta IS e or t ese specla cases, were entroplc considerations play a role, we then project the notion of asymmetric causation outward b time arallelism so that even events mmetricall related to one another by a lawlike connection and for which no entropic between them can still be said to be asymmetrically related by cause and e ec. e asymme ry 0 e causa re a Ion IS now groun e m e mere fact at one event is earlier in our asymmetric time ordering than the other. Similarly, it may be argued that there is something special about those

case of the gravitational theory of up and down. That is, we need some

3. Sketches

0

some entro ic accounts

grounded - explanatorily accounted for - in the entropic asymmetries

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to be thought of as "fundamental," or, indeed, if one of them should be so privileged. One might try first to account for the asymmetry of knowled e b means of the entro ic as met t in to ex lain wh we can remember the past and not the future and why we have records of the

metry of knowledge. The idea might be, perhaps, that we are often in a situation of knowing what has gone before and wondering what will come next. Our desire to infer the future from the ast knowable b memory and records would then be argued to underlie our idea that the

causes - would then be taken to precede in time the events inferred at is, exp aine, at is, e e ects. Again a p ausi e case mig t be made out to ex lain how our ideas of ast as havin determinate reali and of the future as being a "mere realm of possibility" are to be accounted

is that anything we take to be related to something else as cause is to e ect must ISP ay e asymmetry In tIme para e to t at 0 e entroplc increase of s stems. Once havin established the entro ic basis of the cause-effect asymmetry, one might try to account for the others by argu-

happening was causally responsible, thereby building on the initial causee ect asymmetry In tIme a envative aSYmmetry 0 recor ,trace, or even memory. We will look, although only very sketchily, at approaches of both these kinds.

Now we must be careful here, for, as Reichenbach acknowledges, it

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calls "macro-entropy." It is the distribution of the sand grains themselves i a e u i rm, isor r arrangemen 0 san as we s ou n 1 more or less smoothly arrayed on an untrodden beach is a high macroentropy state. But the ordered state of the sand with the footlike pattern in it analo ous to the as on the left-hand side 0 the 0 i taken as a low macro-entropy state.

a system in a state of low macro-entropy, what must we infer? By analogy with the behavior of the micro-entropy of systems, we can take it as hi hI im robable that such a state like the foot tint is one of those rare spontaneous fluctuations of the system that are possible but too im-

past causal interaction of the system (the beach) with some external system, an interaction that generated the transient, low macro-entropy. This is 'ust like the account iven of low micro-entro branch s stems that can be causally split off from the low-entropy main system in the

indicators of their past, but not of their future. They are the result of causa interaction wit externa systems t at rna e em records or traces of that ast causal interaction. But, as J. Earman has noted, there is an obvious objection to this

this non-uniformity of the footprint as an anomaly to be explained by re erring to a past causa interaction. But in cases were we expect e order and coherence corres ondin to low macro-entro to be the norm it may very well be a high macro-entropy, disordered state that requires

interaction with an expected, orderly stack of cans resulted in the mess. more p aUS1 e account 0 w at a recor or trace 1S wou en 00 for states of the world whatever their macro-entro mi ht be that would be otherwise unexpected unless some quite specific past causal

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that play a crucial role in some of the things we call records or traces of the past. When I read of yesterday's Dow Jones average in the paper toda I take it that this record rovides me with reliable information about an event in the past. And I take it so because I believe that an

what that average was. That this account is true of memory is less clear. But it does seem at least lausible to maintain that memories rovide a record of the past because an event now - my seeming to remember a fact about

generate the current seeming memory. And that it is therefore legitimate to infer from what I seem to remem er to t e event actua y having occurred even if of course the inferences can sometimes 0 wron . But, of course, we infer as well from present events that would other-

screen, the operator infers that an explosion will occur. The appearance o e image on e screen is, everyt ing e se eing equa, quite an unlikel event. But iven the nature of the setu it mi ht be ve likel indeed conditional on the future event taking place. Now, of course, we

, causal result of what they are a record, and we think of causes as going rom past to uture. e trou e IS t at we on t want to presuppose any of that when we are t .n to round the ve notion of causal as mmet on that of the epistemological asymmetry, and we are trying to ground

The natural way to proceed at this point, it would seem, would be to rst exp ore In some ept t e nature 0 e pro a 1 lStlC re atlons among events that form the basis of our inferences from events we know to have occurred to those we wish to infer, be they events in the past or

another more probable or highly probable relative to background material wou ave to e lscusse In etal. one wou t en ave to go on to ex lain what it is about the probabilistic relationship among events that makes some events traces or records of events in the past, and never

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m c anica principles underlying the entropic asymmetry of thermodynamics. Such an account remains to be given, but let me suggest one factor that will indicate 'ust how difficult it will be to rovi e uch n that will be satisfying. We know from our earlier explorations in this

for example, such as the Principle of Indifference, are empty without something being provided to provide the right or allowable "partitions" of the event s ace. I think similar issues are bound to surface here. For any probabilistic relationship among the macro-events that one prescribes

sifying events into sorts will be found that will generate the same probabilistic relationship between the record event and some event in its future. I think that this will be true no matter how man facts about the probabilistic relationships among the usual classes of events at the micro-

The problem here is that we have not yet specified any limits to how events are to sorte, c ass' e , or partitione at t e macro- eve . Yet this must be done for it doesn't seem lausible that the entro ic as mmetry in its pure "micro" form will provide the clarification that we want

events for statistical mechanical purposes we did have some grounds, owever s en er, or t n 1ng 0 some categonzat10ns as privi ege - m articular those that used the standard measures over the a ro riate phase spaces. But that resource won't be available to us at the macro-

correlated in the right way with tomorrow's Dow Jones average, so that cou ,were to now 1S present macro- act, 1 er tomorrow s avera e. Would that I knew what this macro-fact is! But such a macro-fact will, presumably, be sufficiently "unnatural" that unlike the "natural" fact

will not give us any way of accessing it. "Natural" facts, facts somehow access1 e to our compre enS10n or grasp, may e pro a 11stica y re ate to one another in a manner a ro riate for our bein able to have records of the past but not of the future. And it may very well be that this

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asymmetric fact about the world other than the great entropic asymmetry would be big enough to do the job? But it remains an unfinished task to fill in the a ro riate details to make such a sto of a ossible reduction of the knowledge asymmetry to the entropic asymmetry a convincing

on an asymmetry that might at least be thought to be connected to the entropic asymmetry focuses on the intuitive asymmetry in time of the causal relation. If we can show that our sense that causation alwa s proceeds from past to future is founded on the entropic asymmetry, we

influence the future but not the past might follow from the fact that actions influence by means of causation. One might even try to ground the e istemolo ical as mmet on the causal. Here one would ar ue that traces or records, or even memories, are, perhaps by definition, the

could be a record of that event only if that event was somehow causally instrumenta in orming t e recor: , weer is is e nitiona 0 w at it is for the record to be a record of that event or not. There is therefore at least plausibility to the claim that if we can show the causal relation

have records of the past and not of the future as well. ne very c ear an very suggest1ve attac on s pro em 1S ue to D. Lewis. Lewis wants to characterize the causal relationshi b the use of counterfactual conditionals, thinking of the cause of an event as an event

from a variety of different directions, including issues of preemptive causes an 0 non-causa epen enC1es t at stl groun appropnate counterfactuals. But the sim Ie version will do for our ur oses. When we think of what would have happened had some events in the world been other

occurred. But, Lewis insists, even if we sometimes initially think that a an event een ot erw1se, t e past re at1ve to t at event wou ave been otherwise as well, on reflection we usually drop our confidence in such "backtracking" counterfactual conditionals. If this is so, then, we

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wa. a e causa i w u i a u event not occurred, its future effect events would not have either. But the absence of that actual event would not have changed what happened in i t h t causal influence 0 ld ro....."'.."t-.o::·r1 into the future. IJ'U'........,

familiar "macro-entropic" facts are noted about how localized distinctive events lead, in the future, to correlated macro-events at separations from one another without there bein an such arallel association of a sin Ie event with a set of correlated separate events in the past. The wave

familiar examples. Now it does seem plausible to argue that had the stone not been tossed in the ond an one of the local ieces of a circular wave propagated out from it would not have occurred. Yet we are more re-

given the remaining disturbances caused by that initial pebble-tossing event, t eir existence, even' one itt e it 0 the wave is t ought out of existence is still ood reason to believe that the stone tossin occurred in their past. One single event at a time, the stone tossing, gives rise

probabilistically by the occurrence of the single past event. 0 ow we eva uate counte actua Now, e argument goes, t ' conditionals. We think of a world like ours exce t that in this new "possible world" the event taken not to have occurred in our counterfac-

veloped ricWy by Lewis, we can think of possible worlds as "closer" or er rom one anot er. 1n 0 e wor 1n w c a glven event, A which actuall does occur has not occurred. What is that world like? To decide this we should ask ourselves what the possible world is like

Of course, there are lots of technicalities involved in these notions. For examp e, 1S 1t c ear t at one an 0 y one 1st1nctlve POSS1 e wor w , in eneral, be closest? A ain, if our aim is to decide which counterfactuals are true, our possible worlds picture isn't yet much help. We need

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the criteria we do in fact use in thinking about possible world nearness. We generally think that worlds with "big, widespread, diverse" violations of the laws of nature are ve far from the actual world. Next he ar ues our intuitive evaluations of the truth of counterfactuals leads us to

, we then seek to avoid even local, small, violations of the laws of nature in our wor d. Fina y, he argues, it is of Ii e or no importance that we secure a roximate a reement with re ard to matters of articular fact even if these facts are of great importance to us. As Lewis demonstrates,

button connected to a Doomsday machine, the counterfactual we take to e true comes out true i we imagine .m pus .ng t e utton. Dooms ay occurs rather than sa his ushin the button and the si nal down the wire violating the laws of nature on the way to the machine or any of the

needed to allow an event that did happen not to have happened, eeping t e past 0 at event constant. But a gigantic one wou e needed to allow that event to have ha ened kee in the future exact! the same. Why? Because events are usually connected in the past to only

button). But events leave an enormous multitude of traces or records of elf occurrence into e ture. or to ave not occurre w en it actua y did would re uire a vast arra of miracles to break the connection of A with all of the multitudinous other future events that were, as a matter

our intuitions about the closeness of possible worlds and hence about e con lions or coun e ac a s, to ge us time-asymmetry. lrst, there will be a time-asymmetry of counterfactual dependence, and then, from the characterization of causation in terms of counterfactual depend-

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iagnosis 0 11 r v u i u as really is the source of our felt time-asymmetries about causation. As usual, the theory is that such macro-entropic spreading situations give us n rf: I n dependence, and that this time-asymmetry is then projected onto the

classes at the macro-level. If we are perverse enough we can reassort the events of the world to generate, even in the stone-in-the-pond cases that are aradi matic for Lewis' ar ment macro-s readin into the ast and not into the future. So, once again, much insight into why some ways of

that I do not know how to connect the several asymmetries I have discussed with the famous asymmetry of entropy." Well su estions can be made. As we noted in Cha ters 2 and 8 Einstein and others have made serious efforts to connect the outbound

converging on it in perfect correlation, with more purely thermodynamic/ statistica mec anica assumptions a out emitters an a sor ers in e universe. It is certainl uite conceivable that with sufficient insi ht we could construct an argument that starts with the standard temporal asym-

sible generation of the conditions in the natural world that ground our intuition 0 causation gOIng rom past to ture or 0 recor s elng 0 y of the ast and not the future. Perha s we could even ex lain the oneway efficacy in time of the process we call memory. It remains to be seen

4. Our inner awareness of time order Let us suppose that we can in fact provide the filling-in of the Boltzmann

isolated systems in the world. Let us suppose, that is, that we can explain, In t e manner c aractenze In t east sectIon, w y It IS t at we ave records and memories in one, arallel, time direction and not the other, and why we intuitively take it that causation goes from the direction of

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Were that so, how would it be best to describe the relationship between the "after" relation and the relation one state of a system bears to another when the former state is in the time direction from the latter in which entropy, in general, increases? Let "alb" mean" a is later than b." Let "aEb" "

, systems." What is the relation of the l-relation to the E-relation? One suggestion that has at least initia p ausibi ity is that there is only one relation involved here a sin Ie relation that can ro erl be called the l-relation or can properly be called the E-relation. That is, that the

9,1,2, in the context of theory simp yare arrays 0 so ium an em irical discoveries. Yet the truths, that each salt crystal is

reduction. We discover that salt crystals c orine ions. Suc micro-reductions are are discoveries that certain identities are the same thing as a specifiable array of

discovery, for example, that light waves are nothing but electromagnetic waves w ose requencies are wit in a certaln speci e range. It isn't that li ht waves are associated with or correlated to the resence of electromagnetic waves (as smoke might be to fire, say), even in a lawlike

direction is just being the direction in which the gravitational gradient is ownness ts Just t e gra lent 0 t e gravltatlona potentla pOlnting an pointingness." Of course, all of this has now become "localized" as noted before, the direction of the gravitational gradient varying from point to

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wave, 1 1S argue , 1 cou no e e case wave that was not an electromagnetic wave. Now none of this is meant to deny that there might be things in the lated retinas, reflection off metals, and so on - but were not electrois suc a e rig ay i u i a ese wou no be light waves at all, because they are not electromagnetic waves, and that is what light waves are. Instead, they are things other than light aves that behaved as li ht waves do. In some other ossible worlds there might be a substance that filled oceans, that was used to make 2

•

like), because "water" is the word we use to refer to that substance that has the appropriate identifying features in this world, and in this world that stuff is H O. Nor of course is an claim bein made that denies the fact that it is an empirical discovery on our part that light waves are, in

the gravitational field, then it is necessarily the case that that is what "down" is. In this context how lausible does the claim look that "after" is 'ust "in the time direction in which entropy generally increases," construed as

An important part of the process of substance identification as it appears in science is t e "secon arizing" 0 properties. "Loo ," it mig t e

ar ed "how can we think of li ht waves as bein identical to electromagnetic waves? Light waves have color, but there is no place in our

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of any physical object. We have seen how such a stripping away of "secondary qualities" from the physical system is instrumental in the i ntificato account of the reduction of thermod namics to statistical mechanics.

philosophical theories of the relation of mind to brain that argue that mental processes - for example, having a pain - can be simply identified with a ro riate h sical rocesses in the brain. In this case there is no place to which to remove the "secondary quality," so that if we are to

quality" of the pain. The implausibility of doing this as e many, in erent ways, to e ske tical that the identificato reduction of mind to brain can be carried out, at least where such mental processes as the experience of qualia

vokes modal intuitions in its formulation. If pain really is a stimulation of neura ers, en is is a necessary trut . But it seems to us t at we can ima ine ain without neural stimulation or neural stimulation without pain, and the possibility of such an imagining is usually taken as good

Now the mental-physical identity theorist can reply that we might think e wrong. ater t at we can unag1ne wat€r t at 1S not Hz , ut we wou bein H 0 it is necessaril H 0 and our vaunted ima ination misleads us as to the apparent contingency between being water and having the z case, we can explain why we are deceived into thinking that we can 1mag1ne water t at 1S not z ecause we can 1mag1ne somet 1ng ot er than water havin those "accidental" or "non-essential" ro erties b which we originally identified stuff as water (such as filling oceans, being

neural stimulation or neural stimulation without pain isn't of that sort. ur 1reet ep1steffilc access to w at pa1n 1S 1 e te s us w at t e essential features of pain are and what they are like. And knowing what we know of these essential features, we see that an explanation of our

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by neural stimulation, and there is no "accidental" feature associated with

going on no pain would be felt. Why is all of that relevant to issues concerning the possible identity of "afterwardness" and "bein in the direction in time of hi her entro ?" To see this, one needs to consider some arguments of A. Eddington,

is at pains to emphasize to his audience the fact that temporal asymmetry reveals itself in the physical world in the form of the entropic asymmetry and in no other uise. But he is also at ains to indicate his rave doubts that one could identify temporal asymmetry "as we experience it" with

thinks, in the usual empiricist way, to the contents of direct awareness. At east t at is so or most aspects 0 t e wor . Lig t, e c aims, is just a "somethin " identifiable b its ultimate henomenal causal effects (hitting our retinas it causes us to have visual flashes). Electromagnetic

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features of that other sort of entity and knowing that two such radically different kinds of things could not be identical with one another. At least, this is the Eddin ton version of an ar ument uite closel related to that used by Kripke to defend a kind of Cartesian argument against the mind/

ascribe to the physical world are of the kind known only indirectly to us through their causal consequences in our world of phenomenal experience. Certainl we will want to sa this is true of such thin s as electric charge or strangeness or electromagnetic field potential. Someone like

things of ordinary life. Even the spatial aspects of things, he would say, are 0 y in ire y own. Even s ape, size, and other geometric features as meant to characterize the realm of the h sical are onl "inferred features" that are, in Eddingtonian terms, "only symbolically known."

the components of our visual awareness have shape and size? Aren't they spatia y re ate to one anot er in posItion? Here the Eddin onian would ar ue that there has been an e uivocation on the notion of "spatial feature." He would argue that one must not

postulating the nature of the former, the two are not to be identified. er a, ow our Vlsua expenence goes IS eVl ence to us concernIng the nature of h sical s ace, but when one considers what h sics now says about the nature of physical space (say, in the general relativistic

tion and reality with which the various alternatives suggested to take its p ace ave never a equa e y ea. uppose IS account IS p aUSl e. Can we then go on to argue for a radical disidentification of all features of the realm of the perceived from all features of the physical realm?

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e ermines i . ime i e p ysica wor ca 0 e, e 1n s, a mere "something," some relation I-know-not-what, identifiable by us only "symbolically" by its place in the explanatory structure we posit to acn

understanding of the world at all, he seems to think, we must take it to xp temporality in the temporal relations our inner experiences have to one another. Then comes the ar ment that we know what "later-than-ness" is like directly and non-inferentially. We know it because the fact of one event's w

what this is like, we know that it is not a relation like that of one state of a system being more disordered than some other state and separated from that other state in time. So we know that whatever the relation between temporal asymmetry and entropic asymmetry may be, it is not " " Perhaps it would be worth ending this discussion with an extended quote rom E ington, one t at espite its 0 scurity, is ric wi suggestiveness as to 'ust wh the issues conneetin time itself with related physical features of the world in time, like asymmetric entropy increase,

physical sides of our nature, Time occupies the key position. I have already which relate it to the other entities of the physical world, and directly through a

is really electromagnetic wavelength, I do not think he would say that the famil1ar movmg on 0 time 1S rea 0' an entropy-gra lent... ur trou e 18 at we have to associate two things, both of which we more or less understand, and, so as we understand them, they are utterly different. It is absurd to pretend that we are ignorant of the nature of organization in the external world in the same way that we are i norant of the nature of otential. It is absurd to retend that we have no justifiable conception of "becoming" in the external world. That dynamic uali - that si nificance that makes a develo ment from the future to ast farcical- has to do much more than pull a trigger of a nerve. It is so welded into have direct insight into "becoming" which sweeps aside all symbolic knowledge

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this is great, and the issues far too deep to explore here. What we can say, though, is this. That although the notion of property identification . ht serve ve well to let us sa what we want to about the relationship between down and the direction of the gravitational gradient, it will

, even if the Boltzmann thesis is made plausible to the degree that we become convinced that all of the intuitive temporal asymmetries can be ex lained on the basis of the entro ic as mmet of s stems in the world. On the other hand, given the role memory plays in our immediate

IV. Further readings direction of the future can be found in Brush (966), Chapter 10. For intro uctions to t e nohon 0 symmetry 0 aws an its re ation to s mmet of s ace-time see Sklar (1974) Cha ter V Section Band Earman (989), Section 3.4 and 3.5.

only lawlike asymmetry would be sufficient for space-time asymmetry is gIven In e erg an In OrwlC apter . For an oudine of the intuitive time as mmetries, see Sklar (974), Chapter V, Section A, Part I.

, (956), Part N. See also Griinbaum (973), Part II, Chapter 8, and Horwich an . ee a so ar apter , apters" On "the princi Ie of the common cause" and "screening off," see Reichenbach (1956), Part IV, Chapter 19. See also Salmon (1984), Chapter

(982). n IClsms 0 e en roplC accoun 0 e ongln 0 e In UI lve asymmetries can be found in Mehlberg (1980), Volume 1, Part 1, Chapter V, and Earman (974). See also Sklar (974), Chapter V, Section F.

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.

.-

.

.-

.

un {fie 1ssue Or oUrutreCl awareness 01 lemporal asymmeuy, see Eddington (1929), Chapter 5, "Becoming." See also Sklar (1984), Chapter 12. For the related argument concerning mind and brain, see Kripke (1 (72) no . .. 144-1';';.

11

At this point, it might be useful to take a retrospective look at some what extent the questions have been answered and to what extent imortant uzzlin issues still remain to be resolved. The reader will not be surprised by now, I expect, to discover that it is the author's view that fundamental and important ways. tions in statistical mechanics requires, I believe, some version of an interpretation of probability that views it as frequency or proportion in the physical world. Although "subjectivist" or "logicist" interpretations certainly of indifference in generating the posited probabilities, and although, as

But, as we have also seen, this understanding of probability is fraught

explain the real behavior of systems. There are also those serious problems

speculation, there has been a continuing necessity in understanding the

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r u pro a . i .es derived by some theoretical schematism or other are to be appropriately associated with actual and experienced proportion in the world. As we have seen one of the most crucial ro Ie s i e of statistical mechanics is the justification or rationalization or explana-

success in the equilibrium portion of the theory. Here, the results of Ergodic Theory do go quite a way toward showing us, on the basis of the constitution of the s stem and the laws overnin its micro-d namics that the standard probability measures have a distinctly privileged status

logical arguments, the neglect of sets of measure zero remains an interesting problem for the theory, a problem not resolvable by resort to structure and micro-d namics alone. A ain the fact that most realistic systems pretty clearly do not meet the conditions for a system to be

the equation of evolution we want, but only at the cost of imposing on t e eterminlstic evo ution 0 t e ensem e a contlnua reran omization that is im ossible to 'ustif iven the fact that the evolution is deterministically fixed by the initial ensemble distribution.

bution - is essential to get the non-equilibrium results we want. These e appropnate netlc lnc u e t e appropnate nlte re axatlon hmes an behavior in the a roach to e uilibrium. Whether we model a roach to equilibrium by one of the mixing approaches or by the alternative Lanford

, concocting proposals that would explain why the familiar probabilistic POSltS are JUSh e ,an w y t e symmetnc POSlts at wou generate anti-thermodynamic behavior are not, no convincing rationale yet exists for why the standard probability assumptions over initial conditions work

The current state of major questions

415

x

case that matters are most clear. Here, as we have seen, Ergodic Theory, combined with elements that rely upon the thermodynamic limit to connect avera e values with overwhelmin I most robable values ives us a framework in which many questions can be answered. It is important

sophical accounts of statistical explanation would have led us to expect. In the equilibrium case, it is neither subsumption of events under statistical eneralities nor robabilistic-causal ex lanations that we derive. Instead it is the "transcendental deductions" of the equilibrium distribution that

the standard probability measure over a phase space is the unique timeinvariant measure absolutely continuous wit e standard measure. We have seen how such results can be a lied to t to round ar uments of the sort, "If equilibrium exists and has a certain statistical character-

why they get there by the route they actually follow. It is a so important to remem er ow many puzz es sf remain even in the e uilibrium case and even when the modest notion of statistical explanation is the one had in mind. The measure zero problem and the

in the non-equilibrium situation, the picture is much less clear. I have argue at some approac es to accountmg or non-equ 1 num e aVlor, for a varie of reasons seem less lausible than the more orthodox accounts. Thus, I have argued, neither Jaynes' subjectivism, nor interven-

equilibrium. ut, as we ave seen, t ere lS not a slng e ort 0 ox approac to the roblem but, instead, a varie of different a roaches. We have also seen that in some cases, as in the opposition between the mixing approach

between the approaches taken. Although both models rely on initial en pro a lty assumptions over t e mlcro-con ltions 0 a system an follow out the evolution of the initial ensemble by using the microdynamical laws, the models offered of approach to equilibrium are quite

416

Physics and chance

It is very important to note how the two approaches rely on distinct ".

" in a r: - p e 0 e r mo ecu es, an 0 en examp e, on sue resort to infinite time limits. The Lanford approach, on the other hand, relies upon utilization, crucial utilization, of the Boltzmann-Grad limit. The role la ed b the man de rees of freedom of the stem i as we have noted, quite different in the two approaches. One can learn

fundamental physics, idealization plays a subtler role. Whereas in simple cases, which idealization is the right one to pick is clear, here it is not. Where in sim Ie cases what errors will be introduced when one oes to the idealization may be calculable, here the wayan idealization may

such idealizations, their success in answering our original "why?' questions always remains in doubt. Within an of the orthodox a roaches however the attern of statistical explanation is that of probabilistic causal explanation familiar to

then follow from the assumed deterministic laws. Of course, what we can say a out ose trajectories 0 interest to us varies rom one mo e to the other. But all will suffer the roblem of accountin for the initial probability distribution. As we have seen, this is the most central and

proportions in initial states, the separation of the system from the cosmo ogica main system, singu ar initia states, among 0 ers it seems uite clear that there is no sin Ie account of wh initial states of s stems are distributed as they are that is generally taken to be satisfactory. Of all

So great is the disagreement in this area that the opponents will often not even agree a out t e appropnate onto ogy to presuppose. oug most of the orthodox adhere to exact micro-states of s stems, there are, as we have seen, radicals who will seek the solution to the initial prob-

The current state of major questions

417

ity of a dispositionalist sort for the new statistical states that have been substituted for the rejected exact micro-states. But, as we have seen, such ro osals need a Ion wa to 0 before we even understand them well enough to judge their plausibility.

increase. From the orthodox perspective, the problem of irreversibility is just a sub-problem of the problem of picking the initial probability distribution. It must be em hasized that the bi er roblem ickin the right distribution and rationalizing it, would still exist even if the asym-

problem is. For now we must find a rationale for the initial probability e parity-o -reasoning distribution t at oes not a prey to one 0 ar ments that informs us that anti-thermod namic behavior is 'ust as inferable as is thermodynamic behavior on the basis of the rationale

It is to resolve the problem of temporal asymmetry that cosmology has

een intro uce into e picture, or at east at is t e primary aim 0 introducin it. We have also noted how cosmolo has been invoked since Boltzmann's time in attempts to reconcile the apparent conflict

very far from equilibrium. As we have seen, cosmology cannot be counte y success even in accomp is lng t is atter tas. or, as we have also seen, does the invocation of cosmolo ical facts, now in the form of one version or another of a Big Bang cosmology, fully resolve

to the quite fascinating problem of trying to account for the temporally aSYmmetflc mcrease 0 entropy 0 t e universe ta en as a woe. s we have seen, there is no clear solution to that problem that has been universally accepted. Indeed, it isn't completely clear just what kind of

one that I believe has not received its due in the previous discussions of cosmo ogy an lrreversl 1 1 . ave argue a 1 is very ar rom c ear how one can derive the kind of temporal asymmetry one desires from the cosmological entropic asymmetry - that is to say, the statistically

418

Physics and chance

empi ing to derive this branch system asymmetry from the temporal asYffimetry of cosmic entropy increase are all fatally flawed. I have also argued a this remains true even when the deriva ion like that f ich is based on its own additional statistical posits, at least so long as those

formulation of his so-called Lattice of Mixtures, and that it is in this trick, not in the time direction of entropic increase in the cosmos, that the time-as met derived in his roof of arallelism for branch s stems is to be found. I think that such hidden asymmetric positing goes on in the

pieces cut off from the main system than spontaneous fluctuations from isolated high-entropy states, the direction of entropy change of the cosmos cannot b itself I have ar ued rovide the full reason for arallelism in entropic increase of the branch systems.

better put by saying that they would argue that what thermodynamics an its un er ying statistica mec anics sows us is at one cannot ope to understand these theories so Ion as one remains committed to the standard ontology of the traditional micro-theory. I am thinking here of

cal picture that seems satisfactory overall is the orthodox one, in which t e exact m1cro-states genu1ne y eXlst, 1nsta 1ty an a. But even from this orthodox ers ective, the issue of reduction is one that remains complex and subtle. From the orthodox standpoint, we

and the reducing statistical mechanical theory are, in their conceptual structure, qU1te un 1 e e more usua eones 0 p YS1CS. or 1S reason, the reductive relationship in question is more complex than the usual reductive relationship grounded on identificatory reduction at its

The current state of major questions

419

minating sorting out of the variety of notions implicit in the original naive versions of the theories. It is also im ortant to remember the one clear sense in which it does seem so far that thermodynamics remains an "autonomous" theory. Even

conditions in the world that are essential to derive its basic non-equilibrium results. It remains very hard to see how such posits can somehow be eliminated or made innocuous. To this extent thermod namics and statistical mechanics do require the positing of fundamental facts about

the systems to which statistical mechanics and thermodynamics are applied. Finall we come to that as ect of the discussion that has seemed of most interest to philosophers - the question of whether our intuitive

example, the debate over whether entropic asymmetry constitutes an asymmetry 0 time or mere y an asymmetry in time - may not e rea issues at all. But the uestion of whether an ex lanato account of the origin of all of our intuitive temporally asymmetric concepts can be found

the final Boltzmann thesis. I have argued that although that final thesis 1S 1nte 191 e an ar rom reJecta e as a sur on 1ts ace, 1t 1S a so ar from bein established. The difficulties encountered in trying to ground all of our intuitive

claim, a basic asymmetry to time itself, and it is this aSYmmetry that accounts t or our 1ntu1tive asymmetnes 0 ow e ge, causation, and concern, and for the entro ic as metry as well. I have not explored these claims in this book. Suffice it to say at this point that al-

and the future being merely a realm of possibility, we are far from having anyaccoun a wou e accep e y mos as even 1n e 191 e. ven if we had one, it is very far from clear how the intuitive asymmetries would really be accounted for on the basis of the metaphysical model.

Physics and chance

420

Boltzmann to the present cannot but help be struck by the way in which

as deep puzzles for over a century. Attempts at solving the profound quandaries at the foundations of statistical mechanics have led to some of the most innovative conce tual develo ments in h sics. Furthermore, whole rich branches of mathematics, such as Ergodic Theory in all

tal issues that arise when probabilistic reasoning is applied to the dynamics of systems. Yet despite the richness of the resources that have been develo ed and des ite the immense clarification of the issues that has been obtained, the most basic questions of the explanatory accounts to

It is impossible to resist ending with a quote from an important worker

in t e e

,J. Le owitz:

Material demanding familiarity with mathematics or physics at the intermediate level is marked e). Material demanding comprehension of more advanced mathematics or h sics is marked

e· .

Benjamin.C··)

Balescu, R. 1975. Equilibrium and Nonequilibrium Statistical Mechanics. New Barbour, J. 1989. Absolute or Relative Motion, Volume 1, The Discovery of Dynamzcs. am n ge: am n ge mverSl ress. Barrow, J. and Tipler, F. 1986. The Anthropic Cosmological Principle. Oxford: or Umverslty Press. Batterman, R. 1990. "Irreversibility and Statistical Mechanics: A New Approach?" Philosophy of Science 57: 395-419. Batterman, R. 1991. "Randomness and Probability in Dynamical Theories: On the Pro osals of the Pri 0 ine School." Philoso h 0 Science 58: 241-263. Berry, M. 1978. "Regular and Irregular Motion." In S. Joma, ed., Topics in Nonlinear k· s· "

Theoretical PhySics 22: 745-755. Harvard University Press.

Studies in Statistical Mechanics. Am-

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Denbigh, K. and Denbigh, J. 1985. Entropy in Relation to Incomplete Knowledge. Devaney, R. 1986. An Introduction to Chaotic Dynamical Systems. Menlo Park, : enJaml ummmgs. Dias, P. and Shimony, A. 1981. "A Critique of Jaynes' Maximum Entropy Princip e." A vances in App ie Mat ematics 2: 172-211. Earman, J. 1974. "An Attempt to Add a Little Direction to 'The Problem of the Direction of Time'." Philosophy of Science 41: 15-47. Earman, J. 1986. A Primer of Determinism. Dordrecht: Reidel. Earman . 1989. World Enou hand S ace-Time. Cambrid e MA: MIT Press. Eddington, A. 1929. The Nature of the Physical World. Cambridge: Cambridge

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~ V.

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,

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... .

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/0"",,1 /"1-.

1

.. .

...

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J

•

.

~

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,;"" n

or,

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.

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'~v,

,

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~

IVI.

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,1

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~'1:0· -~OV •

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In

x

action-at-a-distance e ectro ynamics, 307 a la a IC systems, Adrian, R., 30

,

,

,

.,

astronomy, KAM Theorem application to, 1 5

asymmetry intuitive, 385-396, 411, 419 In mo ecu ar 10 ogy, i - s autonomous theory, thermodynamics as,

Ehrenfests' critique of, 60-67, 415 entro of 40 41 354-355 356 35 H-theorem of. See H-theorem probabilism of, 38-44, 50, 125-126, s ,

,

317, 319, 320, 327, 375-378, 390

, Bacon Francis 17 Baker's Transformation, 273, 274 Bayes' Theorem, 94, 190 approac 0 non-eqU11 num,

,

,

Bernoulli, Daniel, 17 contributions to kinetic theo ,28, 29 Hydrodynamics, 28 Bernoulli sequence, 92, 93, 98

,

,

definition of, 238-239 Big Bang cosmologies, 300-303, 308, 309, 311, 316, 317, 377, 384, 417 Big Crunch, of the universe, 309, 316, 317 biological clocks, self-stabilization in, 28

.

.

,

.,

,

,

,

,

,

90, 156, 200, 201, 202, 207, 252, 288, 415 ensemble transformation of, 208, 209 generalization of, 85, 210, 212

,

Bell Theorems, 144

,

,

287,295 295, 415, 416 Boltzmannian behavior, 225, 226, 227, 288 Boolean algebra, of propositional logic,

, ., Boyle, Robert, 406 branch systems, 318-331, 332, 418 bridge la~s, 337-338, 373 Brillouin, L., 38 Brute posit approach, of Tolman, 195

,

contribution to contemporary ergodic

caloric-substantival theory of heat, 17, 28,

Index

430

,

, ., Law of Thermodynamics by, 21-22 Carnot, S., contributions to thermodynamics, 18, 20, 21 causation, 328, 344 Humean critique and, 128-131 's ss'

asymmetry of, 378-384, 411 C systems, 239, 263, 356 dynamical instability in, 234 Culverwell, R, 42

,

, and, 140-148 Central Limit Theorem, 94 chaotic evolution, varieties of, 235-240 chaotic systems, 202, 223, 224, 283, 29 macroscopic, 244-245

de Broglie, L., 72 F" 11 derivational models, of reduction of thermodynamics to statistical mechanics, 333-337 Descartes, Rene, 129

, Dicke-Brans scalar-tensor theory, 1 DLR condition 81 DNA, 370, 383 Mendelian genes as, 340 Do rus in, R.., 81 Duhem, P., 34, 131

,

, distribution in, 156-157 coarse- rainin in K systems, 237, 238, 241 use in kinetic equation derivation, 212, 2 3, 1, 21 , computer models

,

dynamical laws, time-asymmetric, 246-250 d amical s stems com uter models of 222-224 dynamics, 15 Earman, ].,398

, The Nature of the Physical World, 408 Ehrenfest P. and T. Ehrenfest 212 215 257, 259, 284, 285, 415 criticism of kinetic theory, 59-71, 85,

,

,

critique of Ergodic Hypothesis, 160, 164 cosmic time, 301 Cosmolo kal Ar ments for existence of God, 312 cosmological principle, 301 cosmoogy in approach to time asymmetry, 295, interventionist approach combined with, 296 irreversibility and, 297-332 modem, 331 ra la on asymme ry an , statistical mechanics and, 10, 295

Einstein, Albert, 1, 4, 24, 72, 248, 249, 394 avitational field e uations of 301 Einstein-Rosen-Podolsky thought experiments, 144 e ectromagnettc aws, symmetry 0 , un er reflection in space, 247 philosophical issues of, 3 theory of, 1, 2-3, 4 emergence of new structures, 344 o erma ea ures, empiricism, vs. conventionalism, 1

Index

431

fate in universe, 19

190-194 r

ensemble(s) canonical, 50-52, 54, 83, 204, 350, 351 dynamic evolution of, 54

n

equilibrium velocity distribution law. See Maxwell law Ergodic Hypothesis, 45, 76, 159-162, 163, 167, 168, 186

187, 195 er odici equilibrium statistical mechanics and, 175-194 equilibrium theory and, 188-190, 279 limits of, KAM Theorem and, 169-174 351 non-e uilibrium 82-86 199-207 origins of, 44-48 panta-micro-canonica, 158, 183, 19 probabilities, 176

295 Er odic Theorem 61 62-6 67 76-78, 79, 89, 94, 116, 166, 167, 358, 414, 415, 420 origin 0 , - 8, 12 ergodic theory

,

, Enskog, D., 82, 84, 373 entro asymmetry, change in time of, 382-384, 387-396, 397, 400, 401, 404, 417, 419,

philosophical observations on, 195 errors theo of 30- 1 Euler's equations, 16 evolutionary biology, concepts in,

, Boltzmann's concept of, 40, 41,

,

,

coarse-grained, 56, 57, 58, 68-69, 85, 109-110 212 243 257 278 354-355, 356-358, 374, 415 concepts of, 3, 4, 337, 354-355, 374 ne-gram, , 356, 357, 358-359

expansion, of the universe, 303-305

, Feigl, H., 338 Fermi Enrico 222 Fermi-Dirac distribution, 73 First Law of Thermodynamics, 20, 345, "footprint-in-the-sand" system, 397-398

non-conservation of, 25, 27 non-e uilibrium, 374 of the universe, 303-305, 309-318, 319, 331, 377 eqU11 num revised notion of, 23

Galilean space-time, 380 Galileo's law of fallin bodies, 335, 336 r-space, 63, 66, 67, 68, 79, 85, 199, 202, 207, 212

ergodicity and rationalization of, 175-194 equilibrium theory, 4, 9, 20, 71-76, 156-195

gauge field theory, 3 generalized master equation, 228-232, 295 general relativity theory, 1, 2, 4, 369, 409 geome 1C eones, Gibbs, Josiah W.

432

Index ,

ensemble approach of, 48, 81, 157, 204, statistical mechanics of, 48-59, 90, 191, 262, 263, 278, 285 Ehrenfests' criticism of, 48-59, 67-71, 257, 259, 284, 415

195 intertheoretic reduction, philosophical models of, 333-344, 373 interventionism, 329-330, 332 interventionist approaches o

0

, invariant measure, of probability, 1

-1 2

invariant probability distribution, 47 irreducibility, of thermodynamics, 368 irre uci ile p enomena, 343, 371 irreversibility

distinction between

conservation and, 16-20 cosmolo and 2 7- 2 interpretations of, 246-279 mainstream approach to, 260-262 origin 0, 10 Ising model, 75

, importance to Industrial Revolution, 18 theo of 18-19 21 heat flow, temperature as driving force of, 16 eat ow equation, Heisenberg, Werner K., 72, 266 '0'

,

,

subjective probability approach to non-e uilibrium of 255-260 Jeans, James, 66 Jeffrey, R., 113 Je reys,., , Joule, James P., 29

,

Herepath, W., 29 Hilbert s ace ra behavior in 72 Hobson, A., 194 Hopf, Eo, 78, 168, 262

323,367 statistical a roach to, 63-66 Hubble red-shift, 301 Hume, David, on causation, 128-131, 142,

, Humeanism, 127, 139

Ideal Gas Law, 16, 22, 346 identificatory reductions, 337-341 inductive logic, theory of, 117-118 n us a evo u lon, ea engme importance to, 18 o

0

26 Kelvin-Planck version of Second Law of Thermodynamics, 21 Kemeny, Jo, 334 emeny- ppen elm accoun 0 reduction, 335 333, 335, 336 Keynes, J., 117 Khinchin, A., contribution to equilibrium theory, 162-164, 177, 195, 201, 279, kinematics, 15, 18

Index

433 ,

derivation of, 202

macro-entropy, 398, 402, 403, 411

.

.

kinetic theory in derivation of kinetic behavior, 207-210 early, 28-30 Ehrenfests' criticism of, 59-71 o gases,

,

,

,

Markovian postulate, 214, 215, 218, 232,

objections to, 34-37

Markov process, 85, 284, 285, 287 Markov sequence, 92 Martin, P., 81 Martin-Lof, P., 108-109

revision of, 23 Kirkwood, J., 207 KMS condition, 81 Kolmogorov, A., 78, 168, 171 ,

,

,

,

.,

,

Krylov, N. S., 78, 168, 286, 287, 292, 294,

use in dealing with non-equilibrium,

contributions to non-equilibrium statistical mechanics, 262-269 The Foundations of Statistical Physics, 262

matter fundamental constitution of, 2-3 magnetization of, 22 Maximum Information Theoretic Entropy

,

,

,

,

dynamical instability in, 234

R

1

Landauer, R., 38 Lanford, 0., 81, 12 , 22 261, 277, 414, 416

,252,

420 calculation of u'lib 'um 45-48, 157, 159-160 contributions to kinetic theory, 14, 30-31, 3 , 3 ,37, 1, 4 , 59, 90 probabilism of, 38-39, 50

, 178 Maxwell Maxwell 156 Maxwe Maxwell

,

"

liouville's Theorem, 180, 212, 243 Lobachevskian three-s aces 302 Locke, John, 406 Lorentz invariance, 379 Losc ffil t, J., reverS1 11ty argument 0 , 35, 36, 39, 59, 60, 221, 250, 253, 313,

,

,

,

Loschmidt Paradox, 263, 267 LSZ scatterin e of 2 Luder's Rule, 289 ac, . as opponent of kinetic theory, 34, 44,

Demon 8 88 221 distribution law, 30, 31, 32, 60, potentia, 31, 82 transfer equations, 81, 82

,

,

mechanical work, derived from heat en ines 19 Mehlberg, H., 370, 382, 384 Mendelian genetics, reduction to mo ecu ar 10 ogy, metric indecomposability, 165-166, 181

,

,

,

Minkowski space-time, 297, 335, 369, 381 Misra, B., 242 mixing o msou e U1 s, of systems, 358, 416

434

Index ,.,

83, 84, 85, 224 motion, dynamical theory of, 18 multiple realizability, property of, 352 J.l-space, coarse-graining of, 354 - 355

,

Penrose law of initial smoothness, 317

.

..

phases, phase changes and, 75-76 phase-translation problem, 88 philosophy, foundations of physics and, 1-13

, .,

, .,

Navier-Stokes equations, 16, 244

Planck, Max, 72

neo-Newtonian space-time, 380-381 Newton, Isaac, 146, 297 concept of space-time of, 380 gravitation theory of, 301, 335

Poincare, H., 36, 171 Poincare instability, 363 Poincare map, 245 Poincare section, 172, 173, 222-223

.

.

,

~

,

,

reduction to special relativity, 333 "No Hidden Vari bles" roofs 2 1 non-equilibrium behavior, statistical explanation of, 279-295, 414 non-equi i rium eory, 9-10, 32, 59, 71, 107, 153, 188, 196-218, 416 Zenzen for, 296 lied to s stems far from e uilibrium 26-28 applied to systems near equilibrium, 2 -2 ,88 Boltzmann equation solution in, 81-86 a

,

,

Poincare-Zermelo cycles, 68 Poin i r dic r m Popper, K., 99 positivist models, of reduction of ermo ynamics to statistical mechanics, 333-337, 373 Prigogine, 1., 230, 242, 283, 287, 358, 359, 6 use of singular distribution for initial ensembles, 269-276, 292, 296 pnmor la matter "soup," 302-303 Principal Principle, 114

. .

.

,

,

rationalization of, 86-88, 89, 215-217, 219-296 conflicting, 277-279 non-unitary transformations, of probability lStri utton, Nozick, R., 311

Principle of Indifference, 118 -120, 190, 400 Principle of Sufficient Reason, 129 principle of the common cause, 411 nnclp e 0 e onservatlon 0 nergy, 20

Objective Bayesianism, 117, 127, 259 a roach to e uilibrium theo in 190-194, 195 Onsager, Lars, 26, 75 onto ogy of genetics and evolutionary theory, 370

413-414 basic ostulates of 91-93 consequences, 93-94 dispositional, 99-102, 155 lstn utlons 0 application, 124-125, 157-158, 159,

Oppenheim, P., 133,334

as feature of collections of systems, 281-288 as feature of individual system states, 288-293 orma t eory 0 , 127, 347-348

,

parallelism, of systems, 324, 327, 329, 375, 378,418

Index

435

37-44

theory, 202, 206, 217, 277, 290-291,

invariant, 179-182 logical theories of, 117-120, 127 objective randomness and, 108-110 of outcome, 30 sis

resiliency, probability and, 116-117 reversibility, 420 at ensemble level, 264 Reversibility Objection (Umkebreinwand),

120-127 b' c iv ccounts of 110-11 as theoretical term, 102-108 time and, 176-179 tychism and, 122-124 vs. proportion, 125-126

Robertson-Walker metrics, 301, 302 161 Rosenthal A. Royal Society of London, 29 Ruell, D., 81 Russell, Bertrand, 1 3

,

12

-,

,

,

uanta of radiation ener in 72 quantum electrodynamics, 333 quantum mechanics, 1, 2, 12, 88, 200 o serva es 0 , 290 philosophical questions in, 7

,

, , quantum theory, development of, 72 uarks in elementa articles 2 Quasi-Ergodic Hypothesis, 77, 161, 166, 168, 186, 187, 195 uete et,

'

,

,

,

,

,

,

368, 370, 375, 379, 382, 383, 388 Caratheodo 's version of 21-22 Clausius' version of, 21 "self-organizing" phenomena, 28 separate systems, non-causa interdependence of, 2

, 305-307 radicall autonomous conce ts as problem in intertheoretic reduction, 341-344 ran OmlZatlOn, app lcatlon 0 240-242

""

Schuetz, Dr. - [Boltzmann's assistant], 43, 12 Schwinger, ]., 81 screening-off principle, 411 ermo ynamics, 17, 21, Secon Law 0 22, 25, 27, 32, 38, 39, 150, 151, 255,

,

,

Skyrms, B., 116 Smart . 396 S-matrix theories, 2 space-time asymmetry m, symmetries of, 379-382, 411 space-time theories, 1, 2, 3, 6 hiloso hical uestions in, 7 space-time topology, causal theory of, 1 special relativity, definitions in, 1 spm-ec 0 expenments, 295, 359, 374

branch systems of, 319-320, 322, 324, 325, 327, 331 The Direction of Time, 319-320 notion of "marking," 11, 143 re a Ofilsm, vs. su s n Iva Ism, relativity, reduction of Newtonian

spread-outness, of an ensemble, 56 Stalnaker, R, 402 stars, formation of, 308, 309 statistical explanation, 128 -155 P 1 osop ers on, , in statistical mechanics, 148-154

436

Index ,

autonomous principles of, 5

history of, 14, 15-28, 88

.

.

cosmology and, 10 explanations provided by, 8-9 of Gibbs, 48-59 history of, 14-89 P IOSO I sp

,

,

reduction of thermodynamics to, 10-11, problematic aspects of, 361-373 role of probability in, 8 statistical explanation in, 148-154 theory of, 3

theory of, 3-4 Thomson, W., 35 time direction of, 11, 375-412 Eddington concept of, 409-410, 411,

, Boltzmann's assumption of, 32-33, 60, 122 interventionist, 330 perpetual, 84 Strong Law 0 Large Num ers, 93, 99 subdynamics, 230-233

,

time-asymmetry, 11, 145, 147, 199, 202, 21 2 2 2 2-2 368, 378-384, 402, 411, 417, 418, 419-420 entropic asymmetry an , 382-384 time order, inner awareness of, 404-411

,

substantivalism, vs. relationism, 1 su ers e supervenience cases of reduction, 341, 370, 373, 378 symmetries, 0 aws an space-time, 379-382, 411

of theory of evolution under collision, 42-4 time-symmetry paradox, Gibbs' response to, 58 To man, R., 1 1, 195, 303-30 , 308 topology, readings on, 195

symmetry principles, 413 s stems concentration curve of collection of, 65 Szilard, 1., 38 Teleological Arguments universe d namic model of 301 entropy of, 303-305, 307-309, 309-318, 319, 331, 377, 417 expansIon 0 , "heat death" of, 20 thermodynamic limit, 75, 181, 200-201 thermod amic limit theo ,75, 78-81, 88 thermodynamics as au onomous eory, basic principles of, development, 17

..

.

non-equilibrium in, 298-299 size of, 375 steady-state models of, 301 U systems. See C systems van der Waals, ]., equation of state for

Index

."

VVll

IV.ll~'i;;~,

..,. n., 1"" IVU

ur71vvu~~n_,-~

von Neumann, John, 2, ••

•

tA

theory, 164-167 Wald, A., 108-109 Watanabe, S., 259 water, sUDsrnnce 01, T

.,

,J"

n, 78 .

..

.

un_v,y,

..." ...

..JVI

Wiederkebreinwand. See Reversibility ~

(

...

qUO, '-tUI

Wightmannian axiomatic theories, 2 Yablo, S., 353 Y systems. See C systems I von, J., ~UI

.." , ~7

•••••

Z, as partition function, 74 2 72

weak anthropic principle, 298, 376 weak interactions, systems governed by, 391 Weak Law of Large Numbers, 93 weyt

o •

0'

<7

wave function, 365

~::IVP-

437

U:;l1:)Vl, .)11

7~hplf S

1R~· -1R4 1R7

Zermelo, E., 36, 43, 59, 60 Zeroth Law of Thermodynamics, 20, 24, 345,350 Zustandsumme, as partition function, 74 -0'

.1'..,

~~o

Statistical mechanics' one of the cruciaJ fundame tal theories of

phystcs. wrence Sklar. <me of the preeminent philosophers of physics, offers a comprehensive. non-technicaJ .ntroduetion to the theory and 10 tmltorical attert1JfS to understand its foundational delL MSklar. ab'eady celebrated for his seminal wort on temporality. prese.n • II c:omp~bensive inquiry ifllo oootempomry statistical medlan- • The I'3IIge and depth or this swdy are e traordinat}'. few)(perts . t areas of physics, malhematics. and chcmislJ)' ave extefl' y. familiarity wi other th phi icaJ or ItdlnicaJ developments. ••• one of the most Unponant books in pbi.10s0phy of science of the last SO yean... -CHOICE

man the mysteries of ph)' . C$ are located w' • quantum mcchank:s. Two of the Pl\»t profou lie within the do . of stati:s&ical mc:dJooics I

and are buJ,ely iJldepefIdeM of q

tum theory. They concem the underRmd'jng on pllysiUJ probabm.)' and the origin and upllmation of lImibir y. To these two great w.w:s Lawrence Sklar's excel· III and extensive book IS dev~cd. is fully infonned by pbysic:a'l dteory, and e-.pI '.. careMIy and aetu:nIle y all the necessary physics without making uy of the inBca.Irate or sensaticnal . mar SO IRIIIIY noo-spccia)ist treaunClWS of ph)'s' . Uncluttered by urmecess8J)' tedmical deWl. ", bas the greal meritlhat tbe coooeptual 5 If -ghliglaed. ... Sklar ~ mllnyother issues. There is no doubt Ihat many of them are very hard, and pans of Sklar' lbook are fftcUlt by any standard. It is to h- credit that he has made the conceptual . and lhe plethora of solullions put fonwrd to deal with them as dear as they can he made. The fowxJaOOos of starislical mechanics preserll very deep . cullies. The)' have been with us a ng Ii .' ike 5 I helicve they win be with \IS for a long lime yet... - Peter C1artl., 1imes HisMr Edu.cmicm upplmlMf 0

0

0

•

Lawrence Sklar is th~ Jam B. and . J. Neboo Professor of Philosophy the Universo.y of Mtclli an. is the 11th« of Space. TUM. and pacetinu ( 977). PhiWsophy and SpQct'lim Pit. .( 1985) and the PhilOSDphy (Jf PhyJia (1992).

CAMBRIDGE UNIVEIlSITY PRESS

9