Statistics of knots and entangled random walks

dVn

(A.12)

d*r,

It is clear t h a t the conditional density x (wm | w m _ i , w m _ 2 , . . . , u>i) de­ pends evidently on um a n d w m _ i only. Thus it is proved t h a t VN is the Markov chain a n d its conditional transition density is found □. Eq.(A.lO) shows t h a t A (*+i) =

A (*) e x p {F

( W j t + 1 ) Uk)}

(A.13)

Statistics of Knots and Entangled Random

150

Walks

where F ( « * + ! , « * ) = \ In ( A 2 + I COS2 T ( A ) + A," 2 ! sin 2 7<*>)

(A.14)

Rewriting Eqs.(A.13)-(A.14) in the form m

In A<m) = £ * > * + ! , * * ) *=i

(A.15)

we can attribute to the function \(m> the sense of the Lyapunov exponent of the product of the first m matrices of the product under consideration, gW. Take two numbers oi and 02 and denote by QN conditional probability (induced by VN) of the fact that 01 < l n A ^ < 02. It should be noted that we can drop the simplifying assumption (see above), so all A^ m ' for \ <m< N are no longer bigger than 1. Thus we have all necessary definitions to be able t o formulate our main problem. Fix a number I, 0 < ? < 1, p u t iVi = [ciV] a n d ask the following question. What is the limit probability distribution for the value -?== In A ^ 1 ' as N —» 00 where the distribution of A ^ 1 J is determined by QN-

We use the Cramer's method in the theory of probabilities of large deviations for the sequences of independent random variables. So, let us rewrite t h e density corresponding t o t h e distribution PN in the following form N

T0((v0,u>1,...,wN)

= ir0(w0)

J J 7r(w m |w m _!)

(A.16)

m-l

where 7r(w m |u; m _i) is the transitional density found in Lemma A . l . Now for any b (—00 < 6 < 00) we introduce the new "normalized" probability distribution VN(O) with the density

Trb((jj0,iv1,...,uN)=

xo(w 0 ) I I 5 r ( w m | w m _ i ) 11 (A.17) Using the statistical mechanics analogy we could say that the function in the exponent is the sum of "Boltzmann weights" of random steps with the "inverse temperature" 6 = — ^ ; hence 0 ^ ( 6 ) is nothing else than the eN(b)

Appendix

151

partition function determined by the normalization condition N

y i f(>(wo,<«>!,...,WAT) = 1 m=l

Thus Eq.(A.17) defines the non-homogeneous Markov chain. The joint probability distribution corresponding to Qpj reads N

To(Wo) I I

7r 6 (w 0 ,w 1 ,...,a;jv) =

T(wm|wra_l)

^ A " 18 ^

""sfjV)

where N

N

/ * < > M I J * ( « m | « m - : t ) JldVrn m=l m=0

.

(A.19)

{u>0,...,uN} ai < X W < a 2

Eq.(A.17) can be rewritten as follows x (wo, ■ ■ ■, « * ) = x 6 («„, • • •,«») ©^(6)e- 6 S - - F(<"-w™+>> The quantity E(iV) can be estimated from above N

f /

bai

S(JV) < @N(b)e-

r

dw

(A'2°)

wfc(«oi-.,«Ar) ]\dMm

(A.21)

7T6 (w 0 ,..., UN) I J

»»

m = 0

i

ax < AW < a2 and from below N

f /

Z{N)>eN(b)e-ba> J-

Ol < ^

(JV)

i

m = 0

< °2

Denote w' = ( $ ' , A', v'')" w " - (*">•*"> v") and consider the positive kernel

#6(w V ) = *KI"V F ( u / V ) associated with the integral operator (Kbf) (»") = / 2r(w'V)/(w')«b>'

(A.22)

152

Statistics of Knots and Entangled Random Walks

In turn, the operator adjoint to (A.22) has the following form {K*bf) (u/) = I K{u"\u,')f(w")du>"

(A.23)

L e m m a A.2 The operator Kb has a positive eigenfunction hob(u') = * , ( * ' , A',
P(
A')
where gb satisfies the following conditions: 1. gb($',\>,,X) = 0; 2. for some positive constants c\ and c2 we have c\ < gb(&, A', ip') < C2 ifp( 0. The adjoint operator K^ has a positive eigenfunction h*(u") = hb($", A", V") such that a < g*b(X",
We denote

Proof. Let us begin by analysing of the eigenfunction fej. Take an arbitrary function u(u') in the form u(u') = p{(#', A', ", A") j „(', A', v')p{i>W, A')

dip'

dip1

ebF(u"\u')d&

Put Vl($,
«i(«") = »!(*",
dj^ 6 f e W>d$', dy>'

then Vi satisfies Condition 1 of Lemma A.2. We may assume that

p(V'V,A')for some d. Otherwise we can pass to the operator (Kb)1, where s = s(d).

Appendix

153

For two different values w" = ( $ " , ^ " , A") and w" = ( l > " , ^ " , X " ) we have

fv(*',V',\')ptf\v',\')

vi («*)

Jv(*',)p$\) max <

d

£*""*>')& e

MW)d#'

dip'

d^ d^' e x p | 6 ( m a x F ( w " , w ' ) - min F(u",

d

u'))\

dy> min dp' Here V> a n d -0 are the values of V>' corresponding t o £T" and oi for the same w'. If we p u t now £j,v — Vi, we immediately come t o the conclusion that the operator Lb has a positive kernel on a compact set and therefore has a positive eigenfunction with the positive eigenvalue. The same arguments work for the operator K^. T h e fact that the operators Kb and KI have the common set of eigenvalues can be shown by simple direct calculations □. Now we rewrite 7r&(wo,..., wjv) (see Eq.(A.17)) as follows

fft(wo,- • . , « J V ) = A

(b) —

eN{b)g*{u,N)

The function B

»(«'>')

n 11

A(6) s *(w,„_i)

6F(u'»„(w")

A(6)fe*(W')

can be considered the kernel of a stochastic operator Vb- T h e invariant measure for Vb has the density vb(w') = fc{,(w')/ij(u/). T h e functions gb, g^ are normalized in such a way t h a t hb(u') becomes the density of a probability measure. Thus, we have

N

ba

E(2V) < \ (b)e- >

N r l / rf(uo)**>(«o) I I P»0"m> w »»-i)rw

a

m=1

N

TI I

duk

fc=0

(A.24)

Statistics of Knots and Entangled Random Walks

154

and N

N

N

ba

w

v(N) > \ (b)e- > / 06>o)*oM II »("•»• » - 0 r ^ 7 1 1 1 ^ * (A.25) where fi is the integration domain as in Eqs.(A.20), (A.21). L e m m a A . 3 There exists one and only one value bo for [F{u>",U')pbo(u"tu>')Vba(w')du>du"

which

= 0

(A.26)

Proof. This Lemma is well-known in statistical mechanics. It means that b0 is found under the condition t h a t the expectation of F(UI\UQ) with respect to the stationary Markov measure with transition density Pi, 0 (a/',u/) is equal to zero. It can be shown t h a t j F(u", u>')Pb{u>", w')vb(u')du'dw"

> 0

(A.27)

because we can rewrite the integral in Eq.(A.27) as follows

I

F(u"tw')pb("",v')Mu')du'te"

i d A ^ o Ndb

N

r ln

/

=

7ro Wo

(

b w

F u

( ""u»-1)

£

) I I *( m|wm-i)e " -

1

N

JJ

m=l

dwm

m=0

(A.28) and therefore ^

j

F(U")U')pb(w",u>')ub(a>')du'dU"

=

N^L i ^ 1 6 ( Y, Fi"m,«>m-l)j where Var(,(...) is the variance found with the help of distribution VN (b). Thus Eq.(A.27) is proved. This yields that the expectation (Eq.(A.28)) is a monotone increasing function of 6. It is easy to find such periodic sequences t

{ w m } for which the sum £3 F{um\um-i)

over a period t is strictly positive

m=l

(or strictly negative). Correspondingly, the limit in Eq.(A.28) is positive

Appendix

155

as 6 —+ oo and negative as b —» — oo. Therefore there exists one and only one value of to for which the integral Eq.(A.26) is zero □. Since N m=l

we can use the local central limit theorem for Markov chains with a compact phase space which gives N „ I J P*o(Wm,Wm_l)1—;

ff&0(Wo)*'o(wo)

/

N r TT duk


1 /^T^ J

9b0(uo)Muo)K0{u)N)du)N

(A.29) as N —► oo. The constant a — 0 enters in the asymptotics of the variance N

Jim Var6

V

F(w m ,w m _i)

\ra-l

~ N
Take now two numbers U\, u 2 («i < u 2 ) and consider the probability

We have N 1

9/v

N

fTJ^ / ToM J ! *"(wmk»-l) ]^[ du)m \

/

/

V

' •.

m=l

m=0

^ g ^ / ^NAN n w-- i^-o^y n' n (A.30) where n = { { « 0 l . . .,w/v} : «i < A<w> < a 2 ; m < ^ l n A<"*> < w 2 }. The analogous inequality can be written estimating g^r from below. The probability density p 6o follows from Eq.(A.30) and the local central theorem for Markov chains:

156

Statistics of Knots and Entangled Random Walks

For u — O (\/~N) and v — O (VN) the conditional probability den­ sity reads

This yields N

1

N

/ T0(U>O)KO{V0) Y[ % K K - I ) T ; ri.

; JJ * " * ~

6

m=l

°^ ^ *=0

n "J = / e~ ^-^N" du . I glAwo)ir0{w0)du>o I y/2ir(l - f)er J VZirNa J J

gbo(wN)duN

Combining Eqs.(A.24), (A.29), (A.30), we get e60(o2-ai)

/• „a /• 3 ebo(al-a3) — / e2'('-«)dti
(A.31)

»1

Let us formulate now the main theorem: T h e o r e m A . l ( S i n a i a n d N e c h a e v 2 ) The limit probability bution of the variable -Js= In A ^ 1 ) with respect to the distribution Gaussian with the variance a(l — ?) and zero's mean.

distri­ QN is

Proof. Take an interval [01,02] and decompose it into small parts: a i < aW < o ( 2 ) < . . . < o(r> = o 2 such that a^ - o ( J _ 1 ) < e where e > 0 is a given small number. Prom Eq.(A.31) we have 7>{tii< - ^ = l n A ( J V l ) < u 2

I

OJ
V^i

J

^^{oO-^^lnAW^aO)} ^ ^{01 < l n A W < a 2 } x7> | U l

1

7

^ 2 ^ ( 1 - 5) 7 U1

< 1

« ^

J

mA (",)

< U2 j a 0-i) < AW < a ( » | =

^ 7 ' { a O - 1 ) < l n A ( v v )
7>{oi
L

,,,,„ v

'

;

v1

J

Appendix where |5^{N,e)|

□.

157

< 2e for sufficiently large N. This gives the desired result

R,ci6rGncGS 1. H Fuistenberg, Trans. Amer. Math. Soc, 198 (1963), 377 1. S.Fu Nechaer, Ya.G. Sinai, Bol. Soc. Bras. Mat., 21 (1991), 121 2. V.N. Tutubalin, Prob. Theory and Appl.s .M (1965), 15; ibid 13 (1968), 65

Appendix B P O L Y M E R C H A I N IN R A N D O M A R R A Y OF TOPOLOGICAL OBSTACLES Although the thermodynamic properties of polymers in the regular lattices of topological obstacles have been the subject of multiple studies and researches, the statistics of loops in disordered arrays of immobile ob­ stacles has not received all the attention it deserves. At the same time the absence of translational invariance in the last case might have some influ­ ence on statistics of polymer chains. In particular, this question has been recently analysed in papers 1 and 2 which we find reasonable to reproduce below in a slightly shortened form. 1.1. P h a s e T r a n s i t i o n i n P o l y m e r L o o p s I n d u c e d b y T o p o l o g i c a l Disorder Lately much attention has been devoted to the investigation of equi­ librium and dynamic properties of polymer chains with excluded volume in random environment in connection with valuable technical and biologi­ cal applications. Among the most important problems are the problems of DNA gel-electrophoresis as well as of polymer filtration and adsorption in porous media 3 ' 4 . However, as far as we know, all analytical works are con­ cerned with investigation of disorder only in potential interactions 5 ' 6 ' 7 ' 8 ' 9 ) leaving topological constraints out of consideration. (To be more correct it should be stated t h a t in numerical simulations 9 the volume interactions and entanglements were taken into account simultaneously). At the same time there is a set of problems where the topological disorder plays an ex-

158

Appendix

159

ceptionally important role. First of all it concerns the swelling and the high-elasticity properties of networks prepared in concentrated solutions. At present we distinguish in literature between two groups of works dealing with rubber and swelling properties of polymer networks with topo­ logies! constraints. In one group of works ' n the topological constraints have been mod­ elled by the uniform lattice of rods nonintersectible for the chain. This prob­ lem has been solved analytically and the dependence of stress under strain for the uniaxial extension-compression obtained there is in compliance with corresponding experimental d a t a u . But the principal shortcoming of this model is connected with the regularity of the lattice of obstacles. Below we show t h a t the presence of disorder in distribution of obstacles radically changes the statistics of chains. In another group of works 1 2 ' 1 3 the main attention has been paid to the influence of topological state of network subchain and of preparation conditions on thermodynamic properties of gels. It is clear that fixed topo­ logical structure of polymer gel is the typical example of quenched disorder. Actually, the structure is formed during the network preparation and can not be changed without the network destruction. Because of strict topo­ logical constraints on chain conformations the full phase space of the gel is divided into separated domains—just like the multi-valley structure of the spin glass phase space. T h e application of replica approach allows the author of 12 to obtain reasonable results for rubber properties of networks. But the method elaborated there for identification of the topological state of the chains seems rather doubtful. Here we consider the model of polymer loops with volume interac­ tions randomly entangled with randomly distributed infinite parallel rods. Within the framework of field-theoretical approach we show that the last disorder changes essentially the statistics of the chain and under some con­ ditions induces the collapse transition of the loop. 1 . 1 . 1 . Description

of the

System

Consider the closed 3D-polymer chain with 2D-volume interactions topologically entangled with an array of immobile randomly distributed rods perpendicular to xtj-plane. The density of the rods projected to xyplane is assumed to be Gaussian (which we take instead of Poissonian one for simplicity). We take also the winding number (i.e. number of rods

160

Statistics of Knots and Entangled Random

Walks

Fig. B.l. Polymer loop topologically entangled with random array of rods and its 2D projection where P is a point of contour selfintersection.

enclosed by the chain) to be quenched with Gaussian distribution having nonzero mean (see below). The peculiarity of the volume interactions considered in our model consists in the fact that one chain segment excludes for the others not only one point in the space, but also the line perpendicular to the xyplane. In other words all points of selfintersections in 2D projection are excluded. The typical conformation of closed chain in an array of rods in 3D space and its 2D projection with a selfintersection point P are shown in figs.B.l. Because of 2D character of volume interactions we avoid the points of selfintersections and deal with nonintersecting loops only. Since the disorder in positions of the rods is quenched we would like to calculate the free energy of the chain averaged over the distribution of the rods. Using the concept of the free energy self-averaging, the results obtained will give us the prediction of chain statistics in a given distribution of topological obstacles. Some comments should be made concerning this model:

Appendix

161

(i) We consider the chain attached in one point to the plane projection to avoid the situation of annealed disorder: actually, in absence of a fixed point and with no rods enclosed, the loop can visit all points of the volume sample. This indicates t h a t the disorder in the distribution of obstacles changes effectively from quenched to annealed one; (ii) We introduce such unusual volume interactions pursuing the forth­ coming aims (see Section 1.2 of Appendix B). Namely, our final task concerns reconstruction of the true configuration of the closed chain with ordinary volume interactions embedded in 3D-space with points of selfintersections in the projection and topologically entangled with an array of rods. We will raise it from simple "blocks"—loops, con­ sidered here; (iii) We discuss the possibility of collapse for different "preparation condi­ tions" of the loop expecting that the interplay of disorders in winding number and in spatial distribution of rods can lead to the so-called "noise-induced" phase transition in polymer chain.

To the determine of the topological state of the closed chain with respect to the rods we can use the Gauss linking number. It is well known that this invariant is rather weak because it counts the number of obstacles entangled with the contour in algebraic sense only. For example, it can not distinguish from topological point of view the contour shown in fig.B.l and the trivial unentangled one. But it seems to be obvious t h a t in our model just because of 2D-character of volume interactions and of the absence of contour selfintersections the Gauss invariant can be made complete. To analyze the equilibrium conformation of the loop in a random array of topological obstacles we use the mean-field approximation. T h a t is, we choose the density of the loop's segments, p, as the order parameter and minimize the averaged free energy with respect to it. We find t h a t the topological disorder induces the spontaneous symmetry breaking with p ^ 0 which we interprete as a collapsed phase of polymer loop.

1.1.2. Field

Representation

of the

Model

Define the xi/-coordinates of i's rod (obstacle) as r, = (xi,yi). Then the Gauss invariant represented in terms of contour integral has the usual

162

form

Statistics of Knots and Entangled Random

Walks

14

G,{C}

=

i-/d,^Wvx(ln|R±(«)-r,-|)xi? = c

-{

1

point r, is inside the contour C

0

otherwise

(H 11

where R ( s ) is the 3D radius-vector of the contour C, R x ( s ) is its 2Dprojection to the xy-plane, s is the variable along the contour and t] is normal vector to xj/-plane (7) = (0, 0,1)). The invariant G i ( C ) has the sense of the total angle (normalized by 2x) covered by the radius-vector R x ( « ) . This angle has the sign which depends on clock- or counter clockwise of path integration in Eq.(B.l). Simple generalization of Gj of the form N

'=£
(B.2)

1=1

counts the number of obstacles inside the contour C. It can be easily tested that the 2D-vector field A introduced as follows TV

A = £Vx(ln|Rx(s)-r,|) x ,

(B.3)

i=l

satisfies the conditions divA

=

0

rot A

=

T J [ V ? ( R X ) -
(B.4) where y ( R x ) = ^ £ ( R x — r,-) and y>o is the mean density of the obstacles 1

in xy-pl&ne.

Then,

J

da

We suppose y ( R x ) is randomly distributed with the Gaussian density Pi{
~

expj-^-ydRx[vKRx)-*>o]2j

=

eXP

(B.5)

(r tA)2

{-2^/^ °

}

Appendix

163

where the dispersion (po of the distribution Pi{ip(H^)} coincides with the mean density. The partition function of the loop containing inside fixed number of obstacles, c, reads

Z(c) = J D{K}6 [K(N) - R(0)] 6 (c - j x e x p j - i ^ ^ ? ^

^^lAds\

d3-^-fdsfds'6(B.±(s)-K±(s'))\

(B.6) where a2 is the 2D-excluded volume and I is the length of the chain segment. In consideration the distribution in the number of obstacles inside the contour (below referred to as topological charge) we introduce the proba­ bility density P2{c) which is also assumed to be Gaussian (it is consistent with Eq.(B.l)) 2 '

*=vsM-^l

<*■"

with mean CQ and dispersion A c . We would like to calculate the free energy F — — In Z of the loop aver­ aged over the distributions Eq.(B.5) and Eq.(B.7). Although this problem is rather complex from the computational point of view, we can essentially simplify it by working not with topological charge, c, but with its chemical potential, g, having sense of the flux through the contour. Let us introduce the grand canonical ensemble, Z(g), and the distribution, i^Cff), defined as follows (see Eqs.(B.6) and (B.7) correspondingly):

z(g) = f e-'C9Z{c)dc

(B.8)

J — OO

J —<

and

For the mean free energy of the loop we have < F >= - < In Z >= - f P i M i M f f ) In Z(g)dg

(B.10)

Statistics of Knots and Entangled Random

164

Walks

Within the framework of the replica approach we exploit the identity d

(F)

dn

(B.11)

(zn(g))

n=0

where all calculations are performed for an integer value of n and the con­ tinuation of the value n — 0 is taken only in the final expression. The replica partition function Zn(g) in the field representation has the form Zn

(9)

= /

f l D{MD{ra}exp

ff(») ( * , * ' , A , g)

=

{ - j dWT<"> ( « , * * , A , gr)J

£ > »

(V

- igA.Y + T y VV

(B.12)

+rU:

a= l

\a =l

(B.13) where ^ = {iplt..., V»„}, ** = { ^ J , . . . , $ * } , r is the chemical potential conjugate to the length of the loop, I is the length of the chain segment and L is the mean size of the coil in z-direction. Using the standard representation (see, for example, l s ) , we can rewrite Eqs.(B.12)-(B.13) as follows Zn(g)=

-^|zHx}exp j-^|dRx2(R)jexp(-n)

(B.14)

whe $ = In det

I2

(Vj. -

igXf

V* +

T+IX + lnir

(B.15)

E q . ( B . l l ) becomes rather obvious in the graphical representation. Af­ ter the integration over ^-variable it takes the form in fig.B.2 where each loop carries a factor n and due to the limit n - » 0 only the irreducible dia­ grams of type A remain. Each solid line in this expression corresponds to a polymer propagator and each dotted line represents the volume interaction. In summary, it can be said that the continuation n - » 0 in the parti­ tion function Zn(g) (Eq.(B.14)) being applied to E q . ( B . l l ) makes it possi­ ble:

Appendix

165

Fig. B.2. Illustration to Eq.(B.15)

- to aveiage the logarithm of the partition function (i.e. the free energy) over the disorder; - to extract the nonselfintersecting loops exclusively. It should be emphasized, that in the case of ring chains the averaging of the free energy over the disorder as well as correct describing the volume interactions is governed simultaneously by one index n. From that point of view each field component should be considered as one replica. Averaging of Eq.(B.12) over the spatial disorder (Eq.(B.5)) we obtain

I f [ £»{*,**, A}«[didA] exp {- - JRB& (*,**, A,,)} (B.16) where

a=l

■

j

+ -—(rotA)2 + — 2V>0 4

( ] P VaV>a ) \a = l ) (B.17)

The Hamiltonian Eq.(B.17) is nothing else than the Euclidean version of scalar electrodynamics. The correlator of A-fields in the momentum space has the form (A,(«)Ai(-K))0 = g (fy ~ t^f) (B.18)

Statistics of Knots and Entangled Random

166

£

£-~?

V

ig

hi -&

s1/2

Walks

ig(k+1/2q)f(k)y,m(k+q)A(q)

V

*

= -lz/4g2W'Az (a)

(b) Fig. B.3. a) Two types of vertex operators for the effective Hamiltonian Eq.(B.17); b) Graphic representation of the one-loop series contributing to the effective potential

1.1.3. Effective

Potential

Method

We analyze Eqs.(B.16)-(B.17) using the effective potential treatment which was used for the first time by S. Coleman and E. Weinberg 16 for similar Hamiltonian in quantum scalar electrodynamics. Considering the perturbative series in powers of magnetic flux, g, we can distinguish two kinds of vertices shown in fig.B.3a. The effective potential appears from the summation of one-loop diagrams with external V'-lmes taken at zero's momentum. Thus we have two possible series reproduced in fig.B.3b. Following the arguments of Ref. 16 it can be shown t h a t the series (a) vanishes because of the transversality of A-correlator (Eq.(B.18)) and the remaining series (b) corresponds to the Gaussian integration in Eqs.(B.16)(B.17). Finally, in the one-loop approximation the effective Hamiltonian H\^ can be rewritten as 2

rr(") _ rr(") i r * t t * X La + ^eff — ""one-loop + T * *

( * * * ) 2 -I-const

(B.19)

Appendix

167

where ^one-loop = \ j

J^j2

* («' +
Q = jVoS*

(B.20)

and n

*(R)¥*(R) = 5> a (R)rt;(R) a=l

Because the value of H ^ " ; _ l o o p is ultraviolet-divergent we cut off the integral in Eq.(B.20) at some value K = A. Thus we obtain for #o"e-ioo P the expression: ^ L o o

P

= £

{In [A2 + Q t f ] - 1} - £ * * * In ^

^

^

(B.21)

Expanding Eq.(B.21) in power series in Q\P\P*/A 2 and omitting the terms not depending on QW*, we get

^:Lioop- ^*** ( i - k [§***])

(B-22)

In compliance with the work 16 the replicated effective potential for our system can be written as follows H

is = :p*** (1 - In ^ * * * J+T*** + ^ - ( * * * ) 2 + B * * * (B.23)

where we add the "mass" renormalization counterterm (the last term in Eq.(B.23)). It should be noted t h a t because of the 2D character of the problem the coupling constant La2 remains non-renormalized. It should be emphasized that the 0(n) symmetry of the volume in­ teraction term in Eq.(B.23) is illusory. Diagram series (fig.B.3a)) clearly show t h a t only one-loop diagrams survive at n —♦ 0. The terms of the order of n2 and higher are responsible for the interreplica interactions and under the condition n —» 0 we can neglect them with respect to the linear ones. So, the single-replica terms remain exclusively and the actual symmetry of the volume interaction term changes from 0(n) to the hypercubic one. We define the renormalization of the mass, B, for the replica symmet­ ric solution by the condition d2 dipdip*

g(») eff

=

TO

(B.24)

168

Statistics

of Knots

and Entangled

Random

Walks

where the subtraction point M is completely arbitrary. For the value of the counterterm B we have B =

Si

m

8TT

(%nM2) + $- - La2M2 2 \A / 8T

(B.25)

Substituting Eq.(B.23) for Eq.(B.25) we get

(B.26) The unusual logarithmic dependence of the mass on replica index disappears after renormalization. The application of Eq.(B.ll) to the partition function (Zn{i/)ip", g}) = exp{-n/^/

t f f W l S

}}

allows us to write the effective potential av-

eraged over the spatial disorder in the distribution of the obstacles: /efr(W2,<;) = r ' M 2 - J U o < 7 2 H 2 l n | ^ | 2 + ^ | V | 4

(B.27)

The dependence on the cut-off parameter, M2, is absorbed by the effective chemical potential r*. The value of T* controls the average number of chain segments, N, where N — — g— In G(T*, q = 0). The segment-tosegment correlation function in the loop, G(r*,q = 0), can be defined in the usual way:

G_1

^9 = 0 ^ W / £ f f

=T

*

Thus, in our approximation we have:

N = ~

(B.28)

Averaging Eq.(B.27) over the distribution, i^Cff) we get /eff{|^|2,V5,co,Ac}=r*|V|2-^o^-(l-£)|^|2ln|V|2+^|V-|4 (B.29) Minimising of Eq.(B.29) with respect to the density p = \tj}\2 we obtain

La2po =

Tte,po'be i1 ~ I t ) ( ! + l n ( £ 3 p °)) - 2T*

( B - 3 °)

Appendix

169

density of chain segments, p Fig. B.4. Dependence of the effective potential on the order parameter (i.e. density of chain segments) for different chain lengths.

This equation has one stable and one unstable solutions i.e. the loop un­ dergoes the collapse phase transition of the first order—see fig.B.4. In particular, the binodal curve is determined by the equation

X = 32^° A-e V - ^)

ln

IS* 0 AT V ~ ^)

[a)

(B.31)

where we have eliminated r* by using Eq.(B.28). 1.1.4.

The Role

of

Fluctuations

Let us estimate the validity of the mean-field approximation consid­ ered in the previous section. We can easily expand the effective potential feft{ip,ip*} taking into account the fluctuations of the scalar fields if) and

Statistics of Knots and Entangled Random

170

Walks

V>* n e a r t h e s t a b l e mean-field solution:

S2

1

2 6 # V7

+ fc2A2(Vo,V>5) AV-AV-*

fes{i>,r)

v.=vo^"=^;

(B.32)

where Ai>{k) = i>(k) - Vo,

Aifr*(*) = ^ * ( * ) - i & ;

In expansion Eq.(B.32) the last term in square brackets is described by the diagram series of the type (a) in figB.3b. Performing the summation, we get

k2x2^0,r0,9) = 2

^IH'-WM-) (*•-?)

(B.33)

Vo

2

« + 4£ W M V o irhere -g2

ACV

Aj'

-i2(k± - K)2 + 2\lq2 + T*

4

fcx and 5 are the components of the wave vector

in sy-plane and along 2-axis correspondingly. It can be shown that near the binodal curve l

r{\M ) =

b2 6ip6ip

7/eff

r = r0 where r* is the value of the chemical potential on the binodal curve. The mean square value of fluctuations of the fields ip and rjj* is

a

<' *>=(4 /

d2k r(\^0\2)/2 +

\2k2

\k\
Performing the integration in Eq.(B.33) it can be seen t h a t near the binodal curve the fluctuations weakness is determined by the inequality: (|AV| 2 > |V>o|

, 2A2(X) In , cV /.„ < 1 ;( T * 4TTXA?(X) - T)1

Appendix

171

where x

=

roi292 16*-

2

and the value A on the binodal curve is defined by the relation X2(X\ cV

;

=

t

8xX(i/a)2-ln(X(I/a)2)

,„

^X(l/af

ln{X[L/a)2)

It should be emphasized t h a t X = {a/L)2 and A2 —► oo on the curve in fig.B.5. It means t h a t the result obtained within the framework of the mean-field approach becomes exact on this line. Thus the mean-field ap­ proximation is valid in vicinity of the boundary of shaded area in fig.B.5.

1.1.5.

Comments

and

Conclusions

We understand the collapse of the chain in the following sense. Take an ensemble of polymer loops of length N0 with the mean number of obsta­ cles enclosed Co and dispersion A c . We call the set of values (No, Co, A c , ^o) "the preparation conditions". For example, we could assume that the value Co appears in course of random closure of the chain of length iv"o; c 0 = Co(N0). 1. Let us grow the chain length keeping the preparation conditions fixed. It can be seen that when the loop reaches some critical value Nc, it becomes unstable and collapses according to Eq.(B.29). Actually, the spatial structure of long enough chains (N ^> No) consists of entangled part of length N\, bounded in the finite domain with dimension Do = DQ(CQ, Nc form the collapsed state instead of increasing the dimensions of "octopus pulps-like" parts. 2. Let us discuss the role of dispersion A c . We start with the case c ^ 0. It is clear t h a t the dispersion in the number of obstacles inside the contour relieves the collapse transition, because some contours appear to be unentangled even in preparation conditions. Prom this point of view the equation on the binodal curve (Eq.(B.30)) can be interpreted as follows: for a fixed chain length, Nc > N0, and spatial distribution of obstacles, ip0,

172

Statistics of Knots and Entangled Random

Walks

Fig. B.5. The phase diagram of the collapse transition. The shaded area corresponds to the collapsed state of the loop.

Eq.(B.30) gives the condition on the dispersion A c necessary for producing phase transition (for the fixed mean value c 0 ). Turn now to the case c 0 = 0. In this case all topological constraints are outside the chain. If we do not fix some point of the chain in the projection attached to the surface, the chain can move through the whole volume sample and the disorder should be regarded effectively as annealed. The fact that one point of the chain is kept fixed is not reflected in the mean density but it is so in correlation functions. We believe that in this case the microstructure of the chain resembles parts of condensed drops with tails between them (compare to 7 ) . Eq.(B.30) gives a reasonable answer to the case co = 0: all contours are unentangled and the presence of dispersion (i.e., presence of part of entangled contours) only depresses the collapse transition. The region of the values v>o and A c , where the transition takes place is shown in fig.B.5 and is defined by the inequality

Appendix

173

It should be noted t h a t the above mentioned mechanism of phase transition is a particular case of the so-called noise-induced transitions 1T . At least, for each nontrivial stable solution, p 0 , of Eq.(B.30) the field A acquires a finite screening length, £4: ^ o c ^ i - ^ l - f ^ p o

(B.35)

This is nothing else t h a n the usual Higgs phenomenon 1 6 . In terms of our model it means t h a t the loop has tried to be compactificated in the regions free of obstacles. 1.2. C l a s s i f i c a t i o n a n d S t a t i s t i c s o f C o m p l e x L o o p s In this Section we extend the approach elaborated in Section 1.1 of Appendix B (Ref. 1 ), hereafter referred to as I, to the description of thermodynamic properties of complex loops (i.e. loops with volume interactions and with points of chain selfintersections in ay-projection). We would like to pay particular attention to investigation of the influ­ ence of "preparation condition" on thermodynamic properties of polymer loops. Under the "preparation conditions" we understand the initial topological state of the complex loop with respect to the randomly distributed array of parallel rods. The importance of this question is obvious. Elastic and swelling properties of polymer networks and gels depend strongly on the initial topological configuration of chains in the sample. The corre­ sponding experimental d a t a can be found in 2 0 whereas 2 1 is devoted to some qualitative physical explanations of this phenomenon. Let us consider the following facts: (i) An arbitrary complex loop in a quenched array of topological obsta­ cles can be represented in a "cactus-like form" with "leaves" being the simple loops containing no points of selfintersections on the pro­ jection to the ay-plane. T h e points fastening different simple loops together function as the points of selfintersections. (ii) T h e collapse transition of the complex loop with fixed preparation conditions (see below) can occur independently in different leaves when the chain length is increased, though the simultaneous collapse in all leaves is entropically unfavourable. The methods used here have been proposed in I and represent a com­ bination of the field theoretical efFective potential treatment 1 6 with the

174

Statistics of Knots and Entangled Random

Walks

Fig. B.6. Example of complex loop (with its 2D projection) entangled with random array of topological obstacles.

replica approach 9 . The instability with respect to the collapse transition in different leaves can be described in a self-consistent way by the so-called asymmetric solution corresponding to the simplest (Gaussian) trial func­ tion with a variational parameter. This method corresponds to the well known Feynman variational principal 2 3 . 1.2.1. "Cactus-like"

Representation

of Complex

Loop

Consider an arbitrary polymer loop embedded in 3D space and topologically entangled with an array of immobile randomly distributed rods normal to ay-plane (see fig.B.6). We assume that the loop does not pro­ duce any entanglements and that the difference between the model under consideration and that shown in fig.B.l consists in presence of many points of selfintersections in xj/-projection in fig.B.6. Let us neglect for a moment the topological constraints and pay at­ tention to the shadow graph, obtained by projection of the closed chain onto the xy-plane. Assume t h a t this graph is in general position i.e. con-

Appendix

175

Fig. B.7. Shadow graph (a); its representation by two cactus-like graphs [(b) and (c)] and their topologies [(d) and (e)].

tains double points of path selfintersection only. The triple and higher order points of selfintersections can be removed by means of infinitesimal continuous deformation of the path. An arbitrary shadow graph can be represented in form of equivalent cactus-like graphs where each leaf has no points of selfintersections. This correspondence is not unique as it can be seen from an example shown in fig.B.7: one and the same shadow graph can be represented by topologically different cactus-like graphs (see fig.B.7b,c). Thus it is necessary to distinguish between two types of points in cactus-like graphs: (i) Points joining different cactus leaves. They are called junctions and are marked by bold circles (points 1,2,3 in fig.B.7b and points 1,4,5 in fig.B.7c); (ii) Points of overlapping of different leaves. They are called regular points (points 4,5 in fig.B.7b and points 2,3 in fig.B.7c). The difference in representation of the complex loop by shadow graphs corresponds to different ways of preparation of the initial shadow graph. The way of preparation is a succession of "elementary technological operations" which produce a complex loop with junctions and regular points

176

Statistics of Knots and Entangled Random

Walks

Fig. B.8. Reidemeister moves for shadow diagrams.

starting from a loop without selfintersections. To be more rigorous let us introduce the following definition. D e f i n i t i o n B . l We distinguish between the Reidemeister moves (see Chapter 1) by establishing the equivalence relations of shadow link diagram.. According to our nomenclature the crossing point in the move I is the jjnction whereas the crossing points in moves II and III are the regular points—see fig.B.8. S t a t e m e n t 5 The regular way of preparation of an arbitrary shadow graph corresponding to any complex loop is as follows: (1) Take a simple loop and create the necessary number of junctions means of Reidemeister move of type I.

by

(2) Deform continuously the resulting cactus-like graph employing Reidemeister moves II and III keeping the number of junction points fixed. (3) Sum over all topologically different cactus-like graphs resulting in the complex loop. C o n j e c t u r e 6 We suggest the probability distribution ferent cactus-like graphs to be uniform.

of formation

of dif-

Appendix

177

In this case the fraction of shadow graphs in ensemble is increased proportionally to the number of its different representations by the cactus-like graphs. For example, the fraction of graphs like the one shown in fig.B.7a is increased twofold being a result of the two types of its cactus-like covering. So, this assumption gives us the possibility to determine the correspondence between ensembles of all shadow graphs and all cactus-like representations. Otherwise, for the nonuniform probability distribution it would be necessary to discriminate fractions of specific cactus-like graphs (e.g., fig.B.7d and fig.B.7e) which makes calculations much more complicated but does not change the main conclusion of the Section. It should be emphasized t h a t the uniform probability distribution has nothing in common with the actual statistical weight of cactus-like graphs, resulting from the configurations! partition function of a graph with a fixed number of junction points. The real form of the probability distribution is the problem of preparation conditions of the complex loop. Finally, we emphasize t h a t in principle different preparation conditions could correspond to the cactus-like graphs of the same topology (see, for instance fig.B.9). From the topological point of view the graphs in fig.B.9b and fig.B.9c are identical, both of them should contribute to the shadow graph (fig.B.9a). To distinguish between these equivalent graphs we have to define from very beginning the orientation (i.e. the direction of pathway) on the shadow graph. Hence the topological charge of a simple loop (leaf) can be both positive and negative (as it is in I).

1.2.2. Effective plex

Hamiltonian

and Mean-field

Free Energy

of

Com-

Loops

T h e Hamiltonian describing the complex loop can be written as follows (compare to Eq.(B.17)) H{n,m)i%

**,

A

, ff) = £

£

i>ai ( j ( V

x

- iffA)2 + l-Vf

+ rj

i,ai

(B.36) To discriminate the complex loop structure, we have attributed an additional index i(i= c ... m) to the uompoitents of tae rields dP ana \P* ion in e ( - , . . . , ,

178

Statistics of Knots and Entangled Random

Walks

Fig. B.9. Twofold [(b) and (c)] cactus-like representation of shadow graph (a); [diagrams (d) and (e) are topologicaUy equivalent to each others but have different distribution of topological charges.

enumerating different leaves. The interaction constant w stands for the fugacity of the number of junction points connecting different leaves; r is the mass term, i.e. the chemical potential conjugated to the length of the whole connected graph, {A, g, = exp{-F("r a a >} = I DAD^aiDrai

e x p { - t f {y,Q1; # . , A } }

(B.37) The generating function of the connected graphs (the free energy) has the form F ( » - ) { r , W } = nf;f;<9(fc;P)(-^)P

(B.38)

where 3(Jfe;p) is a contribution of a connected graph constructed from Jb simple loops with p junction points and the combinatorial factor Cm gives

Appendix

179

the number of ways in which such contribution can be accomplished. It is easy to see from Eq.(B.38) t h a t in the limit m —> oo only the cactus-like graphs with p = k - 1 survive. As a result the free energy of the quenched cactus-like system is given by Fcactu,

= lim Urn - — ^ " ' n->om->oo man

m

) = V^fe; k - l)i=^ *-* k\

(B.39)

where (run.) —» 0 and the factor A;! appears from the fact t h a t the junction points are indistinguishable. Now we can perform integration in Eq.(B.38) over the vector-potential A (as in I). T h e effective free energy in the mean-field approximation reads

F(n,m}{Tt

T

w) =

2

n

"

^_ -£{i,airaif - £; E E +*to*tKt a=l i^LJ

a=l Vt2g2

32x

m

/JL ™

EEu,'

i^

\a=li=l


E$>^,-)H <*=1 i = l

n

/ J J

i

+^ E E *«i& + 5 E E ^«>«.a=l i=l

a = l t'=l

(B.40) where -2

? =A irc Vi-x-A j

( B - 41 )

co and A c stand for the mean topological charge of simple loops and its dispersion. In Eq.(B.40) A is a cut-off parameter which has appeared due to the ultraviolet divergence of a resulting integral and the mass renormalization counterterm, B, is introduced to subtract this divergence. Thus one has *'

F(n'm)

,

;.

. =r6aP6ij

(B.42)

with an arbitrary subtraction point rjiai = pai, ij)*^ — n*pj. Let we assume this point to be symmetrical with respect to all indices Mc = M«, = V-

(B-43)

Using Eqs.(B.42) and (B.43) we arrive at the following expression for

Statistics of Knots and Entangled Random Walks

180

the free energy

F<"'ra)(r, w) = E E 2—2 / 32?r

"

/

m

EX>««£i N V.a=l i = l n

v^r^a^

mnfi n

m

n

m

EE^-<

\a=l i= l m

+ (B.44)

^ E E ^.-^i - V E E ^^-Ci+ a=l i=l -2—2 /

i

a = l i^jf n

m

n

m

\

where the fourth-order vertex constant Vjj has the form L a

^'-i^1-^

(B.45)

The validity of the mean-field approximation and the role of the intrareplica fluctuations have been investigated in I. The contribution of interreplica fluctuations as well as the possibility of replica symmetry breaking are much more involved problems which need additional investigation and are beyond the scope of the present analysis. 1.2.3.

Variational

Principle

The Eq.(B.44) for the effective mean-field free energy is symmetric with respect to the permutations of the ij) fields in the leaves and replica spaces. On the other hand the nontrivial structure of the fourth-order vertex (Eq.(B.45)) in the leaf space forces us to look for an asymmetric leave-space-solution in the condensed state. Such solution (symmetric in the replica space and asymmetric in the leaf space) can be written in the form *«.* = ? + « H

< , - = ? + «<*

(B.46)

where V> and ij) are the average values of the order parameters and 6 fields determine the dispersion in the leaf space. Let us suggest for 6, 0* the trial function in the Gaussian form: P(0i,0;)ocexp

m

(B.47)

Appendix

181

where 1

°°

"■2=-Y,eft

(B.48)

•=i

The distribution Eq.(B.47) can be regarded as test function with the vari­ ational parameter a2. Now we are able to utilize variational principle. Substitute Eq.(B.44) for Eqs.(B.46) and (B.48) taking into account Eq.(B.47) and proceed with the limit m —» oo. As a result we obtain the free energy in the asymmetric state fa{Pa, c2, T,W) = lim lim — F("- m ){p a ,
(B.49)

mn

where p = ipij)* stands for the density. The variational parameters p and tr2 are the solution of pair of coupled equations ^ — / a ( p a , T 2 ) = 0;

/a(Pa)O.2)

OPa

=

0

( B 5 0 )

OT2

The asymmetric solution corresponds to the case c 2 > 0. It becomes sym­ metric for tr2 = 0: f,(p„a2

= 0) = lim Um — F^m\p.,a2

= 0)

(B.51)

n—*0m—>oo 77X71

in this case the free energy is only a function of the density. Minimizing ft (p« i o-2 = 0) with respect to p, we find the density of the symmetric state considered in I. Performing the calculations described above we find fa(P, a2)

=

2 \(La 4V

■-

- w)p2 + (La2 - | ) a2p + \ (la2 " -(p a2))ln In ?±Z-=— + (p + o-

32TT

^

VP

V

7

2/

T

(a2)2

- J)

( p + 0-2) _ ^(La2

-

w)p

p.2

16TT

K

32TT

(B.52) A . S Y M M E T R I C SOLUTION. In this case a-2 = 0 and the free energy takes the form f.{p.,

r, v) = \{La2

- w,)p2 - ^ - p . In ^

+ r>.

(B.53)

Statistics of Knots and Entangled Random. Walks

182

where r;=r-M2(Za2 -10,)+ ^

(B-54)

The minimization of Eq.(B.53) with respect to p, yields (La2 - w,)p, = ^

In (jL + l ) - 2T;

(B.55)

As discussed in I this equation has one stable and one unstable solution. Hence the first-order phase transition (loop condensation) occurs and the binodal curve is determined by the equation

JV.bin

32a-

■h(f(^y)

( B - 56 >

where N*la stands for the renormalized loop length at the binodal point. We define (as in I) T*

It is obvious that the symmetric solution corresponds to the case of the simple loop considered in I with the replacement La2 —* La2 — w,. We have w, > 0 what means that N*'m < N$™ , e l o o p . The free energy Eq.(B.53) at the binodal point has the form

B.

A S Y M M E T R I C SOLUTION.

In this case the Eqs.(B.50) takes the

form l

(T

-(La

and

2

N

, IT

W

2

a\

-wa)Pa+(La

-

2

^

-

V< V —

+T-S(La2-wa)+^

f

, c

Pa +
i

L

(B.58)

=0

(B.59) + T -

2 M

( l a

2

- ^ ) = 0

Appendix

183

These equations can be transformed into: (2La2 - wa)*l = ^

(in t + d + i ) _

2T;

_ {La2 _ Wa)pa

(B.60)

where 2

2

r 2 9 Lao= ¥^ - <7 La2 pa = ^16x - + M 2 w a

(B.61)

and n2(La2 - wa) + ?L±-

r*a=r-

(B.62)

loir

The analysis of these equations shows again that the nontrivial values of p and a2 appear as a result of the first order phase transition. For binodal curve we find 2 1 =fl 9 in N* 32x The corresponding free energy (Eq.B.52)) at the binodal point has the form

Finally let us extract the density of the junction points, K. In the symmetric case we have K

= W--3^r

=

T{-W^)

{La*-w.y

( B 6 5 )

whereas in the asymmetric case we have K = Wa

-dW=T\-V^)

{2La? - „ay

( R 6 6 )

The expression of difference between N*in and N*ln follows directly from Eqs.(B.56) and (B.63):

+1 +

ffbin

Wbin

!6

X

(B67)

('"fe ) **"' '


If the densities of junctions in symmetric and asymmetric cases are equal, we can write La2 ~W ' 2La2 — wa

W ' wa

(B.68)

184

Statistics of Knots and Entangled Random Walks

where

, wa = -{ALa2

/ J2La2Y

+ Y)-

1 +

-Y2

'

4« V 16^ /

Supposing the density of junction points to be high, we get La2n > 1 2 2 2

M * )

(B.70)

In this region Eq.(B.67) takes the form 1 #bin

1 #bin

0 16 X

(B.71)

and we arrive at the following inequality

N** < N?* < N&vtau*p

1.2.4. Comments

and

( B - 72 )

Conclusions

Eq.(B.72) has very clear meaning. Let us take an ensemble of complex loops, where each chain has the length No, carries the topological charge with the mean value Co and dispersion A c and has the density of junction points K. The distribution of obstacles is characterized by its 2D-density <po. Call the set {No, Co, A c , K, ipo} the preparation conditions (compare to I, where K = 0). If we increase the chain length keeping the preparation conditions fixed then the loop condensation occurs at N = N„in and the equilibrium thermodynamic state of the loop becomes asymmetric in leaf sizes, i.e. from geometrical point of view the complex loop resembles one long simple loop with many small "subloops" attached. We emphasize that the inequality Eq.(B.70) is crucial for the Eq.(B.72). It is easy to see that in opposite limit in Eq.(B.70) (small density of junc­ tion points), Eqs.(B.60)-(B.62) lead to an unphysical solution a2 < 0. This means t h a t the asymmetric state becomes favourable at high density of junctions or at highly entangle state. It follows from Eq.(B.69) t h a t in

Appendix

185

this region w, < La? and wa < 2La2, so the fourth-order vertex does not change its sign and the collapse is induced by the random distribution of obstacles only. We have tried to generalize the results concerning the problem of loop condensation in disorder array of topological obstacles (considered in Ap­ pendix Section 1.1) to the case of complex loops. We have showed that the general result of I remains unchanged, i.e. random distribution of obstacles induces the collapse transition in the loop prepared in specific topological state. However the phase behavior becomes much more intricate. We have found, for instance, t h a t the collapse transition is accompanied by the re­ building of internal structure of the loops from symmetric to asymmetric distribution in the leaf space (see above). We believe t h a t the proposed model can be also used for the investi­ gation of the stress-strain behaviour of polymer chains in random arrays of topological obstacles what could be regarded as the basis for the description of high elastic properties of irregular gels, which demonstrate very unusual thermodynamic behavior under deformations 24 > 25 . References

1. S.K. Nechaev, V.G. Rostiashvili, J. Phys. II (France), 3 (1993), 91 2. V.G. Rostiashvili, S.K. Nechaev, T.A. Vilgis, Phys. Rev.(E), 48 (1993), 3314 3. F.A.L. Dullein, Porous Media, Fluid Transport and Pore structure, (Aca­ demic Press: N.Y., 1979) 4. W.W. Yan, J.J. Kikland, D.E. Bly, Modem Size Exclusion Liquid Cromatography, (J.Wiley: N.Y., 1979) 5. A. Baumgarthner, M. Muthukumar, J. Chem. Phys., 87 (1987), 3082 6. S.F. Edwards, M. Muthukumar, J. Chem. Phys., 89 (1988), 2435 7. M.E. Cates, R.C. Ball, J. Phys. (France), 49 (1988), 2009 8. T.A. Vilgis, J. Phys. (France), 50 (1989), 3243 9. A. Baumgarthner, M. Moon, Europhys. Lett., 9 (1989), 203 10. F.F. Ternovslrii, A.R. Khokhlov, Zh.Exp.Teor.Fiz., 90 (1986), 1249 11. A.R. Khokhlov, F.F. Ternovskii, E.A. Zheligovskaya, Physica A, 163 (1990), 747 12. S.V. Panykov, Zh.Exp.Te0r.Fi2;., 94 (1988), 174 13. P. Goldbard, N. Goldenfeld, Phys. Rev. (A), 39 (1989), 1402, 1412 14. M.G. Brereton, S. Shah, J.Phys. (A), 13 (1980), 2751; D.J. Elderfield, J. Phys. (A), 15 (1982), 1369; F. Tanaka, Progr. Theor. Phys., 68 (1982), 148, 164. 15. V.J. Emery, Phys.Rev.(B), 11 (1975), 239 16. S. Coleman, E. Weinberg, Phys. Rev. (D), 7 (1973), 1888

186

Statistics of Knots and Entangled Random Walks

17. W. Horsthemke, R. Lefever, Noise-Induced Transitions, (Springer: Berlin, 1984) 18. S. Nechaev, Int. J. Mod. Phys. (B), 4 (1990), 1809 19. A.R. Khokhlov, S.K. Nechaev, Phys. Lett. (A), 112 (1985), 156 20. R.T. Deam, S.F. Edwards, Philos. Trans. R. Soc. London (A), 280 (1976), 317 21. G. Ferry, Viscoelastic Properties of Polymers (Wiley: New York, 1980) 22. S.F. Edwards, in Polymer Networks, edited by A.J. Chompff and S. Newman (Plenum Press: New York, 1972) 23. R.P. Feynman and A.R. Hibbs, Path Integrals and Quantum Mechanics (Academic Press: New York, 1965) 24. Y. Rabin, R. Bruinsma, Europhys. Lett., 20 (1992), 79 25. S.V. Panynkov, ZhETP Letters, 58 (1993), 114 (in Russian)

Subject Index Abelian problem Alexander polynomial Algebraic area Algebraic invariant Annealed disorder Array of obstacles

6 4 7 4 29 43

Beltrami-Laplace operator Bethe ansatz Blob Braid group Brownan bridge

75 31 113 44 43

Cactus-like graph Cayley tree Central charge Central limit theorem Commutation relations Conformal dimension — invariance — mapping Constant negative curvature Critical exponent Crumpled globule

173 51 84 52 68 90 93 85 43 84 125

Dichromatic polynomial Directed polymers

15 134

Effective potential Elasticity of polymer networks Entanglement Eulerian walks

166 115 6 22

Fibre bundle Flory approach Fokker-Plank equation Functional (path) integral Fiirstenberg theorem

86 129 78 8 78 187

188

Statistics of Knots and Entangled Random

Walks

Fractal dimension Free group

128 51

Gauss linking number Gaussian coil Geodesies Group of differential equation Gyration radius

4 130 53 89 112

Hamiltonian walks Hecke algebra Holonomy Homology Homotopy Hyperbolic space — manifolds

37 46 94 12 10 75 80

Isotopy ambient — regular

16 15

Jacobi Theta-function Jacobian of conformal transformation Jones invariant

97 98 13

Kauffman invariant Knot complexity — diagram — entropy — group — quasi— twisting — linking Knotting probability

19 48 15 3 12 133 20 6 12

Lattice animal Large deviations method Lobachevskii plane/space Locally free group Lyapunov exponent

113 146 75 43 74

Magnus representation Markov chain

48 4

Subject Index

189

— move Matrix unimodular Metric tensor Modular function — group — graph Monodromy Multiconnected space

45 146 75 97 89 57 89 42

Nematic ordering Non-abelian problem Non-Euclidean distance Normal order of grnerators

134 10 69 69

Order parameter — in replica space

140 36

Parisi ansatz PCAO model Peano curve Phase diagram Pochhammer contour Potts model Primitive word Probability distribution — conditional — joint Punctured plane

37 108 126 139 10 13 44 21 42 53 43

Quenched disorder

21

Random walk Reidemeister moves/theorem Replica trick Riemann surface — sheet

2 15 34 83 86

Self-similarity Shadow diagram of knot Skein relations Solvent quality

126 176 18 129

190

Statistics

of Knots

Spin glass Splitting State model Swelling Symbolic dynamics Temperley-Lieb algebra Topological invariant

and Entangled

Random

Walks

14 17 20 126 70 1 3

Ultrametric space/potential Universal covering space/surface

104 85

Variational principle Vertex Virasoro algebra Virial expansion

174 13 90 113

Wiener measure

5

Yang-Baxter relations

1