Long-range Interactions, Stochasticity and Fractional Dynamics: Dedicated to George M. Zaslavsky (1935 - 2008)

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NONLINEAR PHYSICAL SCIENCE

NONLINEAR PHYSICAL SCIENCE Nonlinear Physical Science focuses on recent advances of fundamental theories and principles, analytical and symbolic approaches, as well as computational techniques in nonlinear physic al science and nonlinear mathem atics with engineering applications . Topics of interest in Nonl inear Physical Science include but are not limited to: - New finding s and discoveries in nonlinear physics and mathematics - Nonlinearity, complexity and mathematical structures in nonlinear phy sics - Nonlinear phenomena and observ ations in nature and engineering - Computational methods and theories in complex systems - Lie group analysis, new theories and principles in mathematical modeling - Stabil ity, bifurc ation, chaos and fractals in physical science and engineering - Nonlinear chemical and biological physics - Discontinu ity, synchronization and natural complexity in the physic al sciences

SERIES EDITORS Albert C.J . Luo

Nail H. Ibragimov

Department of Mechao ical and Industrial Engineering Southern Illinois University Edwardsville Edwardsville, IL 62026-1 805, USA Email: aluo @siue.edu

Department of Mathematics and Science Blekinge Institute of Technolog y S-371 79 Karlskrona, Sweden Email : [email protected]

INTERNATIONAL ADVISORY BOARD Ping Ao , University of Washington . USA; Email: aoping@ u.was hington.edu Jan Awrejcewicz, The Technic al University of Lodz, Poland ; Email: awrejeew@ p.lodz.pl Eugene Bcnilov, University of Limerick. Ireland ; Email: Eugene.Benilov@ ul.ie Eshel Ben-Jacob, Tel Aviv University, Israel; Email: cshclts' tnmar.tau.ac.il Maurice Courbage, Univcrsitc Paris 7, France; Email: mauriee.eourb [email protected] Marian Gid ea , Northe astern Illinois University, USA; Email: mgidea @neiu.edu James A. Glazier, Indiana University. US A; Email: glazicr @indiana.edu Shijun Liao, Shanghai Jiaotong University. China; Email: [email protected] Jose Antonio Tenreiro Machado, ISEP-Institute of Engineering of Porto , Portugal; Email: jtm @dec.i sep.ipp .pt Nikolai A, Magnitskll, Russian Academy of Sciences. Russia; Email : nmag @isa.ru Josep J. Masdemont, Univcrsitat Poliiccnica de Cata lunya (UPC). Spain ; Email: jos [email protected] Dmitry E. Pelinovsky, McMaster University. Canada; Email: dmpeli @math .memaster.ea Sergey Prants, Vl.Il'i chcv Pacific Occanological Institut e of the Russian Academy of Sciences, Russia; Email: prants @poLdvo.ru Vlctor I. Shrira, Keele University, UK; Email : [email protected]. uk Jian Qiao Sun, University of California. USA; Email: jq sun @uemereed .edu Abdul-Majid \Vazwaz, Saint Xavier University, USA; Email: wazwaz@sx u.edu Pei Yu, The University of Western Ontario . Canada; Email: pyu @uwo.ea

Albert C.l. Luo Valentin Afraimovich

Long-range Interactions, Stochasticity and Fractional Dynamics Dedicated to George M. Zaslavsky (1935-2008)

With 114 figures

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HIGHER EDUCATION PRESS

~ Springer

Edito rs

Albert C.J . Luo Department of Mechanical and Industrial Engineering Southern Illinoi s University Edwardsville Edwardsville, IL 62026-1805 , USA E-mail : aluo @siue.edu

Valent in Afraimovich IICO-UASLP, Av. Karakorum 1470 Loma s 4a Seccion , San Luis Potos i SLP 782 10, Mexico E-mail : [email protected]

ISSN 1867-8440

e-ISSN 1867-8459

Nonline ar Physical Science ISBN 978-7-04-029188-9 Higher Educat ion Press, Beijing ISBN 978-3-642-12342-9

e-ISBN 978-3-642-12343-6

Springer Heidelberg Dordrecht London New York Library of Congre ss Control Number: 2010924294

© Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010 Th is work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproducti on on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permi ssion for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulation s and therefore free for general use. Cover design : Frido Steinen-Bro o, EStudio Calamar, Spain Printed on acid-free paper Springer is part of Springer Science-Business Media (www.springer.com)

To the memory of George M. Zaslavsky

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Preface

George M. Zaslavsky was born in Odessa , Ukraine in 1935 in a family of an artillery officer. He received education at the University of Odessa and moved in 1957 to Novosibirsk, Russia. In 1965, George joined the Institute of Nuclear Physics where he became interested in nonlinear problems of accelerator and plasma physics . Roald Sagdeev and Boris Chirikov were those persons who formed his interest in the theory of dynamical chaos . In 1968 George introduced a separatrix map that became one of the major tools in theoretical study of Hamiltonian chaos. The work "Stochastical instability of nonlinear oscillations" by G. Zaslavsky and B. Chirikov, published in Physics Uspekhi in 1971, was the first review paper "opened the eyes" of many physicists to power of the theory of dynamical systems and modern ergodic theory. It was realized that very complicated behavior is possible in dynamical systems with only a few degrees of freedom . This complexity cannot be adequately described in terms of individual trajectories and requires statistical methods . Typical Hamiltonian systems are not integrable but chaotic , and this chaos is not homogeneous. At the same values of the control parameters, there coexist regions in the phase space with regular and chaotic motion . The results obtained in the 1960s were summarized in the book "Statistical Irreversibility in Nonlinear Systems" (Nauka , Moscow, 1970). The end of the 1960s was a hard time for George . He was forced to leave the Institute of Nuclear Physics in Novosibirsk for signing letters in defense of some Soviet dissidents. George got a position at the Institute of Physics in Krasnoyarsk , not far away from Novosibirsk. There he founded a laboratory of the theory of nonlinear processes which exists up to now. In Krasnoyarsk George became interested in the theory of quantum chaos . The first rigorous theory of quantum resonance was developed in 1977 in collaboration with his co-workers. They introduced the important notion of quantum break time (the Ehrenfest time) after which quantum evolution begins to deviate from a semiclassical one. The results obtained in Krasnoyarsk were summarized in the book "Chaos in Dynamical Systems " (Nauka, Moscow and Harwood, Amsterdam, 1985). In 1984, R. Sagdeev invited George to the Institute of Space Research in Moscow. There he has worked on the theory of degenerate and almost degenerate Hamilto-

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Preface

nian systems, anomalous chaotic transport, plasma physics, and theory of chaos in waveguides . The book "Nonlinear Physics : from the Pendulum to Turbulence and Chaos" (Nauka, Moscow and Harwood, New York, 1988), written with R. Sagdeev, is now a classical textbook for everybody who studies chaos theory. When studying interaction of a charged particle with a wave packet, George with colleagues from the Institute discovered that stochastic layers of different separatrices in degenerated Hamiltonian systems may merge producing a stochastic web. Unlike the famous Arnold diffusion in non-degenerated Hamiltonian systems, that appears only if the number of degrees of freedom exceeds 2, diffusion in the Zaslavsky webs is possible at one and half degrees of freedom . This diffusion is rather universal phenomenon and its speed is much greater than that of Arnold diffusion . Beautiful symmetries of the Zaslavsky webs and their properties in different branches of physics have been described in the book "Weak chaos and Quasi-Regular Structures" (Nauka , Moscow, 1991 and Cambridge University Press, Cambridge, 1991) coauthored with R. Sagdeev, D. Usikov, and A. Chernikov . In 1991, George emigrated to the USA and became a Professor of Physics and Mathematics at Physical Department of the New York University and at the Courant Institute of Mathematical Sciences. The last 17 years of his life he devoted to principal problems of Hamiltonian chaos connected with anomalous kinetics and fractional dynamics, foundations of statistical mechanics, chaotic advection, quantum chaos , and long-range propagation of acoustic waves in the ocean . In his New York period George published two important books on the Hamiltonian chaos: "Physics of Chaos in Hamiltonian Systems " (Imperial College Press, London, 1998) and "Hamiltonian chaos and Fractional Dynamics" (Oxford University Press , NY, 2005) . His last book "Ray and wave chaos in ocean acoustics : chaos in waveguides " (World Scientific Press, Singapore, 2010) , written with D. Makarov , S. Prants , and A. Virovlynsky, reviews original results on chaos with acoustic waves in the underwater sound channel. George was a very creative scientist and a very good teacher whose former students and collaborators are working now in America, Europe and Asia. He authored and coauthored 9 books and more than 300 papers in journals. Many of his works are widely cited . George worked hard all his life. He loved music, theater, literature and was an expert in good vines and food . Only a few people knew that he loved to paint. In the last years he has spent every summer in Provence , France, working, writing books and papers and painting in water colors. The album with his water colors was issued in 2009 in Moscow. George Zaslavsky was one of the key persons in the theory of dynamical chaos and made many important contributions to a variety of other subjects . His books and papers influenced very much in advancing modern nonlinear science . Sergey Prants Albert C.J. Luo Valentin Afraimovich March, 2010

Contents

1 Fractional Zaslavsky and Henon Discrete Maps Vasily E. Tarasov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1 1.1 1.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I Fract ional derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3 1.2.1 Fractional Riemann-Liouville derivative s . . . . . . . . . . . . . . . . . .. 3 1.2.2 Fractional Caputo derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4 1.2.3 Fraction al Liouville derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.4 Interpretation of equations with fraction al derivatives . . . . . . . . 6 1.2.5 Discrete maps with memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 8 1.3 Fractional Zaslavsky maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9 1.3.1 Discrete Chirikov and Zaslavsky maps . . . . . . . . . . . . . . . . . . . .. 9 1.3.2 Fractional univers al and Zaslavsky map . . . . . . . . . . . . . . . . . . . 10 1.3.3 Kicked damped rotator map .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3.4 Fractional Zaslav sky map from fractional differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4 Fract ional Henon map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4.1 Henon map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 14 1.4.2 Fractional Henon map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IS 1.5 Fractional derivative in the kicked term and Zaslavsky map . . . . . . . . 16 1.5.1 Fractional equation and discrete map . . . . . . . . . . . . . . . . . . . . . . 17 1.5.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.6 Fract ional derivative in the kicked damped term and generalizations of Zaslavsky and Henon maps. . . . . . . . . . . . . . . . . . . . 21 1.6.1 Fractional equation and discrete map . . . . . . . . . . . . . . . . . . . . . . 21 1.6.2 Fractional Zaslavsky and Henon maps . . . . . . . . . . . . . . . . . . .. 24 1.7 Conclusion. . . . . . .. . . . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . . . .. . . .. 25 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2 Self-similarity, Stochasticity and Fractionality Vladimir V Uchaikin

27

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Contents

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.1.1 Ten years ago . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Two kinds of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Dynamic self-similarity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.1.4 Stochastic self-similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5 Self-similarity and stationarity . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.2 From Brownian motion to Levy motion. . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Self-similar Brownian motion in nonstationary nonhomogeneou s environment . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Stable laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Discrete time Levy motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 Continuous time Levy motion . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.2.6 Fractional equations for continuous time Levy motion . . . . . . 2.3 Fractional Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Differential Brownian motion process . . . . . . . . . . . . . . . . . . . . . 2.3.2 Integral Brownian motion process . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Fractional Brownian motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Fractional Gaussian noises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.3.5 Barne s and Allan model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.6 Fractional Levy motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Fractional Poisson motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Renewal processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.4.2 Self-similar renewal processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Three form s of fractal dust generator . . . . . . . . . . . . . . . . . . . . .. 2.4.4 nth arrival time distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.5 Fractional Poisson distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Fractional compound Poisson process . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Compound Poisson process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Levy-Poisson motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Fractional compound Poisson motion . . . . . . . . . . . . . . . . . . . .. 2.5.4 A link between solutions 2.5.5 Fractional generali zation of the Levy motion . . . . . . . . . . . . . . Ackno wledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Appendix. Fractional operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Reference s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

Long-range Interactions and Diluted Networks Antonia Ciani, Duccio Fanelli and Stefano Ruffo 3.1 Long-range interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3.1.1 Lack of additivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Equilibrium anomalies: Ensemble inequivalence, negative specific heat and temperature jumps . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Non-equ ilibrium dynamical propertie s. . . . . . . . . . . . . . . . . . . . 3.1.4 Quasi Stationary States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27 27 28 29 30 31 32 32 35 40 45 50 51 54 55 56 58 60 61 62 64 64 65 66 68 68 70 70 71 73 74 75 77 77 79

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3.1.5 Physical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.1.6 General remarks and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 97 3.2 Hamiltonian Mean Field model: equilibr ium and out-ofequilibrium features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 97 3.2.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.2.2 Equilibrium statistical mechanics . . . . . . . . . . . . . . . . . . . . . . . . 100 3.2.3 On the emergence of Quasi Stationary States: Nonequilibrium dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.3 Introducing dilution in the Hamiltonian Mean Field model . . . . . . . 120 3.3.1 Hamiltonian Mean Field model on a diluted network . . . . .. 120 3.3.2 On equilibrium solution of diluted Hamiltonian Mean Field 121 3.3.3 On Quasi Stationary States in presence of dilution . . . . . . . . . 123 3.3.4 Phase transition 129 3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 130 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4

Metastability and Transients in Brain Dynamics: Problems and Rigorous Results Valentin S. Afraim ovich. Mehmet K. Muezzinoglu and Mikhail f. Rabinovich . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.1 Introduction: what we discuss and why now 4.1.1 Dynamical modeling of cognition . . . . . . . . . . . . . . . . . . . . . .. 4.1.2 Brain imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Dynamics of emotions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.2 Mental modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.2.1 State space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.2.2 Functional networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.2.3 Emotion-cognition tandem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.2.4 Dynamical model of consciousness . . . . . . . . . . . . . . . . . . . . . . 4.3 Competition-robustness and sensitivity. . . . . . . . . . . . . . . . . . . . . . .. 4.3.1 Transients versus attractors in brain . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Cognitive variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.3.3 Emotional variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.3.4 Metastability and dynamical principles . . . . . . . . . . . . . . . . . .. 4.3.5 Winnerless competition-structural stability of transients . . 4.3.6 Examples: competitive dynamics in sensory systems . . . . .. 4.3.7 Stable heteroclinic channels 4.4 Basic ecological model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.4.1 The Lotka-Volterra system . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.4.2 Stress and hysteresis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Mood and cognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.4.4 Intermittent heteroclinic channel . . . . . . . . . . . . . . . . . . . . . . .. 4.5 Conclusion . . . .. . . . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . . . .. . . . . . .

133 134 134 135 136 137 137 137 140 142 144 145 146 147 148 148 ISO lSI 153 153 ISS 157 160 161

Contents

xiv

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Appendix 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163 163 167 169

5 Dynamics of Soliton Chains: From Simple to Complex and Chaotic Motions Konstantin A. Gorshkov, Lev A. Ostrovsky and Yury A. Stepanyants . . . . 5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Stable soliton lattices and a hierarchy of envelope solitons . . . . . . . . 5.3 Chains of solitons within the framework of the Gardner model . . .. 5.4 Unstable soliton lattices and stochastisation . . . . . . . . . . . . . . . . . . . . . 5.5 Soliton stochastisation and strong wave turbulence in a resonator with external sinusoidal pumping . . . . . . . . . . . . . . . . . . . .. 5.6 Chains of two-dimensional solitons in positive-dispersion media 5.7 Conclusion Few words in memory of George M. Zaslavsky . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

177 177 179 188 193 202 204 212 212 214

6 What is Control of Turbulence in Crossed Fields'!-Don't Even Think of Eliminating All Vortexes! Dimitri Volchenkov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Stochastic theory of turbulence in crossed fields : vortexes of all sizes die out, but one . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 The method of renormalization group . . . . . . . . . . . . . . . . . . .. 6.2.2 Phenomenology of fully developed isotropic turbulence . .. 6.2.3 Quantum field theory formulation of stochastic Navier-Stokes turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Analytical properties of Feynman diagrams . . . . . . . . . . . . . .. 6.2.5 Ultraviolet renormalization and RG-equations . . . . . . . . . . . . 6.2.6 What do the RG representations sum ? . . . . . . . . . . . . . . . . . . . 6.2.7 Stochastic magnetic hydrodynamics . . . . . . . . . . . . . . . . . . . .. 6.2.8 Renormalization group in magnetic hydrodynamics . . . . . .. 6.2.9 Critical dimensions in magnetic hydrodynamics. . . . . . . . .. 6.2.10 Critical dimensions of composite operators in magnetic hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6.2.11 Operators of the canonical dimension d = 2. . . . . . . . . . . . .. 6.2.12 Vector operators of the canonical dimension d = 3. . . . . . .. 6.2.13 Instability in magnetic hydrodynamics. . . . . . . . . . . . . . . . .. 6.2.14 Long life to eddies of a preferable size. . . . . . . . . . . . . . . . .. 6.3 In search oflost stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Phenomenology of long-range turbulent transport in the scrape-off layer (SOL) of thermonuclear reactors . . . . . . . . . .

219 220 221 221 224 226 229 229 233 233 235 238 240 241 242 243 244 249 249

Cont ents

xv

6.3.2

Stochastic models of turbulent transport in cross-field systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6.3.3 Iterative solutions in crossed fields . . . . . . . . . . . . . . . . . . . . . .. 6.3.4 Functional integral formulation of cross-field turbulent transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.5 Large-scale instability of iterative solutions . . . . . . . . . . . . . . . 6.3.6 Turbulence stabilization by the poloidal electric drift . . . . . . 6.3.7 Qualitative discrete time model of anomalous transport in the SOL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6.4 Conclusion . . . .. . . . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . . . .. . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

252 256 259 262 266 267 272 272

Entropy and Transport in Billiards M. Courbag e and S.M. Saberi Fathi 7.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7.2 Entropy . .. . . . . . ... . . . . . ... . . . . ... . . . . . ... . . . . ... . . . . . ... . . . . 7.2.1 Entropy in the Lorentz gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Some dynamical properties of the barrier billiard model. . . 7.3 Transport. 7.3.1 Transport in Lorentz gas 7.3.2 Transport in the barrier billiard . . . . . . . . . . . . . . . . . . . . . . . . .. 7.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Index

277 277 278 283 290 297 299 301 306 307 310

Contributors

Valentin S. Afraimovich Instituto de Investigacion en Comun icacion Optica, Universidad Autonoma de San Luis Potosi, Karakorum 1470, Lomas 4a 78220 , San Luis Potosi, S.L.P., Mexico, e-mail: [email protected] Antonia Ciani Dipartimento di Fisica, Universita di Firenze, and INFN, Via Sansone I, 50019 Sesto Eno (Firenze), Italy, e-mail: [email protected] M. Courbage Laboratoire Matiere et Systemes Complexe s (MSC), UMR 7057 CNRS et Universite Paris 7- Denis Diderot, Case 7056, Batiment Condorcet, 10, rue Alice Domon et Lonie Duquet, 75205 Paris Cedex 13, France, e-ma il: courbage @ccr.jussieu.fr S.M. Saberi Fathi Department of Physics, University of Wisconsin-M ilwaukee , 1900 E. Kenwood Blvd., Milwaukee, WI 53211 , USA, e-mail : saberi @uwm.edu Duccio Fanelli Dipartimento di Energetica and CSDC, Universita di Firenze, and INFN, via S. Marta, 3, 50139 Firenze, Italy, e-mail : duccio .fanelli @unifi.it K.A. Gorshkov lAP RAS, Nizhny Novgorod , Russia, e-mail : Gorshkov @hydro.appl. sci-nnov.ru Mehmet K. Muezzinoglu BioCircuit Institute, University of Californ ia, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0328, USA, e-mail : kerem .muezzinoglu @ gmail.com L.A. Ostrovsky ZeITech/NOAA ETL, Boulder, USA and lAP RAS, Nizhny Novgorod, Russia, e-mail : Lev.A.Ostrovsky @noaa.gov M.I Rabinovich BioCircuit Institute, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0328, USA, e-mail : mrabinovich @ucsd.edu

Contributors

xvii

Stefano Ruffo Dipartimento di Energetica and CSDC, Universita di Firenze, and INFN, via S. Marta, 3, 50139 Firenze , Italy, e-mail: [email protected] Y.A. Stepanyants: Department of Mathematics and Computing, USQ, Toowoomba, Australia, e-mail : [email protected] Vasily E. Tarasov Skobeltsyn Institute of Nuclear Physics , Moscow State University, Moscow 119991, Russia, e-mail : tarasov @theory.sinp.msu.ru Vladimir V. Uchaikin Ulyanovsk State University, Ulyanovsk, Russia, e-mail : [email protected] Dimitri Volchenkov The Center of Excellence Cognitive Interaction Technology (ClTEC) , University of Bielefeld , Postfach 100131, D-3350 I, Bielefeld, Germany, e-mail : [email protected]

Chapter 1

Fractional Zaslavsky and Henon Discrete Maps Vasily E. Tarasov

Abstract This paper is devoted to the memory of Professor George M. Zaslavsky passed away on November 25, 2008 . In the field of discrete maps, George M. Zaslavsky introduced a dissipative standard map which is called now the Zaslavsky map. G. Zaslavsky initialized many fundamental concepts and ideas in the fractional dynamics and kinetics . In this chapter, starting from kicked damped equa tions with derivatives of non-integer orders we derive a fractional generalization of discrete maps. These fractional maps are generalizations of the Zaslavsky map and the Henon map. The main property of the fractional differential equations and the correspondent fractional maps is a long-term memory and dissipation. The memory is realized by the fact that their present state evolution depends on all past states with special forms of weights .

1.1 Introduction There are a number of distinct areas of mechanics and physics where the basic problems can be reduced to the study of simple discrete maps. Discrete maps have been used for the study of dynamical problems, possibly as a substitute of differential equations (Sagdeev et aI., 1988; Zaslavsky, 2005; Chirikov, 1979; Schuster, 1988; Collet and Eckman, 1980). They lead to a much simpler formalism, which is particularly useful in computer simulations. In this chapter, we consider discrete maps that can be used to study the evolution described by fractional differential equations (Samko et aI., 1993; Podlubny, 1999; Kilbas et aI., 2006). The treatment of nonlinear dynamics in terms of discrete maps is a very important step in understanding the qualitative behavior of continuous systems described by differential equations. The derivatives of non-integer orders (Samko et aI., 1993) are Vastly E. Iarasov Skobeltsyn Institute of Nuclear Physics, Moscow State Univer sity, Moscow 119991, Russia, e-mail: [email protected]

A. C. J. Luo et al. (eds.), Long-range Interactions, Stochasticity and Fractional Dynamics © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010

2

Vasily E. Tarasov

a natural generalization of the ordinary differentiation of integer order. Note that the continuous limit of discrete systems with power-law long-range interactions gives differential equations with derivatives of non-integer orders with respect to coordi nates (Tarasov and Zaslavsky, 2006; Tarasov, 2006). Fractional differentiation with respect to time is characterized by long-term memory effects that correspond to intrinsic dissipative processes in the physical systems. The memory effects to discrete maps mean that their present state evolution depends on all past states. The discrete maps with memory are considered in the papers (Fulinski and Kleczkowski, 1987; Fick et aI., 1991; Giona, 1991; Hartwich and Fick, 1993; Gallas, 1993; Stanislavsky, 2006; Tarasov and Zaslavsky, 2008 ; Tarasov, 2009; Edelman and Tarasov, 2009) . The interesting question is a connection of fractional equations of motion and the discrete maps with memory. This derivation is realized for universal and standard maps in (Tarasov and Zaslavsky, 2008; Tarasov, 2009) . It is important to derive discrete maps with memory from equations of motion with fractional derivatives. It was shown (Zaslavsky et al., 2006) that perturbed by a periodic force, the nonlinear system with fractional derivative exhibits a new type of chaotic motion called the fractional chaotic attractor. The fractional discrete maps (Tarasov and Zaslavsky, 2008; Tarasov, 2009) can be used to study a new type of attractors that are called pseudochaotic (Zaslavsky et aI., 2006). In this chapter, fractional equations of motion for kicked systems with dissipation are considered . Correspondent discrete maps are derived . The fractional generalizations of the Zaslavsky map and the Henon map are suggested . In Sect. 1.2, we give a brief review of fractional derivatives to fix notation and provide a convenient reference . In Sect. 1.3, the fractional generalizations of the Zaslavsky map are suggested . A brief review of well-known discrete maps is considered to fix notations and provide convenient references . In Sect. lA, the fractional generalizations of the Henon map are considered. The differential equations with derivatives of non-integer orders with respect to time are used to derive general izations of the discrete maps. In Sect. 1.5, a fractional generalization of differential equation in which we use a fractional derivative of the order 0 ::; f3 < I in the kicked term, i.e. the term of a periodic sequence of delta-function type pulses (kicks). The other generalization is suggested in (Tarasov and Zaslavsky, 2008). The discrete map that corresponds to the suggested fractional equation of order 0 ::; f3 < I is derived. This map can be considered as a generalization of universal map for the case 0 < f3 < I. In Sect. 1.6, a fractional generalization of differential equation for a kicked damped rotator is suggested . In this generalization, we use a fractional derivative in the kicked damped term, i.e. the term of a periodic sequence of deltafunction type pulses (kicks) . The other generalization is also suggested in (Tarasov and Zaslavsky, 2008) . The discrete map that corresponds to the suggested fractional differential equation is derived. Finally, a short conclusion is given in Sect. 1.7.

I Fractional Zaslavsky and Henan Discrete Maps

3

1.2 Fractional derivatives In this section a brief introduction to fractional derivatives are suggested. Fractional calculus is a theory of integrals and derivatives of any arbitrary order. It has a long history from 1695, when the derivative of order a = 1/2 has been described by Gottfried Leibniz. The fractional differentiation and fractional integration goes back to many mathematicians such as Leibn iz, Liouville, Grunwald, Letnikov, Riemann, Abel, Riesz, Weyl. The integrals and derivatives of non-integer order, and the fractional integro-differential equations have found many applications in recent studies in theoretical physics, mechanics and applied mathematics. There exists the remarkably comprehensive encyclopedic-type monograph by Samko, Kilbus and Marichev, which was published in Russian in 1987 and in English in 1993. The works devoted substantially to fractional differential equations are the book by Miller and Ross (1993), and the book by Podlubny (1999) . In 2006 Kilbas , Srivastava and Trujillo published a very important and remarkable book, where one can find a modem encyclopedic, detailed and rigorous theory of fractional differential equations. The first book devoted exclusively to the fractional dynamics and application of fractional calculus to chaos is the book by Zaslavsky published in 2005 . Let us give a brief review of fractional derivatives to fix notation and provide a convenient reference.

1.2.1 Fractional Riemann-Liouville derivatives Let [a,b] be a finite interval on the real axis lit The fractional Riemann-Liouville derivatives Dg+ and D~_ of order a > 0 are defined (Kilbas et aI., 2006) by (D~+j)(x)

=

D';(i~+a)(x)

D11

1

r(n-a)

1 x

x a

j(z)dz (x-z)a -11+1

(x > a),

(DLj)(x) = (-I )"D~(i~=a)(x) (-1)" D11 r(n - a) x

1· (z-j(z)dz x)a-11+1 b

x

where n = [aj + 1 and [aj means the integral part of tive of order n. In particular, when a = n E N, then (D~+j)(x)

(x
a. Here D': is the usual deriva-

= (DLj)(x) = j(x) ,

(D~+j)(x) =D~j(x) ,

(D'b _j)(x) =

(-I)"D~j(x) .

The fractional Riemann-Lioville differentiation of the power functions (x - a)13 and (b - x)13 yields power functions of the same form :

4

Vasily E. Tarasov

=

r(f3 + 1) (x-a)f3 -a r( a + f3 + I) ,

Da (b - x)f3 =

r(f3 + I) (b _ x)f3 - a r( a + f3 + I) ,

Da (x-a)f3 a+ b-

where f3 > -I and a > O. In particular, if f3 = 0 and a > 0, then the fractional Riemann-Liouville derivatives of a constant C are not equal to zero:

On the other hand, for k = 1,2 , ... , [aj + 1, we have D~+(x-a)a -k=O ,

Dg_(b-x)a -k=O.

The equality is valid if and only if, II

f(x)

=

E Ck(x-a)a -k , k=l

where n = [aj + I and Ck are real arbitrary constants. The equation

is satisfied if and only if, II

f(x) = ECk(b-x)a -k, k=]

where n = [aj + I and Ck are real arbitrary constants.

1.2.2 Fractional Caputo derivatives The fractional Caputo derivatives CD~+ and C Dg_ are defined for functions for which the Riemann-Liouville derivatives exists . Let a > 0 and let n be given by n = [aj + I for a 'i N, and n = a for a E N. If a 'i N, then the fractional Caputo derivatives is defined by the equations ( C

a f)() Da+ X

( ll -a ll f )( )

= la+ D

X

lx

I d D~f(z) = r(n-a) a Z(x_Z)a -ll+l '

I Fractional Zaslavsky and Henon Discrete Maps

5

f) (x) = (-I)I1(f" - aDl1f)(x) = (-IY' ( cDa bbr(n - a) where n = [a] + I. If a

l

x

bdz

D~f(z)

(z-x)a -I1+] ,

= n E N, then

If a 'i N and n = [a] + I, then fractional Caputo derivatives coincide with the fractional Riemann-Liouville derivatives in the following cases:

if

f(a) = (D~f)(a) = ... = (DI1 - 1f)(a) = 0,

and if

f(b) = (D'J) (b) = ... = (DI1 - lf)(b) = 0.

It can be directly verified that the fractional Caputo differentiation of the power functions (x - a)13 and (b - x)13 yields power functions of the form

c Da (x _ a)13 = a+

c Da (b _ x)13 = b-

1(f3 + I) (x _ a)13 -a 1( a + f3 + I) , 1(f3 + I) (b _ x)13 -a 1( a + f3 + I) ,

°

where f3 > -I and a > 0. In particular, if f3 = and Caputo derivatives of a constant C are equal to zero :

a > 0, then the fractional

For k = 0 , 1,2 , ... , n - I,

The Mittag-Leffler function Ea[A(X - a)a] is invariant with respect to the Caputo derivatives c D~+,

but it is not the case for the Caputo derivative C

Dg_.

6

Vasily E. Tarasov

1.2.3 Fractional Liouville derivatives Let us define the fractional Liouville derivatives on the whole real axis 1Ft The fractional Liouville derivatives D~ and D~ of order a > 0 are defined (Kilbas et al., 2006) by (D~f)(x)

= D~(I~-a)(x)

jX

a"

I

= r(n - a) axil (D~f)(x)

-

00

f(z)d z (x - z)a- Il+1 '

= (-I )IlDl:(i~-a)(x)

a"

(-I )11

= r(n - a) axil

r oo

Jx

f(z)dz (z-x)a -Il+] ,

where n = [a J+ I and [a J means the integral part of tive of order n. In particular, when a = n E N , then (D~f)(x)

a. Here D~ is the usual deriva-

= (D~f)(x) = f(x) ,

(D'!t-f)(x) = D':f(x),

(D~f)(x)

If f(x) is an integrable function and

f3 > a >

= (-I )IlD':f(x) .

I, then

(DV!fJ)(x) = f(x),

(D~I£f)(x) = (I£-a f)(x) , (DkD~f)(x) If a

= (±ll(D~+kf)(x) .

> 0, then the following relations hold : (§D~f)(k)

=

(~ik)a(§f)(k) ,

where (~ik)a

= Iklaexp{±ani sgn (x)/2}.

Here § is the Fourier transform .

1.2.4 Interpretation of equations with fractional derivatives To describe the physical interpretation of equations with fractional derivatives and integrals with respect to time, we consider the memory effects and limiting cases widely used in physics : (I) the absence of the memory ; (2) the complete memory; (3) the power-like memory.

Fractional Zaslavsky and Henan Discrete Maps

7

Let us consider the evolution of a dynamical system in which some quantity F (r) is related to another quantity f(t) through a memory function M(t) :

F(t) = l M(t-7:)f(7:)d7:.

(I.I )

Equation (1.1) means that the value F(t) is related with f(t) by the convolution operation

F(t) = M(t) * f(t) .

Equation (1.1) is a typical non-Markovian equation obta ined in studying the systems coupled to an environment, with environmental degrees of freedom being averaged. Let us consider special cases of Eq. (1 .1) . (I) For a system without memory, we have the Markov processes, and the time dependence of the memory function is

M(t - 7:) = 8(t - 7:) ,

(1.2)

where 8(t - 7:) is the Dirac delta-function. The absence of the memory mean s that the function F (r) is defined by f(t) at the only instant t. For this limiting case, the system loses all its states except for one with infinitely high density. Using (1 .1) and (1.2), we have

F(t) = l8(t-7:)f(7:)d7:=f(t) .

(1.3)

The expression (1.3) corresponds to the process with complete absence of memory. This process relates all subsequent states to previous states through the single current state at each time t . (2) If memory effects are introduced into the system the delta-function turns into some function, with the time interval during which f(t) affects on the function F(t) . Let M (r) be the step function

M(t - 7:) = r",

(O <7:
M(t -7:) = 0,

(r c- r).

The factor t - I is chosen to get normalization of the memory function to unity :

Then in the evolution process the system passes through all states continuously without any loss. In this case,

F(t)=~ t

(f(7:)d7:,

Jo

and this corresponds to complete memory. (3) The power-like memory function

8

Vasily E. Tarasov

M(t--r) =MO(t--rr- 1

(1.4)

indicates the presence of the fractional derivative or integral. Substitution of (1.4) into (1.1) gives the temporal fractional integral of order e:

I\. t F(t) = r(e) Jo (t--ry~-lf(-r)d'r,

(0 < e < 1),

( 1.5)

where r(e) is the Gamma function, and I\. = r(e)Mo . The parameter I\. can be regarded as the strength of the perturbation induced by the environment of the system. The physical interpretation of the fractional integration is an existence of a memory effect with power-like memory function . The memory determines an interval t during which the function f( r) affects on the function F (r). Equation (1.1) is a special type of equations for F (z ) and f(t) , where F is directly proportional to M * f . In the general case, the values F (z ) and f(t) can be related by the equation

E(F,M*f) =0,

(1.6)

where E is a smooth function . Relation (1.6) is a fractional equation for a dynamical system . This equation describes the memory effect. If F is a coordinate q(t), either velocity q(t) , or acceleration q(t), and f is a derivative q(m )(r), then Eq. (1.6) can be considered as an equation of motion with memory. For a power-like memory function M(t ), we present (1.6) as a equation with fractional derivatives: E(q,q ,q(t), D~q) = 0. As the result, we can use the fractional calculus (Kilbas et a1., 2006) to describe the motion of systems.

1.2.5 Discrete maps with memory The mapping Xn+l = f(x n) does not have any memory, as the value Xn+l only depends on Xn. The introduction of memory means that the discrete value Xn +l is connected with the previous values Xn, Xn -I , . .. ,XI . Particularly, any system, which is described by a discrete map, will have a full memory, if each state of the system is a simple sum of all previous states: n

Xn+l

=

E f(Xk) ,

(1.7)

k=l

where f(x) is a function that define the discrete map. In general, the expression of Eq. (1.7) can tend to infinity. Note that the full memory exists for functions that give a finite sum in Eq. (1.7). The full memory is ideal because has the same action upon the next states as all the others in memory. The map with a long-term memory can be expressed as n

Xn +l

=

E Va(n,k)j(Xk) , k= l

(1.8)

I Fractional Zaslavsky and Henon Discrete Maps

9

where the weights Va(n,k) , and the parameter ex characterize the non-ideal memory effects . The forms of the functions f(x) and Va(n,k) in Eq. (1.8) are obtained by the differential equation. Note that linear differential (kicked) equations gives the linear function f(x) = x. The discrete maps with memory is considered, for example, in the papers (Fulinski and Kleczkowski, 1987; Fick et al., 1991; Hartwich and Fick, 1993; Giona , 1991; Gallas, 1993; Stanislavsky, 2006 ; Tarasov and Zaslavsky, 2008; Tarasov, 2009 ; Edelman and Tarasov, 2009). The interesting question is a connection of fractional equation of motion and the discrete maps with memory (Tarasov and Zaslavsky, 2008 ; Tarasov, 2009) . It is important to derive discrete maps with memory from equation of motion with fractional derivatives .

1.3 Fractional Zaslavsky maps In this section, a brief review of discrete maps is considered to fix notations and provide convenient references (Sagdeev et al., 1988; Zaslavsky, 2005; Chirikov, 1979; Schuster, 1988; Collet and Eckman, 1980). Fractional generalizations of these maps are discussed . The fractional Zaslavsky maps are suggested.

1.3.1 Discrete Chirikov and Zaslavsky maps Let us consider the equations of motion t

00

x+KG(x) L 8(T -n) = 0,

(1.9)

n=O

in which perturbation is a periodic sequence of delta-function type pulses (kicks) following with period T = 2TC /V, K is an amplitude of the pulses, and G(x) is a some function . This equation can be presented in the Hamiltonian form 00

x=p,

t

P+KG(X)L8(T-n)=0.

( 1.10)

n=O

It is well-known that these equations can be represented in the form of discrete map, see for example in Chapter 5 (Zaslavsky, 2005) . The solution of the left side of the n-th kick

xn = xit; - 0) = lim x(nT - e), 10-.0+

P«

= p(tn -

0)

=

lim p(nT - e) ,

10-.0+

tn = nT

(1.11) is connected with the solution on the right hand side of the kick x(tn + 0), p(tn + 0) by Eq. (1.10) , and the condition of continuity x(tn + 0) = x(tn - 0) . Using notation (1.1 I), we can derive the iteration equations

10

Vasily E. Tarasov

= X + PIl+1 T ,

( 1.12)

= Pn - KT G(XIl) .

( 1.13)

XIl +1 PIl+]

il

Equ ations (1.12) and (1.13) are called the universal map . 1fT = I and G(x) = sin( x) , then Eqs . (1.12) and (1.13) give the Chirikov map . The Zaslavsky map (Zaslavsky, 1978 ; Schmidt and Wang , 1985 ; Zaslavsky and Edelman, 2008) is defined by (1.14)

= e- q[Y" + E sin (XIl)],

Y,,+I

( 1.15)

where we use the parameter

J1 = (e q -I )/q . Note that a shift 12 does not play an important role and it can be put zero (12 = 0). The Zasl avsky map with 12 = 0 can be represented by the equ ations XIl+1 P,,+]

=

= X; + P,,+I ,

( 1.16) ( 1.17)

-bp" - Z sin (XIl).

For b = -I and Z = K , we get the stand ard map that is described by Eqs . (1.12) and (1.13) with T = I . For the parameters ( 1.18) Equ ation s (1.16) and (1.17) give Eqs. (1.14) and (1.15) with 12 = O. For large q ----7 00 (for small b ----7 0) Eqs . (1.16) and (1.17) with Z = - K shrink to the one-dimensional sine-map ( 1.19) XIl +] = Xil + K sin(XIl) proposed in (Arnold, 1965) and studied in many papers.

1.3.2 Fractional universal and Zaslavsky map Let us consider a fractional generalization of (1.9) in the form (1.20) where aD~ is the Riemann-Liouville fractional derivative (Samko et al., 1993 ; Podlubny, 1999 ; Kilbas et al., 2006), which is defined by a

2

2-a

aDt x = Dt alt

I

d

2

r

x(r)dr

X= r(2-a) dt2 Ja (t - r )a - I'

(I < a ::; 2).

(1.21 )

II

I Fractional Zaslavsky and Henon Discrete Maps

Here we use the notation D; = d 2 / dt 2, and 011a is a fractional integration (Samko et al., 1993; Podlubny, 1999; Kilbas et al., 2006). The discrete map that corresponds to the fractional differential equation of order I < a :::; 2 is derived in (Tarasov and Zaslavsky, 2008; Tarasov, 2009). This map can be considered as a generalization of universal map for the case I < a :::; 2.

Theorem 1.1 Fractional differential equation of kicked system (1.20) the initial conditions

is equivalent to the discrete map PIl+1

= P«-

Ta -I Il x ll +] = 1(a) ~/k+1 Va(n -k+ I) where I

(1.22)

KT G(XIl) , CTa -2

+ 1(a -I) (n + i)a-2,

( 1.23)

< « < 2 and the/unction Va(z) is defined by ( 1.24)

Proof of this Proposition has been realized for C = 0, (Tarasov and Zaslavsky, 2008) and for C =I- O. (Tarasov, 2009) Equations (1.22) and (1.23) define a fractional universal map in the phase space (XIl ,PIl), which is a generalization of the map (1.12) and (1.13) . Note that Eqs. (1.22) and (1.23) with G(x) = sin(x) give

PIl+1

= Pil -

KT sin(x ll ) ,

Ta -I Il XIl +] = 1(a) ~/k+1 Va(n -k+ I)

< a :::; 2) ,

(I

( 1.25)

CTa -2

+ 1(a -I) (n + i)a-2.

(1 .26)

These equations define a fractional standard map on the phase space , which can be called the fractional Chirikov-Taylor map. The fractional generalization of Zaslavsky map (Zaslavsky, 1978; Schmidt and Wang, 1985) can be defined by

PIl+] = -bpil - Z sin(x ll ) ,

I Il XIl +] = 1(a) ~/k+1 Va(n -k+ I)

< a :::; 2),

(I

C

+ 1(a -I) (n + i)a-2,

(1 .27)

( 1.28)

where the parameters are defined by conditions (1.18) . For b = -I and Z = K, Eqs . (1.27) and (1.28) give the fractional standard map that is described by Eqs . (1.25) and (1.26) with T = I. For for small b ----+ 0 and C = 0, Eqs. (1.27) and (1.28) with Z = -K shrink to a one-dimensional map

K

XIl +1 = -(-) T

a

Il

L Va(n k=O

k) sin(x ll ) ,

(I

< a :::; 2) .

(1 .29)

12

Vasi1y E. Tarasov

This map can be considered as a fractional generalization of one-dimensional sinemap (1.19). The fractional sine-map (1.29) is characterized by a long-term memory such that the present state evolution depends on all past states with special forms of weights function (1.24) .

1.3.3 Kicked damped rotator map Let us consider a kicked damped rotator (Schuster, 1988). The equation of motion for this rotator is

x+qi = KG(x)

f O(t -nT) .

(1.30)

n=O

It is well-known that Eq . (1.30) gives the two-dimensional map (Schuster, 1988):

Pn+l = e-qT [Pn + KG(x n)], x n+] = Xn +

I -e- qT

q

(1.31)

[Pn + KG(xn)].

(1.32)

This map is known as the kicked damped rotator map . The phase volume shrinks each time step by a factor exp( -q) . The map is defined by two important parameters, dissipation constant q and force amplitude K . These equations can be rewritten in the form (1.33) Pn+l = e-qT [Pn + KG(x n)], e qT -I (1.34) Xn+l =xn+---Pn+]. q It is easy to see that these equations give the Zaslavsky map (1.16) and (1.17) with = 0 if

.Q

X n = Xn ,

Y" = p« , E = K ,

If we consider the limit q ----+

00

and K ----+

00

T

=

I,

G(x)

such that q / K

= sin(x) .

= I and the funct ion

G(x) = (r-l)x-rx 2, then Eqs . (1.32) and (1.32) give the logistic map

Xn+l = rxn(1 -xn) . Note that the evolution of logistic maps with a long-term memory is considered in (Stanislavsky, 2006) .

13

I Fractional Zaslavsky and Henon Discrete Maps

1.3.4 Fractional Zaslavsky map from fractional differential equations Let us consider a fractional generalization of differential equation for a kicked damped rotator (1.30) in the form

f

OD~x-qODfx=KG(x)

8(t-nT) ,

( 1.35)

11=0

where q E R and sidered with

oD~

is derivative of non-integer order. This equation will be con-

I < a ::; 2,

f3 = a -

I,

and oD~ is the Riemann-Liouville fractional derivative (Samko et aI., 1993; Podlubny, 1999; Kilbas et aI., 2006) defined by (1.21). Note that we use the minus in the left hand side of Eq. (1.35), where q can be positive and negative value. The discrete map that corresponds to the fractional differential equation has been derived in (Tarasov and Zaslavsky, 2008).

Theorem 1.2 Fractional differential equation of kicked system (1.35) is equivalent to the discrete map PIl+l

= eqT [PIl + KG(xn ) ] ,

I

Xl1 +1

( 1.36)

11

= r(a-I) k~/k+l Wa(q,T ,n+ I-k),

( 1.37)

where the functions Wa(a ,b) are defined by

( 1.38) Proof of this statement has been realized in (Tarasov and Zaslavsky, 2008) . Equations (1.36) and (1.37) define a fractional kicked damped rotator map. This Theorem can be used to derive a fractional generalization of the Zaslavsky map. If we use the conditions XIl = x n,

Y" = Pn ,

e

= K,

T

=

I,

G(x)

= sin(x) ,

(1.39)

then Eqs. (1.36) and (1.37) give the map ( 1.40)

XIl+l

= ( T

I

a -I

11

) LYk+l Wa(q , l ,n+ I-k) .

(1.41)

k=O

These equations can be considered as a second fractional generalization of the Zaslavsky map (1.14) and (1.15) with.Q = O. Note that this fractional Zaslavsky map is

14

Vasily E. Tarasov

derived from fractional nonlinear differential equation (1.35) with condition (1.39). If we use the variables

then discrete equations ~, +1

Xn+l

=

=

u:'

(

-b~,-Zsin(X,,),

"

T a-I

) [Pk+1 Wa(q, I ,n+ I-k).

( IA2) (1.43)

k=O

These equations can be considered as a fractional generalization of the Zaslavsky map Eqs. (1.16) and (1.17) with .Q = O. For a = 2. This fractional Zaslavsky map gives the Zaslavsky map that is described by Eqs. (1.16), and (1.17). For exp{ q} -+ 0 Eqs. (1 AD) and (1 AD) shrink to a one-dimensional map that can be considered as a fractional generalization of one-dimensional sine-map (1.19).

1.4 Fractional Henon map In this section, a brief review of Henon discrete maps is considered to fix notations and provide convenient references (Schuster, 1988; Henon, 1976). Fractional generalizations of this map are suggested . And the Henon maps with memory are discussed .

1.4.1 Henan map The Henon map (Henon, 1976; Russel and Hanson, 1980) can be considered as a two-dimensional generalization of the logistic map (Schuster, 1988). Then the Henon map is presented as a map given by the coupled equations

=

X,,+l

2

l-ax"+Y,,,

Y,,+l

= bx.;

( 1.44) ( lAS)

Substitution of Eq. (1 AS) into Eq. (1A4) gives Xn +l

=

I-ax;'+bxn -l .

( 1.46)

This equation is equivalent to the Henon map that is defined by Eqs. (1.44) and (1.45). Note that the Henon map in the form (1.46) can be derived from the generalized dissipative map. Let us consider the generalized dissipative map

Fractional Zaslavsky and Henon Discrete Maps

15

Xn+l = xn + Pn+]T , Pn+l

(1.47)

= -bpn + KTG(xn), Ibl ~

Using Eqs . (1.47) and (1.48) with K

(1.48)

1.

= T = I, and the function

G(X) = 1- (l-b)x-ax 2,

( 1.49)

the Henon map in the form (1.46) can be obtained. Let us prove this statement. Substitution of (1.47) in the form Pn+] = x n+] - Xn into (1.48) gives

Xn+l -Xn

= -b(xn -Xn -l) + G(x n).

(1.50)

Then we have

Xn+l -bXn -l

= (l-b)xn+G(Xn ) .

(1.51)

Substitution of (1.49) into (1.51) gives the Henon map (1.46) .

1.4.2 Fractional Henon map To derive a fractional generalization of the Henon map we can use the fractional generalized dissipative map and the fact that the Henon map in the form (1.46) can be derived from the generalized dissipative map. Let us consider a fractional generalization of the map (1.47) and (1.48) in the form Ta -1 n Xn+l=qa)k~/k+]Va(n-k+I) , (l
Pn+l

= -bpn+KTG(xn) .

(1.53)

This is the fractional generalized dissipative map. If a = 2, then Eqs . (1.52) and (1.53) give (1.47) and (1.48). If G(x) = sin(x), T = I and K = -Z, C = 0, then Eqs . (1.52) and (1.53) give the fractional Zaslavsky map equations (1.27) and (1.28). To derive a fractional generalization of the Henon map, we can use the fractional dissipative map (1.52) and (1.53) with K = T = I, and the function (1.49) . As a result , we obtain the equations I

n

Xn+l = qa) k~/k+1Va(n-k+ I) , Pn+l = -bpn + I - (I - b)xn- ax~ ,

(Ibl

(1.54)

~ I),

(1.55)

where I < a ~ 2 and Va(z) is defined by (1.24) . These equations can be considered as a first fractional generalization of the Henon map (1.44), (1.45) . Fract ional Henon map (1.54) and (1.55) is derived from Eq. (1.52) and (1.53) , where the dissipation has been introduced into the iteration equation. In this case

16

Vasily E. Tarasov

the fractional equation of motion with dissipation is not used. Equations (1.54) and (1.55) are not derived from fractional differential equations of kicked damped system. We can suggest the second possible getting of a fractional generalization of the Henon map. This map can be derived from the fractional equation of kicked damped rotator (1.35). It is known (see Sect. 1.2 of (Schuster, 1988)) that the Henon map can be derived from equation of motion (1.30) with G(X)

where b = - exp( -q) and T the map equation :

= --q- [I + (I +b)x+ax2] I+b

=

(1.56)

,

I. Using Eq. (1.30) with function (1.56), we obtain

I+b

2

Xn+] =Xn+--Pn-I-(I +b)xn-axn , q _ e _ q Pn+ qb Pn+l-

l+ b

[ 1+ ( I+bxn+ax ) 2n] ,

(1.57)

(1.58)

where T = I, and b = - exp{ -q }. These equations are equivalent to the Henon map equation (1.46) . Proof of this proposition was realized in (Schuster, 1988). Let us consider the fractional differential equation (1.35) that are generalization of Eq. (1.30) by non-integer order derivatives . In order to derive fractional generalization of Henon map, we use Eq. (1.35) with the function (1.56). As a result, we obtain the equations I

Xn+l

n

= r(a-I) t ; /k+l Wa(q,T,n+ I-k) ,

(1.59)

+(I-b)xn+ax~)],

(1.60)

Pn+] =eq[Pn+ I :b (I

where the functions Wa(a,b) are defined by Eq. (1.38) and q = In( -b) . These equations define a second fractional generalization of the Henon map. This map is derived from nonlinear differential equations with derivatives of non-integer order. Equations (1.59) and (1.60) can be called fractional Henon map.

1.5 Fractional derivative in the kicked term and Zaslavsky map In this section, a fractional generalization of differential equation (1.9) is suggested. In this generalization, we use a fractional derivative of the order 0 :::; f3 < I in the kicked term, i.e. the term of a periodic sequence of delta-function type pulses (kicks). The other generalization that is described by Eq. (1.20) is suggested in (Tarasov and Zaslavsky, 2008). The discrete map that corresponds to the suggested

17

I Fractional Zaslavsky and Henon Discrete Maps

fractional equation of order 0 :::; f3 < 1 is derived . This map can be considered as a generalization of univers al map for the case 0 < f3 < I.

1.5.1 Fractional equation and discrete map Let us consider a fractional generalization of (1.9) in the form (0 :::;f3 <1) .

(1.61 )

where ~Df is the Caputo fractional derivative 1-/3 1 _ I t d-r d.x(-r) oD I x-oIl Dl x - r(1 _ f3) Jo (t - -r)/3 ct:r '

c /3 _

(0 :::; f3 < 1).

( 1.62)

Here we use the notation Df = d/dt , and olla is a fractional integration (Samko et al., 1993; Podlubny, 1999; Kilbas et al., 2006) . For f3 = 0 fractional equation (1.61) gives Eq. (1.9) .

Theorem 1.3 Fractional differential equation of kicked system (1.61) is equivalent to the discrete map (1.63) Xn+l = Xn + Pn+1 T , T1-/3 n- I ( 1.64) Pn+1 = Pn - KTC(r(2 _ f3) tr/k+1 V2 -/3(n - k)) , (0 :::; f3 < I), where the fun ction V2 - /3 (z) is defined by (1.65) Proof The fraction al equation (1.61) can be presented in the Hamiltonian form

x=p, ( 1.66)

Between any two kicks there is a free motion P = const ,

x

= pt + const .

( 1.67)

The integration of (1.66) over (tn + e , tn + 1 - s) gives X(tn +1 - 0) = xit; +0) - Pn+,T,

(1.68)

18

Vasily E. Tarasov

p(t"+1 - 0)

= pit; + 0) .

( 1.69)

The solution of the left side of the n-th kick

x"

= x(t" - 0) = limx(nT - f) ,

( 1.70)

1' ~ 0

p" = pit; - 0)

=

lim p(nT - f) ,

1' ~0

til

= nT,

(1.71)

is connected with the solution on the right hand side of the kick x(t" + 0), p(t" + 0) by Eq . (1.66) , and the continuity condition

x(t" + 0)

= x(t" - 0) .

( 1.72)

The integration of (1.66) over the interval (t" - e, til + f) gives

p(t" + 0) = p(t" - 0) - KTG(5D~x) .

(1.73)

Using notations (1.70) and (1.71), and the solution (1.67), we get

pit; + 0) = p(t"+1 - 0) = P,,+I.

( 1.74)

Substituting of (1.72) and (1.73) into (1.68) and (1.69), we derive the iteration equations ( 1.75) X,,+I =x" + p,,+IT ,

P,,+I = p" - KT G(5D~x) .

( 1.76)

To derive a map, we should express the fractional derivative c f3

D nx -

o

t

-

I

=-0-"-----::::-;-

r(1 - {3)

lt

n

0

dr dx('f) (t" - 'f)f3 dr

( 1.77)

through the variables (1.70) and (1.71) . Using x = p, we have (1.78) It can be presented as

c f3 _

oDtnx-

I

"-'

Ll

r(1-{3)k=O

tH '

tk

p('f)d'f (t,,-'f)f3

'

where tk+1 = tk + T = (k+ I )T , tk = kT, and to = O. For the interval (tk,tk+I), Eqs . (1.67), (1.70) and (1.71) give

Then

( 1.79)

19

I Fractional Zaslavsky and Henon Discrete Maps

( 1.81) Using (I - f3)1( I - f3) can be presented as

= 1(2 - f3 ) and Eq. (1.81) , the fraction al derivative (1.79)

where ( 1.83)

•

Thi s end s of the proof.

As a result, Eq. (1.76) take s the form of Eqs. (1.63) and (1.64) . The se equations define the fraction al generalization of univers al map. Let us con sider the following Corollary.

Theorem 1.4 For f3 = 0 thefra ctional universal map, which is defined by Eqs. (1 .63) and (1.64), gives the usual universal map.

Proof Let us cons ider that the fractional universal map (1.63) and (1.64) for f3 Substitution of Eq. (1.63) in the form

= O.

( 1.84) into iteration equation (1.64) gives

Then , the fraction al map (1.63) , (1.64) is defined by Xn +l

= Xn + Pn+l T ,

E

T -/3 n Pn+] = Pn - KTC(1(2 _ f3) (Xk -Xk-l)V2-/3(n -

where 0 ::::; f3

< I. For f3 = 0, we have V2 - /3 (z) = I, and

(1.86)

k)) .

20

Vasily E. Tarasov 11 -1

[,(Xk+l-Xk) =XIl-XO· k=O

(1.87)

PIl+l =PIl-KTG(xll-xo).

(1.88)

Then Eq. (1.85) gives

As a result, Eqs . (1.86) with This ends of the proof.

f3 = 0 give the usual map for the case Xo = O. _

1.5.2 Examples Let us consider some examples of the fractional map Eqs . (1.63) and (1.64) .

Example I.l.lf G(x)

= -x, then Eqs . (1.63) and (1.64) are XIl +l

=XIl + PIl+I T,

where V2 - !3 (z) is defined in (1.24) . Equations (1.89) can be considered as a fractional generalization of the Anosov type system.

Example 1.2.lf G(x)

= sin(x), XIl +l

Eqs. (1.63) , (1.64) give

= XIl + PIl+] T, ( 1.90)

This map can be considered as a fractional generalization of standard map. The other possible form of Eq. (1.90) is XIl +l

= XIl + PIl+l ,

where we use Eqs. (1.84) and (1.24), T = I, and 0 ::; f3 < I. These equations define a fractional standard map that can be called the fractional Chirikov map.

Example 1.3. The fractional generalization of Zaslavsky map (Zaslavsky, 1978; Schmidt and Wang, 1985) can be defined by

Fractional Zaslavsky and Henon Discrete Maps

X,,+I ,,-I (

P,,+I

= x" + P,,+I ,

21

(1.92)

)

= -bp,,- K sin(~ ~(d ~;) [(n - k)I -13 - (n - k- 1)1 - 13]).

(1.93)

For b = I, we get the fractional standard map (1.91) . Th is map is one of possible fractional generalizations of the Zaslavsky map. In this generalization we introduce a dissipation by the change of the variable p" ---+ -bp". Equation (1.93) is not directly connected with a fract ional equ ation of motion . A generalization of the Zaslavsky map that is derived from the differential equation with fractional damped kicks is suggested in the next section.

1.6 Fractional derivative in the kicked damped term and generalizations of Zaslavsky and Henon maps In this section , a fractional generalization of differential equ ation (1.30) for a kicked damped rotator is suggested. In this generalization, we use a fract ional derivative in the kicked damped term , i.e. the term of a periodic sequence of delta-function type pul ses (kick s). The other generalization, which is described by Eq. (1.35), is suggested in (Tarasov and Zaslavsky, 2008). The discrete map that corresponds to the suggested fractional differential equ ation is derived .

1.6.1 Fractional equation and discrete map Let us con sider the fraction al generalization of equation (1.30) in the form

D?x-qDlx=KG(~Dfx) [,8 (t- nT ), (0 ~ f3 < I),

( 1.94)

,,=0

where q E R, and ~ Df is the Caputo fractional derivative (Kilbas et al., 2006) of the order 0 ~ f3 < I defined by (1.62) . Note that we use the minus in the left hand side of Eq. (1.35), where q can be positive and negative value . The fraction al der ivative ~ Df x is presented in the kicked damp ed term . Theorem 1.5 Fractional differential equation of kicked system (1.94) is equivalent to the discrete map I -e- qT (1 .95) X,,+] = x" + P,,+I , (1 .96)

22

Vasily E. Tarasov

where the funct ions W2- /3 are defined by

Proof. Fractional equation (1.94) can be presented in the Hamiltonian form

.x = P, p-qp =

KG(~Dfx)

f 8(t -nT) ,

(0 <

f3 < 0,

(1.98)

q E R) .

n=O

Between any two kicks

p-qp = O.

(1.99)

For t E (tn + O,tn+] - 0), the solution of Eq. (1.99) is

p(tn+l - 0) = p(tn + O)eqT . Let us use the notations tn

(1.100)

= nT, and Xn = x(t n - 0) = limx(nT - f) , 1' ~ 0

(I.IO!)

Pn = p(tn - 0) = lim p(nT - f ). 1' ~ 0

For t E (tn - £ , tn+I

-

f), the general solution of (1. 98) is ( 1.102)

Then

Pn+l = e

qT

[Pn + KG(~DeX)] .

(1.103)

Using (1.103), the integration of the first equation of (1.98) gives qT

_ l-e [ Pn +KG(oDtx) C /3 ] Xn+l-X . n-qn

(1.104)

Let us consid er the Caputo fractional derivative from Eqs. (1.103) and (1.1 04) . Then (0 :::; f3< 1). Using r

= p, we have (1.105)

23

I Fractional Zaslavsky and Henon Discrete Maps

This relation can be rewritten as (1.106) where tk+1 = tk + T = (k+ 1)T, and tk = kT, such that to = O. For r E (tk,tk+d , Eqs. (1.100) and (1.101) give

p( r) = P(tk + O)eq(r- lk) = p(tk+l - O)e- qTeq(r-1k) -- p k+l eq(r-1k- T) -- pk+ 1eq(r-1k+l ). Then

(1.107) As a result, Eq. (I . I07) can be represented by r)d~

p( jlk1k+ 1 (tn-r)

f3 l = Pk+lqf3 - eq(n -k -l )T [r( 1- f3 ,q(tn - tk+d) - r(1 - f3 .qtt; - td)] . (1.108)

Here r(a ,b) is the incomplete Gamma function , where a.b are complex numbers. This function can be defined by ba r(a ,b) = r(a) - - lFl(l , I + a; - b ). a

Here

1FI

is the confluent hypergeometric Kummer function (Erdelyi, 1953),

(a,c;z) =

(a )kzk

L -() k" 00

IFl

k=O C k

.

and (a)k is the Pochhammer symbol

(a )k=a(a +I) · · ·(a +k-I ),

k EN.

Using (1.106) and (1.108), we obtain ( 1.109)

24

Vasily E. Tarasov

where W2- /3 (q, T ,m) is defined by (1.97) . Substitution of (1.109) into (1.103) and (1.104) gives (1.110)

x n+ ]

= Xn -

l-e

qT

--

q

[

PIl + KG

(r(I1- f3) L PHI W - /3 (q, T ,n - k) n-I

2

)]

.

(1.111)

k=O

Equations (1.110) and (1.111) can be presented in the form of Eqs. (1.95) and (1.95) . This ends of the proof. _

1.6.2 Fractional Zaslavsky and Henan maps The iteration equations (1.95) and (1.96) define a fractional generalization of the kicked damped rotator map (1.30) . If we use the conditions (1.39), then Eqs. (1.95) and (1.96) give the map (1.112) (1.113) where

I-e- q

J1' = - - q

These equations can be considered as a fractional generalization of the Zaslavsky map (1.14) and (1.15) with Q = 0 and J1 = J1' . This fractional Zaslavsky map is derived from fractional differential equation (1.94) with condition (1.39). For f3 = 0 this map gives the Zaslavsky map (1.14) and (1.15) with J1 = J1' and Q = O. By analogy with Sect. 3, we can derive a fractional generalization of the Henon map from equation of motion (1.94) with

G(X) = -~b [I + (1 +b)x+ax2] , 1+

(1.114)

where b = - exp( -q) and T = I. As a result, this generalization has the form of Eqs. (1.95) and (1.96) with the function (1.114) . This fractional Henon map is derived from the differential equations with fractional derivative in the kicked damped term. For f3 = 0 this map gives the usual Henon map. Computer simulations of the fractional generalizations of discrete maps prove that the nonlinear dynamical systems, which are described by the equations with fractional derivatives, exhibit a new type of chaotic motion . This type of motion can be considered as a fractional generalization of chaotic attractor. As a result ,

I Fractional Zaslavsky and Henon Discrete Maps

25

the fractional discrete maps allow to study a new type of attractors that are called pseudochaotic.

1.7 Conclusion In many areas of mechanics and physics the problems can be reduced to the study of discrete maps. In particular the special case of discrete maps has been studied to describe properties of regular and strange attractors . Under a wide range of circumstances such maps give rise to chaotic behavior. The suggested fractional maps can be considered as a fractional generalization of discrete map. Note that a wide class of these maps can be derived from kicked fractional differential equations. The sugge sted fractional Zaslavsky map and the fractional Henon map can demon strate a chaotic behavior with a new type of attractors . The interesting property of these fractional discrete maps is a long-term memory. As a result, a present state evolution depends on all past states with the fractional power-law weights functions Va (z) and Wa(q, T,z) defined by Eqs . (1.24) and (1.38) . Note that the fractional maps are equivalent to the correspondent fractional kicked differential equations. An approximation for fractional derivatives of these equations is not used. This fact can be used to study the evolution that is described by fraction al differential equations and to describe pseudochaotic attractors.

References Arnold v., 1965, Small denominators. I: On the mappings of the circumference onto itself, Amer. Math. Soc. Transl. (2), 46, 213-284. Bateman H. and Erdelyi A, 1953, Higher Transcendental Functions VoL 1., McGraw-Hili, New York. Chirikov B. v., 1979, A universal instability of many dimensional oscillator systems , Phys. Rep., 52,263-379. Collet P. and Eckman J.P., 1980, Iterated Maps on the Interval as Dynamical System , Birkhauser, Basel. Edelman M. and Tarasov V.E., 2009, Fractional standard map, Phys. Lett. A , 374, 279-285. Erdelyi A., 1953, Higher Transcendental Functions, Vol. I and 2., McGraw-Hili, New York. Fick E., Fick M. and Hausmann G., Logistic equation with memory , 1991, Phys. Rev. A, 44, 2469-2473 . Fulinski A and Kleczkowski AS., 1987, Nonlinear maps with memory, Physica Scripta , 35, 119-122. Gallas lAC., 1993, Simulating memory effects with discrete dynamical systems, Physica A , 195,417-430; Erratum, Physica A , 198,339-339.

26

Vasily E. Tarasov

Giona M., 1991 , Dynamics and relaxation properties of complex systems with memory, Nonlinearity, 4, 991-925. Hartwich K. and Fick E., 1993 , Hopf bifurcations in the log istic map with oscillating memory, Phys. Lett. A, 177,305-310. Henon M ., 1976, A two-dimensional mapping with a strange attractor, Commun . Math. Phys., 50, 69-77. Kilbas A.A ., Srivastava H.M . and Trujillo J.1., 2006, Theory and Application of Fractional Differential Equations, Elsevier, Amsterdam . Miller K.S . and Ross B., 1993, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York. Podlubny 1., 1999, Fractional Differential Equations, Academic Press, San Diego. Russel D.A., Hanson J.D . and Ott E ., 1980, Dimension of strange attractors, Phys. Rev. Lett., 45,1175-1178. Sagdeev R.Z., Usikov D.A . and Zaslavsky G.M ., 1998, Nonlinear Physics : From the Pendulum to Turbulence and Chaos, Harwood Academic Publishers, New York. Samko S.G ., Kilbas A.A. and Marichev 0.1., 1993, Fractional Integrals and Derivatives Theory and Applications, Gordon and Breach, New York. Schmidt G. and Wang B. w., 1985, Dissipative standards map , Phys. Rev. A, 32, 2994-2999. Schuster H.G., 1988, Deterministic Chaos : An Introduction, 2nd edn ., Physik Verlag, Weinheim. Stanislavsky A.A., 2006, Long-term memory contribution as applied to the motion of discrete dynamical system, Chaos, 16,043105. Tarasov Y.E., 2006, Continuous limit of discrete systems with long-range interaction , J. Phys. A, 39,14895-14910. Tarasov Y.E., 2009 , Differential equations with fractional derivative and universal map with memory, J. Phys. A, 42, 465102. Tarasov Y.E. and Zaslavsky G.M., 2006, Fractional dynamics of coupled oscillators with long-range interaction, Chaos, 16,023110. Tarasov Y.E. and Zaslavsky G .M., 2008, Fractional equations of kicked systems and discrete maps, 1. Phys. A , 41, 435101. Zaslavsky G .M., 1978, Simplest case of a strange attractor, Phys. Let. A, 69, 145147. Zaslavsky G.M ., 2005, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, Oxford. Zaslavsky G.M ., Stanislavsky A.A. and Edelman M ., 2006, Chaotic and pseudochaotic attractors of perturbed fractional oscillator, Chaos, 16,013102. Zaslavsky G.M . and Edelman M., Superdiffusion in the di ssipative standard map, Chaos, 18,033116. (arXiv:0805.l952).

Chapter 2

Self-similarity, Stochasticity and Fractionality Vladimir V. Uchaikin

Abstract Self-similar generalizations of Brownian motion, Levy motion, Poisson process and compound Poisson process are considered. A deep link between selfsimilarity of stochastic processes and fractional differential calculus is recovered and a new class of probability distributions called fractionally stable distributions is described. This opens a way to more wide and effective applications ofthe fractional calculus to investigation of various real processes.

2.1 Introduction Two kinds of motion, dynamic and stochastic are considered from point of view of self-similarity concept.

2.1.1 Ten years ago I got to know George Zaslavsky in Poland , at the Seminar on Stability Problems for Stochastic Models (1999) . I (with Prof. V. Zolotarev) had just published the book "Chance and Stability " (Uchaikin and Zolotarev, 1999) devoted to description of the Levy stable laws and their applications to various problems, where we touched fractional differential equations. Two years earlier, George (with Prof. A. Saichev) published the article (Saichev and Zaslavsky, 1997) where the stable distributions appeared as a result of solution of some fractional differential equation. So we had a large field for talking and discussing when we walked on walk-ways instead of sitting in the restaurant and drinking an excellent wine... Our attention was attracted Villihmtr Y. Ochatkm U1yanovsk State University, Ulyanovsk, Russia, e-mail : [email protected]

A. C. J. Luo et al. (eds.), Long-range Interactions, Stochasticity and Fractional Dynamics © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010

28

Vladimir V. Uchaikin

to interplay between self-similarity, stochasticity and fractionality. The fate has not allowed us to cooperate for making a joint work, but when I read the last George book (Zaslavsky, 2005), I heard the continuation of our talk taken place in that remote autumn evening .... The aim of my article is to arrange that arguments in such a way that the idea on unity of stochasticity, self-similarity and fractional kind of differential operators become as evidential as possible .

2.1.2 Two kinds of motion There exist two kinds of motion in classical physics : deterministic motion with smooth trajectories obeying Newtonian law, and stochastic one with highly irregular broken trajectories described in terms of random processes theory. An ocular demonstration of the last kind processes gives us Brownian motion. This chapter is devoted to stochastic processes. For simplicity, we will mainly focus on the onedimensional case, but most of statements have higher-dimensional analogues. A standard Hamiltonian dynamic system is characterized by absence of memory. This means that if we know the state x(to) of the system at the present instant to we are able to predict its motion in the future (t > to) without knowing its past (t < to). When repeating observation of the system with the same initial condition x(to) = xo, we obtain the same trajectory of motion x(t), t > to. Stochastic systems differ from dynamic ones by randomness: repeating the experiment with the same initial condition Xo generates a whole ensemble trajectories and we can talk about coordinate at instant t as about a random variable or random vector X(t) with conditional probability distribution P(X(t) E dxIX(to) =xo) . This is a stochastic analogue of the dynamic causality principle, leading to the concept of Markov processes. In this case, we must know the pdf po(x,to) at an initial moment to and the transition pdf p(x ,to ---+ x,t) in order to predict the pdf p(x ,t) at observation moment t:

p(x,t) =

1:

po(x',to)p(x' ,to ---+ x,t)dx' .

A Markov process homogeneous with respect to space and time is called the Levy process (L-process), which can be seen in the Levy and Bertoin monographs (Levy, 1965; Bertoin , 1996)., its transition pdf depends only on differences of arguments:

p(XI,t' ---+x,t) = p(O,O ---+ x-x',t - t') == p(x-x',t -t') . At first glance , there are no common features between Newtonian (deterministic) kind of motion when a particle, for example, a planet moves along a smooth line in space and Brownian (random) kind of motion when the trajectory of a particle is a highly irregular everywhere broken random line. However, such a common feature exists, at least at small scales : both motions are self-similar.

2 Self-similarity, Stochasticity and Fractionality

29

2.1.3 Dynamic self-similarity The self-similarity (scaling) is a special kind of symmetry such that a change in scale of some variables (say, spatial coordinates) can be compensated by a corresponding rescaling others (say, time). The order of the self-sim ilarity is determined by constant H called often the Hurst exponent. A determin istic process x(t) , t :2: 0 is said to be the self-similar process oforder H (in short H -ss process) with respect to point xa and time ta if there exists such a positive H that x(at) -x(ata) = aH[x(t) -x(ta)], for any a > 0 and t > O. When a body falls from height xa = x(ta) in homogeneous gravitational field g then x(t) -Xa = -g(t-ta)2 , and

x(at) - x(ata) = -g(at - ata)2 = a2 [x (t ) - x(ta)].

Therefore, the motion is a 2-ss process with respect to the initial coordinate xa and time ta when the body had zero-velocity. Taking another point on the trajectory as an initial one, say t( > ta, when the velocity is -VI and

x(t) - X(tl)

= -VI (t - tl) - g(t - tl)2 ,

we arrive at the relation

Thus the point (t] ,X(tl )) is not a center of self-similarity, because there is no such H satisfying the condition

However, if we consider the motion of a free body, then

x(t) - x(ta) = va(t - ta), and the cond ition of self-similarity holds for any point of the trajectory and any time, H = I. It is conven ient to choose ta = 0, xa = 0 and write the self-similarity property in the form The solution of the equation is

When H is an integer the following equation takes place

Vladimir V. Uchaikin

30 dHxH _ , () H -H,XH I .

dt

Extending this on non-integer values of H we arrive to the simplest fractional differential equation

2.1.4 Stochastic self-similarity The next representation was inspired from (Bertoin, 1996; Samorodnitzky and Taqqu, 1994). A stochastic (random) process {X (t)} is called self-similar with index H > 0 in the strict sense if for any a > 0 and any n ;::: I, tl, tz ,... ,til the joint distribution of the random variables X (atd , ... ,X (atll ) is identical with the joint distribution of aH X (tl ), ... ,a HX (til):

In many cases, it is more suitable to use the following modification of this criterion: for any positive tl , ... .t; and a should hold the relation n

n

j=l

j=l

L ejX(atj) ~ L ejaHX(tj).

This relation will be expressed succinctly as follows :

{X(at)} ~ {aHX(t)}. We will write

X(at) ~ aHX(t)

to mean that the random processes X (at) and aH X (t) have identical one-dimensional distributions. This property will be termed the self-similarity in the wide sense. For an L-process to be self-similar, it is enough self-similarity of the onedimensional distribution: or equivalently

In terms of pdf p(x ,t), a one-dimensional H-ss L-process is characterized by the relation p(x ,t) = r" p(xt - H , I) following from the chain of evident equalities

31

2 Self-similarity, Stochasticity and Fractionality n- H

x

[ oop(x' ,t)dx' = P(X(t) < x) = P(tHX(I) < x ) = P(X(I) < xt - H) =

[ 00

p(x' ,I)dx'

after differentiating it with respect to x. Formally, the Newtonian motion of a free particle with a constant velocity v can be described as well in terms of 8-pdf p(x, r) = 8 (x - vt) and we have

p (x,t)

= 8(x- vt) = 8((x/t - v)t) = t - 18(x /t - v) = i :" p(xt - H, I) , H = I

as before. We will call a process X (r) self-similar in the mean if the ss property takes place only for the average value:

(X (at )) ~aH (X (t )).

2.1.5 Self-similarity and stationarity A non-degenerate stochastic H -ss process cannot be stationary (Samorodnitzky and Taqqu, 1994) . Really, if it were, we would have for any a > 0 and t > 0

from one side and

X(at) ~ X(t)

from the other. However, there is no such an H

> 0 to equality

be fulfilled for any t > O. There exists the following important interrelation between self-similar and stationary processes: if {X (r), t > O}is H -ss, then the process {Y (r) = e- HtX (et) , - 0 0 < t < oo} is stationary. Conversely, if {Y(t), - 0 0 < t < oo} is stationary, then {X(t) = t HY (in t) , t > O} is H -ss. Really, using the second criterion of stationarity we can satisfy ourselves in validity of these propositions: if {X(t), t > O} is H-ss, then for any 8 1, ••• , 8" and 8 > 0

L 8jY(tj + r) = L 8je-Htje-H'X(e'etj) ~ L 8je-HtjX(etj) = L 8jY(tj) "

"

j=l

j=l

and if {Y (r),

-00

I

"

"

j=l

j=l

< t < oo} is stationary, then for any positive tl , ... ,tIl

and a

" " " L" 8jX(atj) = L 8ja HtfY(lna + lnrj) ~I L 8ja HtfY(lntj) = L 8ja HX(tj) .

j=l

j=l

j=l

j=l

32

Vladimir V. Uchaikin

Notice, however, that every stationary process is O-ss in the mean.

2.2 From Brownian motion to Levy motion Brownian motion in stationary homogeneous environment and self-similar diffusion with a time- and space-dependent diffusion coefficient are described . Levy-motion is derived as a self-similar generalization of Brownian motion . As a result, we come to fractional differential equations for Levy-stable probability density functions .

2.2.1 Brownian motion The most known self-similar L-process is Brownian motion (Bm) that is the Lprocess with transition pdf

{x - -2 (J t 2

p(x, t)

=

V

I ;;=;-:=

2nt(J

exp

2 } .

It can also be determined as follows . Definition of Bm, The random process {X (r), t ~ O} is called (standard) Brow-

nian motion (Bm), if I) X (0) = 0 almost certainly; 2) {X (r}, t ~ O} is a process with independent increments; 3) X(t+r) -X(t)~rI/2G at any t and r where G is a normally distributed random variable with the variance 2:

Recall that a random process is called Gaussian process if, for any n ~ I and t",t,,_I , ... ,tl, the density p"(x,,,t,,; ... ;XI ,tI) is an n-dimensional normal (Gaussian) distribution. Consequently, the 8m is a homogeneous Gaussian process with independent increments, starting from the origin and having no drift. Let us list first properties of Bm .

1. Finite-dimensional densities It is joint pdf's of B(td , ... ,B(t,,) for 0

< tl < ... < tIl

are given by the products

f(x" ,t,,;... ;x2,t2;xI ,t I) = f(x",t"lx,, -I ,t,,- I) ' " f(X2 ,t2IxI ,tI )f(XI ,tI)

= f(x

ll -

Xll -I

,til -

til -I)'"

exp{ -(I / 2(JJ )[(xll

f(X2 -

XI

.t: - tI)j(Xl ,tI)

x ll _d 2/ (tll - tn-d + ...+ (X2 - xd 2/ (t2 - td +xT;td} (JJ(2n)Il/2[(tll - til -I) ' " (t2 - tl)td I/ 2 -

2 Self-similarity, Stochasticity and Fractionality

33

One can directly verify that the final-dimensional distributions of {B(at) , t ~ O} are identical to the finite-dimensional distributions of {a l / 2B(t), t ~ O} and consequently the process is self-similar with H = 1/2.

2. Covariance function As a Gaussian process the Bm-process is completely defined by its mean values

(B(t )) = 0 and covariance function Cov(B(td ,B(t2)). The latter can be computed directly from the definition. Letting 0 < tl < t: and taking into account (B(t )) = 0 we get

CoV(B(tI),B(t2 )) = (B(tI)B(t2)) = (B(tI) [B(tI) + B(t2) - B(tl)]) = (B(tl )B(tI) ) + (B(td [B(t2) - B(tl)]) = (B2(td) + (B(tl))([B(t2) - B(tdD = cr6t" Consequently, correlation of Bm coordinates at an arbitrary pair of times described by the auto-covariance function

t( , t:

is

The function can be rewritten in a form more convenient for algebra:

When time :

tl

= t: = t then the covariance becomes the variance linearly COV(B(t) ,B(t)) = (B2(t )) =

increasing with

cr6ltl .

Since the function Itll + It21-ltl - t21 is non-negative, the correlations between B(tl) and B(t2) are always positive . This results from independence of Bm increments : the more B(tl) the more (in average) B(t2) = B(tl) + [B(t2) - B(tl )], becau se the increment in square brackets does not depend on the first summand.

3. Hitting time Let us denote by Ta the first time of Brownian motion process hits the point ~ a} and conditioning on whether or not Ta ::; t :

a > 0 and compute P(Ta ::; t) by considering the event {B(t) P(B(t) ~ a) = P(B(t) ~ alTa ::; t)P(Ta ::; r) + P(B(t)

~

alTa > t)P(Ta > t).

Taking into account that

P(IB(t)1 ~ alTa ::; t) =

I,

P(B(t)

~

alTa ::; t) = 1/2 ,

we obtain for the cumulative distribution function

1

00

F7;,(t) = P(Ta ::; t) = 2P(B(t) ~ a) =

2~

cry2nt

a

exp( _x2/ 2cr6t )d.x

34

Vladimir V. Uchaikin

=

(fl""

V7r

a/cro lt

exp(

-Z2 / 2)dz

and for the probability density

a;;=;-=exp [-a2/(2(j6t)]t -3/2, a > 0, t > O.

PTa(t) =

(jOY 21t'

This is the Smirnov-Levy stable distribution density . Because of the symmetry of Bm-process this formula can be extended to the a of an arbitrary sign:

PTc,(t)=

la1-::exp[-a2 /(2(j6t)]t -3 /2,lal >0,t >0. (jOY 21t'

The integral of this density converges to I for any a i- 0, this means that the Brownian particle sooner or later hits any point x E (-00,00). The Laplace transform of the Smirnov-Levy density is of the form

pdA) =

la1-:: ["" exp[-At - a2 /(2(j6t)]t - 3/2dt = exp[-(a / (jo)J2I]

(jOY 21t'

Jo

which shows that the case a = 0 is characterized by degenerated D-distribution:

4. Brownian Sample Paths (I) The Brownian particle being at point x = a at time t will with unit probability visit both regions X a and x < a during any small time interval (t, t + h). (2) With a unit probability, the Brownian particle sooner or later hits any point

XE(-OO,oo) . (3) With a unit probability, the Bm-trajectories are continuous. Mathematically, it is expressed in terms of the Lindeberg condition

lim P(IX(t + r) -X(t)1 ~ .1) /-r = 0 for all A > O.

, ~o

Indeed, on substituting here the one-dimensional Bm density,

P(IB(t + -r) - B(t)1

~ .1 )

-r

- - dx 1"" (-2(j2-r = - - 1"" exp (- Z2) - dz V27i-r 2 2)

=

exp I V27i(jO-r 3/ 2 L1

X

I

L1 / (cro vr)

and applying the rule of L' Hospital lim P(IB(t+-r)-B(t)I r H O

~.1) =Iim_l_~ ["" H OV27i dr JL1 /(crovr)

exp

(_Z2) dz 2

35

2 Self-similarity, Stoch asticity and Fractionality

. L1 exp [-L12 / (2O

2J2ir3 / 2

'

we verify the Lindeberg condition.

2.2.2 Self-similar Brownian motion in nonstationary nonhomogeneous environment 2.2.2.1 Three types of the stochastic integral Let us consider the stochastic equation dX (t) dt

= b(X (r), t g (r),

where b(x ,t) is a non-random function of the - 0 0 < x Broeck, 1997), we take the shock model of the noise

~ (t) =

<

00.

Following (Van den

L D(t - Tj) , j

where Tj are random times generated by the Poisson process with unit intensity. At time T] , the variable X will jump from one value to another, and each time a new term appears in right side of the stochastic equation. However, the amplitude of the jump, b, depends on x. Which x-value should be taken then : the value, before the jump, after the jump, or something else ? There exist three different interpretations of stochastic integral of the equation. The first of them, the fto interpretation, corresponds to taking x-value before the jump and leads to the following deterministic equation for probability density (as before, we restrict ourselves by the one-dimensional case)

af

T

ot

a

2

I 2 ox

2

= -~ {b (x ,t)j(x,t)} .

This interpretation is more suitable for theoretical investigations (with the use, for example, of the martingale formalism) and also desirable when simulating of the process. But the price is that the normal rules of calculus become not valid and must be replaced by less familiar Ito stochastic calculus. The second approach is based on the Stratonovich interpretation, where the average ofthe value prior to and after the jump is taking. It leads to equation

af = 2:I ax a { a at b(x,t ) ax [b(x, t) f (x, t )]} . The argument in favor of this rule is that one can show that it is a correct interpretation if ~ (r) is in fact the limit of a shot noise with very small correlation time.

Vladimir V. Uchaikin

36

The third approach is tightly connected with the physical way of thinking based on the fundamental continuity equation af at

aJ ax

added by the empirical Fick low : J(x)

af

= -K(x,t) ax .

Here, J(x) denotes the current density (more strictly, the x-projection of its vector) and K (x,r) is the coefficient called diffus ivity . As a result, we have af

a {

af}

7ft = ax K(x,t) ax

.

-ix {

-ix}

An advantage of the latter equation is that the spatial operator K (x,t) is a self-adjoint in the Lagrange sense differential operator. An addition argument in favor of this approach is that namely the latter form appears as a reduced equation from the more detailed Boltzmann kinetic equation. The diffusion equation (2.3) with variable diffusivity in space and/or in time has been considered by several authors . Richardson (Richardson, 1926) considered the case K(x , t) oc Ix1 4 / 3 in order to describe turbulent diffusion. Batchelor (Batchelor, 1953) used K(x ,t) oc t 2 for the same problem. Later, Okubo (1962) and Hentschel and Procaccia (1984) solved the equation with algebraic time-dependence K(x,t) = const · t Y. Following works (Fa and Lenzi, 2003 , 2005a,b), we'll bring below recent results in this area.

2.2.2.2 Separation of variables Assuming the factorized diffusivity K(x,t)

= K(x)8(t)

and introducing new variable

r

= l8(t)dt

we reduce the three equations listed in preceeding section to the case of timeindependent diffusivity K(x) : af(x, r)

a-r

=

a2 ax2 {K(x)f(x, -r)} ,

(2.1)

37

2 Self-similarity, Stochasticity and Fractionality

(2.2) and

df(x,-r) = ~ { ()df(X ,-r)} d-r dX K x dX .

(2.3)

Evidently, the three equations coincide with each other in case of homogeneous medium K(x) = const and have the some fundamental solution

{x -4K-r 2

f(x ,t) =

I ~exp

2 rcK-r

}

,f(x,O) = o(x) .

We will consider their solutions for power-type diffusivity

K(x) =

lxi -II

(2.4)

under the same initial condition

f(x ,O) = o(x) . 2.2.2.3 Three analytical solutions Solutions of Eq. (2.1). Inserting (2.4) in Eq. (2.1) yields

dfl(X ,-r) = ~ d-r dX2

{I x I-

li f

By representing the spatial variable in the form:

x

z=-g(-r)

and setting

the equation is transformed to I

- g2(-r)

dc{>( r) d[zc{> (z)] dz

d:r

Separating t and z, we obtain two equations:

and

I

( )} x, -r .

Vladimir V. Uchaikin

38

_c [zq,(z)] = K d2 [1 zl- 8q, (z)] dz

dz2

'

'

where C is a constant of separation. Their solutions are of the form

ger) = [C(2 + 8)r] 1/ (2+8 ) and

{C1 ZI 2

q,(z) = c,lzl 8 exp - (2 + +8 8) } , respectively, where C, is one of the integration constants and another one is put to be zero . As a result, we obtain distribution

with

8 E (-oo ,-2)U(-I,oo). Its moments are

In particular, the variance is given by

Solution of Eq. (2.2). By introducing new spatial variable Y=

l

x

dx

-- =

o jK(x)

I

8 /2+ I

IxI 8 /2+'sign(x)

'

the equation

is reduced to the standard diffusion equation with unit diffusivity. As a result we get

Spatial moments related to this process are given by

(X 211 (t)h = (l /vn)(2+8)411/ (2+8) .1((4n+2+8) /(4+28))r 211 / (2+ 8),

n= 1,2 ,3 , ....

39

2 Self-similarity, Stochasticity and Fractionality

In particular, the second moment yields

Solution of Eq, (2.3). The solution of Eq. (2.3) with K(x)

= Ixl - e,

reads

The moments of the distributions are given by (

X 211 (t) ) 3

=

(2 + e)411/(2+e )r((2n + 1)/(2 + e)) r 211 / (2+ e ) n = I 2 3 r(1 /(2+e)) ' " , ...

2.2.2.4 Self-similarity of the distributions Observe that all the three solutions for a power-type diffusivity possess the selfsimilarity property:

f,,(x,t)dx= 4'1l(~)d~,

~

V2

Ixl l + e/ 2

.

= 2 + e l Vi stgn(x), 1

where ,n

(J:) _

'1'1 ~

-

2- (1 +8)/ (2+8) _ ~ 2 /2 1 J: 1(2+2e )/ (2+e )-1 r(( I+e) /( 2+e)) e ~ ,

4'2(~) = he-~2 /2, and

The variance of the distributions increases with time, the law of the increasing is determined by the function r(t) . Taking for it the power law, that is assuming

K(x,t) = KllxIYet e- l, t > 0, e > 0, so that r lutions

= KI i",

we recover the family of OHP (Okubo- Hentschel-Procaccia) so-

40

Vladimir V. Uchaikin

to the equation

it =

K[

ei":'

:x {lxIY~~} .

with mean squared displacements (X 2(t )) 0<: t 28/(2 -y). This family contains the known results of Richardson (y = 4/3, = I)

e

and of Batchelor (y= 0, e

= 3) :

Observe that these two distributions derived to describe relative diffusion of pair of particles in a turbulent media, are different in the form, although their mean squared replacements follows the power type law with the same exponent 3. Eq. (2.3) has also been solved at the presence of absorbing boundaries and the first passage time distribution has been derived for a stationary diffusivity (e = I) :

a 1- y exp{ -a 2- Y/ (2 - y)2K[t} PTc,(t) = trw _ y)/(2 _ y)) [(2 _ y)2K,t](l -y)/(2 -y) , t > O. In a homogeneous medium, when y = I, this density becomes the Levy-Smirnov density with <J6 = 2K[ .

2.2.3 Stable laws 2.2.3.1 Stable random variables Brownian motion is a special case of the self-similar Levy process with H = 1/2. Let's now consider an arbitrary self-similar Levy process {X(t), t > O} . The random coordinates of a particle performing motion of such kind at times t[ and t[ + t: are linked via the relation Provided that

x (0) = 0, the random variables X (tl) and X (t2) are increments of process in non-intersecting intervals (0, t[) and (tl' tl + t2) and, hence, are independent. The probability density function of their sum is given by convolution of densities :

2 Self-similarity, Stochasticity and Fractionality

41

Let us pass from densities to characteristic functions :

In this case the condition of self-similarity becomes

and the convolution is transformed to the product of characteristic functions

Combining two last formulas leads to the functional equation

Let S(a), a

= 1/ H, be a random variable described by characteristic function

so that As one can see from above, the random variable satisfies the relation

where sia l and S~a) are independent copies of the random variable S(a). This relation is the definition property of a-stable random variables with a characteristic index a. The term covers its characteristic function gta)(k) and the related probability density function ia )(x).

2.2.3.2 Stable characteristic functions To find an explicit formula for the stable characteristic functions, we shall introduce the "second characteristics" (Levy, 1965) If/ (a ) (k)

= Ing(a) (k),

for which the property of stability takes the form

Vladimir V. Uchaikin

42

Extending this relation on the sum of arbitrary number n of identically distributed (t, = t: = ... = til = I) terms , we shall obtain

According to the property If/ (a ) (-k)

=

[If/ (a ) (k)]

*,

it is enough to determine the function If/ (a ) (k) for positive arguments. Taking into account its continuity in a neighbourhood of the origin of coordinates and the conditions resulting from definition of characteristic functions If/ (a ) (0)

= 0,

we discover that and If/ (a ) (k)

= _k a [co-

icd .

Since the characteristic function satisfies the requirement

Ig(k)1 < I , then Re

If/ (a ) (k)

~0

and the real constant Co should be positive . On the other hand, from the condition

it follows that Computing the second derivative

and assuming k ---+ 0, one can find out that the variance is finite if and only if ex = 2. Moreover, the variance is a real number, thus should be equal to O. When ex < 2 doesn't play any role). Finally, for the variance is infinite (in this case the value ex > 2 the limit value of the derivative is zero and

c, c,

This condition can only be satisfied by o-distribution concentrated at the orig in:

2 Self-similarity, Stochasticity and Fractionality

43

g(x) = 8(x). Non-degenerate function g(x) should oscillate to have such a second moment and for this reason it can not serve as a probability density function . Therefore, the domain of admissible values of parameter a is (0,2] . The constants Co C I can be chosen in various ways. This leads to various forms of characteristic functions (Uchaikin and Zolotarev, 1999; Zolotarev V, 1986). Form A is obtained by choosing co=l,

cl={3tan(an/2),

where the second real parameter {3 E [- I , I] characterizes the degree of the distribution asymmetry (the skew parameter). Thus, the characteristic function of onedimensional stable density on a positive semi-axis is of the form g(a,{3 )(k) = exp{ -ka[i - i{3tan(an/2)]},

k > O.

Similar calculations for the negative semi-axes yield g(a,{3 )(k) = exp{ - (_k)a [i + i{3 tan( an / 2)]) ,

k < O.

Combining the two formulas, we obtain g(a,{3 )(k) = exp{ -Ikla[i

-

i{3tan(wr/2)signk]} ,

-00

00 .

(A)

This is a standard representation of the stable characteristic function in form A. There is often used another way of representation so-called form C, obtained by introducing into argument of exponential function such a scale factor to get the form g(k; a, B) = exp{ -Ikl a exp{ -i( Bwr/2)sign k}} . (C) Here a E (0,2] is the characteristic exponent of the stable law, {3 E [- I , I] and BE [- Ba, BaJ, Ba = min{I ,2 /a-l} - are skew parameters in forms A and C respectively. In what follows, we will use the notations shown in Table 2.1.

Table 2.1 Characteristics Probability density function Characteristic function Random variable

FormA g(a.m(x) gia.j3 1(k) s (a .j31

FormC g(x;a ,e) g(k;a ,e ) S(a ,e )

Observe, that characteristic exponent a is the same in both forms A and C, the skewness parameters {3 and B are linked by the formula {3 = tan(Bwr/2) /tan(an/2),

44

Vladimir V. Uchaikin

and the random variables themselves obey the interrelation S( a , B) ~ [cos( Ban / 2)] l/a S(a,{3 J.

2.2.3.3 Stable probability densities The details of computing stable pdf's by inverting Fourier transform

1

00

g(x,t) = - I

2n

-

e-lkXg(k)dk •

00

can be found in books (Uchaikin and Zolotarev , 1999; Zolotarev V, 1986; Lukacs, 1960). We restrict ourselves by listing the most important properties of stable distributions (in form C). I . All stable densities are unimodal. 2. Variances of all stable distributions except Gaussian (a = 2) are infinite. 3. Mean values of stable distributions with exponents a ::::; I do not exist. 4. The stable densities satisfy the inversion relation :

g(x;a , B) = g(-x; a , -B) . 5. The stable densities satisfy the duality relation : for

a2

I

g(x;a ,B) =x-1-ag(x-a ;a' ,B') , where a' = l /a ,B' = a(1 + B)-I. 6. At the origin, the cumulative distribution function

G(x;a ,B) =

L~g(x' ;a,B)dx' ,

pdf and its derivative have the following values:

G(O;a, B) = (I - B) /2, g(O;a , B) = n -1r(1 + l /a)cos(Bn/2) , g'(O;a,B) = (2n)-Ir(1 +2/a)sin(Bn) . 7. If the pdf g(x;a, B) is not extreme (i.e. B -I- ±Ba), both its tails fall according to inverse power laws Ixl - a - 1 ("heavy" or "long" tails):

g(±lxl;a,B) rv

r(1 +a) . sm[a(1 ±B)n/2]lxl - 1- a ,

x

8. For extreme densities, when

x ----+

00 .

2 Self-similarity, Stochasticity and Fractionality

45

e=±I ,

a
e = ±(2/a-I) ,

a > I,

one of the tails falls exponentially ("short" tail). 9. As follow s from the formula for G(O; a , e), the most part of probability is distributed over the positive semiaxis in case > 0 and over the negative semiaxis in case < O. When a ~ I G(O; a , I) = 0 and G(O; a , -I) = I, i.e. the extreme distributions become one-sided, distributed exclusively over one half of the axis. For a --+ I

e

e

G(x;a , I ) --+ G(x; I, I) = {

~: ~ ~ : '

and one- sided densit ies become degenerate:

g(x; I,±I ) = 8(x=r= I ). 10. The following stable den sities can be repre sented in term s of elementary function s: the Gaussian distribution

{-4x 2

I p g(x,2,0) = 2jnex

}

'

the Cauchy distribution I

g(x, 1,0)

= 1r(1 +x2 )

and the Levy-Smirnov distribution

g(x, I / 2, 1) =

I ~exp{- (4x) -I} .

2v 1rX3

2.2.4 Discrete time Levy motion Considering random states of Levy motion (Lm), at equally spaced times t = 0, I, 2, ... we are coming to the problem of summation of identically distributed independent standard stable variable s: /1

L/1 =

~ (a ) s: Sj

j= l

As shown above, for any stable random variable the following relation takes place

46

Vladimir V. Uchaikin

Step by step, it can be extended to an arbitrary integer number of terms. Making this and putting tl = t: = ... = til = I, we obtain the general relation of equivalence for sums LIl of the independent identically distributed stable random variables S/ Il

L Il

=

[.sja ) ~nl jaS(a) .

(2.5)

j =l

This relation is valid for any form of representation, A and C. The correspondent pdf of the random variable X (n) == LIl is

To emphasize intrinsic features of stable r.v.s we refer to Feller (Feller, 1971) who noted important and unexpected consequences of the result. Let's consider, for example, a stable distribution with ex < 1. The arithmetic mean (SI + ... + SIl) /n has the same distribution as Sln - I + 1j a . (to avoid cumbersome expressions we will omit characteristic exponent ex.) Meanwhile, the factor n - I + l j a tends to infinity as n grows. Without pursuing of rigor, one can say, that the average of n variables SI , ,SIl turns out to be cons iderably greater than almost any summand Sj , j = 1,2 , , n. This is possible only in the case where the largest term Mil

= max {SI , ... ,SIl}

(2.6)

grows extremely fast and gives a leading contribution to the sum Sn. We do not perform detailed analysis here, using only simple arguments of rather heuristic nature . The cumulative distr ibution function for the maximum term (2.6) can readily be written . Indeed, the event {Mil < x} implies {S I < x,... , SIl < x} and vice versa; therefore, the cumulative probability function

FM

I1(X)

= P{Mn < x } = P{S, < x, ... .S« < x }.

Since S, are independent, the right-hand side of this expression can be transformed to the product of probabilities of individual events {S j < x} . In view of coincidence of the distributions ofthe distributions of S] , it is the mere nth power of this probability: (2.7) If the maximum term gives a primary contribution to the sum, the following relation should be satisfied (if f3 -I- - I) (2.8)

where

F(x) == I - F(x). By virtue of (2.5),

2 Self-similarity, Stochasticity and Fractionality

FLn(x)

47

= P {.[Il ?: x} = P {S ?: n-1 /ax} = Fs (n -l /ax).

(2.9)

Subst ituting (2.9) in the left-hand side of (2.8), and (2.7) in the right-hand one, we obtain Fs(n -1 /ax) rv I-Fs'(x) . Changing the variable n-

l

/ ax

= y, we arrive at the equation (2.10)

solution of which, provided that

f3 i- -

I, is (2.11 )

Similarly, we obtain (2.12) Differentiating (2.11) and (2.12) with respect to x yields the following asymptotic expressions for the density: Ps ( x )

= - P,-s' ( X )

rv

acx -

a- l

x

,

x

---+

---+

00 ,

-00 .

It is clear that this does not correspond to the normal law (a = 2), and that there are no a > 2: the normal law is the only stable law with finite variance. Thus, the characteristic exponent a < 2 takes values from the interval (0, 2]; for a = 2 we have the Gaussian law; with a < 2 we have stable laws whose probabilities of large deviations are power functions ("long " tails). In extreme cases, only one of the tails may be long. The results discussed above admit a very important generalization known as Generalized limit theorem: Let Xl, ... ,XIl be independent identically distributed random variables with the dist ribution fun ction Fx (x) obeying the conditions

I - Fx(x)

rv

cx- /l,

x

---+

00 ,

with /l > O. Then there exist such sequen ces all and b« > 0 that the distribution of the centered and normalized sums 11

2 11 = ([Xi- a11 )/ bll i= l

weakly converges to the reduced stable distribution with parameters

48

Vladimir V. Uchaikin

a = {J1' J1 ::; 2, 2, J1 > 2, and

c-d

f3 = c-s d' as n

--+

00 :

FzlI(x) ~

d a,{3 ) (x).

The coefficients all and b; can be taken in the form given in Table 2.2:

Table 2.2 Centering and normali zation coefficients all and b; (Form A)

p O

2

(X

all

P p P 2 2

0 f3 (c +d)nlnn n(X ) n(X ) n(X )

b; [n (c+ d )]l/a [2r( (X) sinemr/2 )tI /a nl /a (n / 2)(c + d) n [n (c+ d )]l/a [2r( (X) sine(Xn/2) ]-I /a nl /a (c + d )I/2[n lnn] 1/2 [(1/2)0}] 1/ 2 nl/2

We will not prove this theorem but cite some heuristic reasoning following the work (Bouchaud and Georges, 1990) for the case of non-negative summands with the power asymptotic

leading to one-sided stable distributions. Accordingly to (2.7), the probability density function of a maximum term Mil in the sum 1:1l has the form : dF:Mil () X

PMII(X) =

dx

= n [ 1-

1 ' 00

x

,

,

px(x)dx

]"-1 px(x) .

(2.13)

We denote the most probable value Mil by x ll: (2.14) Differentiating (2.13) with respect to x and substituting the result in (2.14), we obtain

[1-1 00

XII

As x

--+

pX(X')dx'] dpx(x ll ) +(n-I)pi(xll ) =0. dxll

(2.15)

00

[" tix (x')dx' px(x)

'V

'V

c[a ,

cax- a - I ,

dpx(x) /dx'V -ca(a+ l)x - a- 2.

(2.16)

49

2 Self-similarity, Stochasticity and Fractionality

and if n

>> 1 Eq . (2.15) yields

the asymptotic relation

a+I

rv

ncax;;a ,

whence

c=

(~)'/a a+1

Estimating a characteristic value of the sum L il Il

LIl=[Xi i=1

with the help of the "truncated average"

one obtains

1

rv

(LIl)x = n lI

Let A > 0 be such that as x sion (2.16) . Then

1

cll l / a

X II

Lil

xpx(x)dx =

xpx(x)dx.

> A the density px(x) follows its asymptotic expres-

where

1

CIl I / a

(L:')x = acn lI

[adx.

From here it can be seen, that as n ---+ 00

a < 1; a = 1, where Co and c, are positive constants. Estimating similarly a truncated variance of the sum L Il , with

(2.17)

a > 1 we obtain

a < 2, a=2.

(2.18)

Conclusions (2.17) and (2.18) show qualitative agreement with the generalized limit theorem. Those readers who are interested in applications of stable laws rather than in their mathematical aspects has a possibility now to pass readily to the second part of our book (Uchaikin and Zolotarev, 1999).

Vladimir V. Uch aikin

50

2.2.5 Continuous time Levy motion Definition of Continuous time Levy motion (CTLM) A random process {X (r), t :2: O} is called a (standard) CTLM with 0 -I ::; f3 ::; I , if I) X (0) =0 almost certainly; 2) {X (r) , t :2: O} is a process with independent increments; 3) X (r + r) -X (t) ~ r l / aS (a,f3 ) at any t and r.

< a ::; 2,

For the sake a/brevity we shall call it L(a,f3Lprocess, then the Brownian motion will be designated as 2-stable Levy motion. As one can directly see from this definition the CTLM pdf is given by the formula

In order to make the difference between sample paths of Levy motion and Brownian motion more clear, let us consider the behavior of funct ion

Q(r ,L1) == P(IX(t+ r) -X(t )l :2: L1 )/r as r

---+

O. For Bm-process

Qw ( r ,L1 ) -_

1

00

I

r:;;. 3/ 2

v tct

LI

2/ 4,

e- x

1_1

00

dx -_ _r:;;. e-z2/ 4 dz. v tct LI / /i

Applying L'Hospital's rule, we get the expression lim Qw(r ,L1)

, ->0

= 0,

reflecting the property of continuity of Bm's sample paths . In the case of Lm with

a <2 It is known, that

x therefore lim Qw(r,L1)

, ->0

= const L1 - a > 0 ,

---+

00,

r

---+

O.

When a < 2, sample paths ofU a ,f3 )m process are not continuous anymore and have jump-like form; this is a main their difference. The jump amplitudes are independent random variables depending on a : the less a, the longer jumps. The width of a diffusion packet grows with time proportionally t l / a . When a < 2 the speed of widening of Lm packet is greater then Bm and is not of the Gaussian form . Its variance is infinite now, and we should use some other measure of the packet width .

2 Self-similarity, Stochasticity and Fractionality

51

Bm is not the only non-degenerate Lm process possessing continuous sample paths. Besides, two degenerate Lm processes with continuous paths exist, representing ballistic non-random motion with a con stant velocity to both directions of the real axis ({3 = ± 1).

2.2.6 Fractional equations for continuous time Levy motion 2.2.6.1 Equations with superposition of Riemann-Liouville operators As we saw above, the characteristic function of CTLM in A-representation has the form

p<,a ,{3 )(k,t)

= g1-a,{3 )(kt l / a ) = exp{t[(a,{3 )(k)} ,

where

[ (a,{3 )(k) =

{-lk1a[i -Ikl,

i{3tan(exJr/2)sign k],

ex -1= 1,1{31 ::; I; , , ex=I{3=O

can be considered as the Fourier image of some linear operator L(a,{3) . We shall call it the Levy operator. Differentiating these expressions with respect to time yields the evolution equations

with the initial condition For

ex = 2 ,{3 = 0,

and we recognize here the image of ordinary diffusion equation

ap (2,0) (x, r) at

with one-d imensional Laplacian .dl ==

= .dIP

(2 0)

,

(x,t)

D; and initial condition

p (2,0) (x,O)

= 8(x).

In the general case, the Levy operator can be expressed through a linear superposition of left-sided and right-sided Riemann-Liouville differential operators. Indeed , writing its Fourier image in the form

-Ikla[i -

i{3tan(exJr/2)sign k] = - [A( -ik)a +B(ik)a] ,

52

Vladimir V. Uchaikin

after elementary algebra we obta in:

A( -ik)a + B(ik) a

Ikl a [Ae - i (an/2)Si gn k+ Bei (an/2)sign k]

=

(a27r) - i(A -B) sin (a27r) sign k] A-B (a7r) . z] . a7r )[l-iA+Btan = Ikla(A+B)cos (2 2 sign Ikl a [(A+B)cos

=

Equating correspondent coefficients from both sides, we arrive at the system of equations I A + B - ---,----,--...,... - cos(a7r/2) '

(13 - I)A + (13 + I)B = 0, solution of which is A

=

I + 13 B 2cos(a7r/2)'

= __1 .,.----'-13--,---,2cos(a7r/2)

The power functions (=fik)a are Fourier images of Riemann-Liuville operators _oo D~ and xD~ respectively, therefore

and the original equation takes form ap (a,f3 )(x t) at '

= - [A _oo D~ + B x D~] p(a,f3 )(x,t), t > O.

2.2.6.2 Equations with the inverse Feller operator Let us come back to the evolution equation for the characteristic function and represent it in the form 1 + if3tan(a7r/2)sign k ap(a ,f3 )(k,t) Ikla[I + f32 tan2(a7r /2)] at Assuming and writing § p for

= _;:;(a,f3 )(k t) p

t

> O.

, ,

8 2 = [I + f32tan2(a7r/2)]cos(a7r/2)

p (see (A.14) in attached Appendix of this chapter) we get

cos(a7r/2) + if3 sin(a 7r/ 2)sign k e (a p (a,f3 )(x,t)) __ e (a,f3) ( ) Ikla82 :# at :# P x,t .

53

2 Self-similarity, Stochasticity and Fractionality

Comparing the left hand side of this equality with the Fourier transform of Feller 's potential (A.17) and inverting the transform we arrive at the equation Ma ap (a,{3 )(x,t)

at

il , \'

t >0

= _p(a,{3 )(x t)

' ,

or

t>0

ap (a,{3 )(x,t) = _(Ma ,)- lp(a,{3) (x t)

at

ll ,l

,

(2.19)

,

with 1+f3 u = 28 2 and 1-f3 v = 28 2 • According to (A. 1 I), evolution Eq. (2.19) can be written in the following explicit forms:

a (2.20)

and

ap (a,{3 )(x,t) at -

a Cr(I-a)

x l "' [2 PA(x ,t ;a,f3) - (1

+ f3)p(a,{3 )(x- ~ ,t) - (1- f3)p(a,{3)(x+~,t)g -l -ad~ (2.21)

where C = [I + f32tan(an /2)rl . In the case of a symmetrical process (f3 = 0) the operator in the right side of (2.21) coincides with the Riesz derivative (A.9)

ap (a,O)at (X,t ) = _D a p(a,o)(x,t) . When f3

= I we have the one-sided stable process with the evolution equation ap (a,I)(X t) at '

= - [cos(an/2)rlD~p (a,I )(x,t) ,

where D~p is the fractional Marchaud derivative (A.6) .

Vladimir V. Uchaikin

54

2.2.6.3 Equation with compositions of fractional operators Observe that the last equation of the preceding section is not applicable to the case with a = I because cos( rr/ 2) = O. This inconvenience can be avoided by using form C. The correspondent characteristic function ji(x,t; a , e) obeys the equation

aji(k,;~ a , e) = -Ikla exp{ -

iae(rr/2)sign k}ji(k,t ; a , e)

(2.22)

with the initial condition ji(k,O;a,e)

=

I.

To pass from Eq. (2.22) for characteristic function to the corresponding equation for the density p(x ,t; a , e) we rewrite (2.22) in the form :

and use (AI6) and (AI9). As a result we obtain l

a(l - f) )ap (x,t ; a , e ) __ Oaf) ( . e) at + P x.t:a,

or ap(x,t;a,e) __ Oa (I -f))O af) ( . e) + P x,t,a , . at -

In the symmetrical case (e

(2.23)

= 0)

ap(x,t ;a ,O) __ Oa ( 0) at p x,t ,a , .

In the extremely asymmetrical case (a < I, e = 1), X(t;a , I) > 0 Eq. (2.23) takes the form ap(x,t;a , 1) __ Oa ( . ) at Orpx,t ,a ,l, . where oO~ is given by (AS). It describes the one-sided Levy motion in the positive direction of x-axis .

2.3 Fractional Brownian motion In this section , Brownian motion is interpreted as a random function obeying some stochastic differential equation with white random noise in right hand side. Replacing integer order of the derivative by fractional order opens another way to generalization of Brownian and Levy motions . We obtain fractional Brownian and Levy motions . Their main property is the memory.

55

2 Self-similarity, Stochasticity and Fractionality

2.3.1 Differential Brownian motion process As shown above, correlations of Bm coordinates at an arbitrary pair of times tl .t: are described by the covariance function

Consider the differential Bm process(dBm), i.e. the process of Bm increments

dB(t)

= B(t + dr) -

B(t) , dt = const.

Evidently,

{dB(t)} ~ B(dt), and therefore,

(dB(t )) = 0,

O}B = er6dt .

Autocorrelations in dB(t) are described by the covariance function Cov( dB(tl) ,dB(t2)) which can easily be calculated from correspondent expression for Bm:

Cov(dB(t,) ,dB(t2))

= (dB(t l )dB(t2)) =

a2(B(tl a )B(t2)) a dtldt2 t]

2

d 2 a l (t ] - t2 )d d 25:( = -er6a2(ltll+lt21-lt]-t21)d :l :l tl t: = era :l tl t: = era u tl 2

otlot2

at,

ti

)d d t]

ti -

The differential Bm process dB(t) is an example of stochastic differentials dX (z ). Many authors prefer to write dB(t)=~(t)dt

or even

dB(t) = ~ (r) dt and call equations of such kind stochastic equations, and the "functions" ~ (r) random noises. In this special case, when B(t) represents Brownian motion, the noise ~ (z ) is called the white noise. We shall use for it the notation

As follows from above, the dBm process possesses the following properties. I) Its mean value is zero : 2) It is delta-correlated:

3) The white noise is a stationary stochastic process. 4) Its stochastic integral

56

Vladimir V. Uchaikin

B(t) =

t Jo

dB(t') = lim

maxL1t ~O

[,L1B(ti) = 11

t Jo

B(I )(t')dt'

is a Gaussi an random variable :

l

B(I )(t' )dt'

~ G(O, cr6t ).

The latter property can be generalized to integration of any arbitrary integrable function, namely : the integral

is a Gaussian random variable with the mean

and the variance

b crl = ([l f(t )B(I )(t)dtf) b

= l dt'lb dt2f(td f (tz) \ B(I)(tl )B(I)(tZ)) = cr61bfZ (t)dt .

2.3.2 Integral Brownian motion process A stochastic process

B(-I )(t ) = oltB(t)

=l

B(t' )dt'

is called the integral Bm (iBm) . The iBm proce ss is also a Gaussian proce ss. One can easily verify it by repre senting the integral as a limit of approximation sums

and taking into account that any set of linear superpositions of independent norm ally distributed random variables L1B(tk) = B(tk) - B(tk- l), k = 1,2 ,3 , ... , is jointly normal. At the limit, we have

2 Self-similarity, Stochasticity and Fractionality

57

Since {B (- I) (r), t ?: O} is Gaussian process, it follows that its distribution is completely determined by its mean value and covariance function . They are easily computed and have the form :

(B(-l) (t )) = . : B(t')dt') =

l

(B(t' ))dt' = 0;

Note, that the process {B (- I) (r), t ?: O} is not a Markov process , however, the vector process { {B(- I) (r), B(t) },t ?: O} is again a Markov process . It is a jointly Gaussian with zero mean and covariance

The concepts of stochastic integrals and differentials are generalized to operating with arbitrary (in some sense) random functionsX(t) , Y(t) , Z(t) , W(t) :

1=

lb

Y(t)dX(t),

dW(t) = X(t)dt

+ Y(t)dZ(t).

The simplest (after the Bm) example of such equation is

~;t) = -pX(t) + ~(t) . Interpreting X as the velocity of a Brownian particle we can recognize in - pX the Stocks viscous force . The solution of the equation under condition X(O) = XQ

is the Gaussian process with the mean

and variance

58

VarX(t) =

Vladimir V. Uchaikin

([l e-,u(t-t')~(t')dt'r) a6 l =

e- 2,u(t - t' )dt' =

At each t , X (t) has the normal distribution . In the limit t an equilibrium distribution -00

----+

<x <

00

(I _e- 2,ut)

;~ .

it take on the sense of

00 .

Three-dimensional analog of the distr ibution is known in statistical physics as the Maxwell distribution. If we define the process by integral

X(t) =

j -

t

,

00

e-,u(t - t ) ~ (t' ) dt' ,

-00

00,

that is remove the initial moment to - 0 0 , we obtain the Ornstein-Uhlenbeck process. Note that for any fixed tl < t

X(t) _e-,u(t -t IlX(tl)

=

i

t

e-,u(t -t

,

) ~ (t' ) dt'

t1

is independent of X(t;), t( < tl . This implies that the Ornstein-Uhlenbeck proce ss is , in fact, a Markov process, moreover it is the only stationery Gaussian process possessing the Markovian property. The covariance function of the process is easily computed:

2.3.3 Fractional Brownian motion Three kinds of stochastic processes considered above and written in terms of derivatives,

B(-l )(t) = OD;IB(t) , B(t) = oD?B(t) ,

59

2 Self-similarity, Stoch asticity and Fractionality

provoke to introduc e a gener al kind of fractional Brownian motion (fBm)

oB(v)(t) = oD~B(t ) = ~DtB(t) =

( I ) T I-v

j't (t - t' )-VdB(t' ). °

It is easy to see, that fBm is a self-similar Gaussian process with the Hurst exponent

H=I /2-v , and the exponent is usually used in notation of fBm : I

j't

(r) == B(I /2- H)(r) = 0 v B(t) = (t - t, )H °BH O t 1(H +I /2) °

- I /2dB(t' ).

Though the process is self-similar, its increments are stationary only when H when it becomes the ordinary Bm:

OB 1/ 2(t ) =

l

dB(t')

= 1/2

= B(t ).

Mandelbrot and van Ness (Mandelbrot and Van Ness J, 1968) gave the now widely accepted version of fBm using a modified fractional integral of Weyl type, I

BH (t ) = 1(H + 1/2)

.{LOoo [(t -

t')H-I /2 - (_t')H -l /2] dB(t') +

l

(t - t')H-l /2dB(t') } ,

where for negative t the notation J~ should be interpreted as form of the expression is

J;o. Another, shorter

By direct computation, one can find the following representation for the autocovariance function :

60

Vladimir V. Uchaikin

where 2 (JH

=

/ 2 ( )) \ BH I

=

1(1-2H)cos(Hn) 2 (Jo' Hn

By definition, the Hurst exponent is a self-similarity index which should be positive. From the other side, if H < I , the fBm is the only self-s imilar Gaussian pro cess with stationary increments (Samorodnitzky and Taqqu , 1994) . For these reasons, the Hurst exponent values are bounded by the region < H ::; I and the fBm is defined as follow s: A Gaussian H -ss process {X (t) } with (X (t) ) = 0, < H ::; I and stationary increments is called fra ctional Brownian motion(fBm), and standard fB m if, in addition,

°

°

0'0

= 1.

When H = 1/2, fBm becomes the ordinary Bm : {B 1/ 2 (t)} = {B (t)} . The case 1/2 < H < I relates to persistent or fra ctional super-diffusion (enhanced diffusion) , the process with H < 1/2 describes antipersistent or fra ctional subdiffusion. Note that all these processes are characterized by Gaus sian one-dimen sional distribution :

{x

2

I ex p -4O'2 --- } p(x , t ) -- 2fim H t2H

'

2.3.4 Fractional Gaussian noises The sequence of stationary increments offBm Yj = BH(j + I) - BH (j) , ... , -I , 0, I , ... forms the fra ctional Gaussian noise (fGn), or the standard fGn if, in addition, (Y]) = I . Direct calculations yield the following integral repre sentation of the fGn : (Jo

Yj = 1(H + 1/2)

j j+l [ . -00

( )

H-l / 2

+ 1- t) +

.

H-l / 2]

- () - t) +

dB(t ).

From the foregoing, some remarkable properties of the fGn process follows. I. The fBm is a stationary Gau ssian sequence with mean zero and variance

(Y]) = (B~ (1 )) = (J6· 2. The covariance function of fGn R(j)

= (YoYj) is

If H < 1/2 the Yj are negatively correlated; in case of H > 1/2 they are positively correlated . When H = 1/2 we have a seq uence of independent random variables. 3. The fGn spectral den sity 5(1), -n < f < n connected to R(j) via relation s

61

2 Self-similarity, Stochasticity and Fractionality

is of the form (Samorodnitzky and Taqqu, 1994)

S(f) = 0"6 C2(H)leij

11 2

-

00

L

-n ~ f ~ n,

If + 2njl -2H-I ,

j=- oo where

2

H(I-2H)

C (H)

= 2r(2-2H)cos(Hn)

is a normalizing constant. Notice that C2 (1/ 2)

= (2n)-I .

It is not hard to conclude that both functions R(j) as j --+ 00 and S(f) as If I --+ 0 behave like power functions (Samorodnitzky and Taqqu , 1994) :

R(j) '" 0"6 H(2H _I)/H-I ,

j

--+

00,

and

S(f) '" 0"6C2(H)lfll -2H,

f

--+

O.

When H = 1/2 we deal with a white noise , the case H = I reveal us the 1/ f-flicker noise . Notice, that R(j) goes to 0 for all values of HE (0 , I), but when H > 1/2 it goes so slowly that the sum [ }=- oo R(j) diverges. Such a behavior of Yj is interpreted as long-range dependence . The case H i- 1/2 provides a counterexample to the central limit theorem. Although the variance of Yj is finite , a non-degenerated limit distribution of

Z/1 =

I

/1

r.;; LYj, n --+

00,

yn j=l

does not exist. To get a non-trivial limit one must take normalized factors n:" instead of n - 1/ 2, because Yj are dependent. In terms of hereditarity concept, one can say that the H parameter regulates the presence or absence the memory: long-memory for 1/2 < H < I, no memory at H = 1/2, and short memory in the case 0 < H < 1/2.

2.3.5 Barnes and Allan model Barnes and Allan (Barnes and Allan, 1966) have developed another model able to characterize the 1/ f noise (see also (Magre and Guglielmi, 1997» . They consider the filtering of a white Gaussian noise by the system described by its impulse response H -I /2

h( ) _

t+ t - =['-,-(H'--+-I-:-/2-'-)

The output is

62

Vladimir V. Uchaikin

This model possesses the following properties. I . Self-similarity For all a > 0 and for all t, we observe

Z(at) ~ aHZ(t). 2. Non-stationarity The autocorrelation function is

2.3.6 Fractional Levy motion Further generalization of the way of inserting hereditarity into self-similar processes is based on using stochastic integrals with respect to the random measure

that describes the random increment of the Levy motion process in (t,t + dz) and

X(t + r) - X(t)

=

lH

dL(a)( r')

~ r ll a S(a,!3 ).

Here, the hereditarity is introduced using the function h(t , r), which determines the contribution of a unite measure at time r to the state of the process at time t :

X(t)

=

i:

h(t ,r)dL(a)(r) .

If the function h (z ,r) is invariant with respect to shift in time,

h(t, r) = h(t - r), such a process is referred to as a moving-average process(MA process). The Ornstein-Uhlenbeck-Levy process can serve as an example of MA process:

63

2 Self-similarity, Stochasticity and Fractionality

Constructed on the same principle, process

with 0 < H < I and H -I- 1/ a, is called the fra ctional Levy motion(tLm), since it is obtained from a Levy motion process by fractional-order integration. Note two important properties of the process {Xfj (z )}. First, it is self-similar with the exponent H, i.e., for any a > 0 and tl , ... .t;

Second, its increments are stationary,

Xfj (r) - Xfj (0) ~Xfj(t + r) - Xfj( r) . In the particular case a = 2, H = 1/2, tLm turns to an ordinary Brownian motion , in the case a = 2, H -I- 1/2 we deal with fra ctional Brownian motion. Its mean value is zero, the variance is

and the covariation function is

The case of H

= 1/2 and

C1 /2( 2

tl, t:

) = { (J2mintr} , t2) , if t] and t: are of the same sign , 0, if t] and t: are of opposite signs.

corre sponds to the ordinary Brownian motion. Since the Bm has stationary increments, the sequence

{z, = xf (j + I) - xf (j) ,

j

= ...,- I ,0, I , ... }

is stationary and is called the fractional Gaussian noise . Its auto-covariance function is

Vladimir V. Uchaikin

64

2.4 Fractional Poisson motion The self-similarity condition being applied to renewal processes leads to fractional generalization of the Poisson process . The link between fractional character of the differential equation and fractal kind of random point distribution is discussed.

2.4.1 Renewal processes The above scheme of the anomalous diffusion process is based on the self-similar generalization of Brownian motion . Historically, it was developed in a different way, using asymptotic analysis of jump processes. The groundwork for this approach was laid by Montroll and Weiss (Montroll and Schlesinger, 1984), and none of the review articles on anomalous diffusion has avoided making a reference to their study (see also an excellent review by Montroll and Shlesinger(Samko et al., 1993)). We note here the main milestones on this avenue using the terminology of the renewal theory (Repin and Saichev, 2000) . Being less formal, this way is more ocular and more productive for physical interpretations in different problems. Let T called the waiting time or interarrival time, be a positive random variable with pdf q(t) and TI, T2 ,' " be a sequence of its independent copies . The new sequence n

T(n)

=

L Tj,

T(O) = 0,

j =1

will be referred to as the renewal timesor arrival times. In physical processes, some transitions from one state of a system to another, collisions of particles, emission or absorbtion of photons, etc, take such a little time that can be considered as instant transitions. The registered transitions of this kind generate in a measuring electric device a correspondent sequence of current pulses of a very short duration. In many cases, they can be considered as zero-duration pulses. We will call these zero-duration phenomena events or jumps. Let N(t) denote a random number of the events in the interval (O ,t]. In this case, the difference N(t2) - N(tl) means the number of events in the interval (tl' t2]' The random process {N(t) ,t :2: O} is said to be a counting process if it satisfies: (a) N(t) :2: 0; (b) N(t) is integer valued ; (c) N(tl) ~ N(t2) if tl < t2. The function N(t) jump-like increasing at each arrival time is called a counting function . Thus, TN(t ) denotes the arrival time of the last event before t while TN(t )+1 is the first arrival time after t. In these terms, N(t) can be determined as a largest value of n for which the nth event occurs before or at time t :

N(t)

= max{n: T"

~

t} .

2 Self-similarity, Stochasticity and Fractionality

65

In other words, the number of events by time t is greater than or equal to n if and only if the nth event occurs before or at time T: N(t) 2n~T,, ~t .

Feller noted that considering renewal processes we deal merely with sums of independent identically distributed random variables, and the only reason for introducing a special term is using such a power analytic tool as the renewal equation. Let us call the mean number of events by time t (N (t) ) the renewalfunction. It can be represented in the form

(N (t )) =

L P(T(n) < t) = 11 >0

l°

q*" (t') dt' ,

q*o(t) = o(t).

The renewal function is a non-decreasing, finite-valued, non-negative and semi additive function :

(N (t + s)) ~ (N(t )) + (N(s)),

t,s 2 0,

It obeys the renewalequation

(N (t )) =

l

[I + (N (t - t' ))]q(t' )dt'.

Its interpretation is very clear: the mean number of events within (O ,t) is equal to the contribution of the first event plus the mean number of subsequent events . For the mean frequency of the events

we obtain from here the similar equation:

f(t)

= q(t) +

l

f(t - t')q(t')dt '.

(2.24)

2.4.2 Self-similar renewal processes Let us try to answer the following question: what form should have transition pdf

q(t) for the process N(t) to be v-ss in medium? In other words, we want to find such q(t ) == 1JIy (r) that (2.25) Following B. Mandelbrot (Jumarie, 200 I), we will call such a set of random points on t-axis the fractal dust and the pdf 1JIy(x) the fractal dust generator (fdg) . As

66

Vladimir V. Uch aikin

follows from (2.24), it is linked with the mean fractal dust density Iv (x) via equation

lJfv (t) = Iv (t) -

l

Iv (t - t') lJfv (t')dt ' .

(2.26)

Applying the Laplace transform

yields the expression

~ ( /\.' ) -_ lJfv

lv(?) ~ 1+ Iv(?)

J1 J1+?v '

(2.27)

which for v = I coincides with the corresponding expression for the ordinary Poisson process :

{2'lJf, (t)}(?) =

---.1!:..." J1+/\.

lJf,(t) =J1e- tu .

Wang and wen of (2003) used formula (2.27) for introducing fractional Poisson processes and derived the fractional integral and differential equations for this density

and oD~lJfv(t)

+ J1lJfv(t) = 8(t) .

2.4.3 Three forms offractal dust generator The solution of the above equations was represented in three forms, two of them were obtained in (Wang and Wen, 2003 ;Wang et al., 2006) . First of them is obtained by performing the backward transition

with the use of the geometrical progression formula

and the relation

67

2 Self-similarity, Stochasticity and Fractionality

Th is leads to the first representation of IfIv (r) in terms of two-parameter MittagLeffler function :

(2.28)

In particular,

1fI1 /2(t) =

~-

2IErfc(p

p2 e /1

vt),

yTCt

where Erfc(t) is the complementary error function : Erfc(t) making use of the formula

=

};r j/'" e-

z2

dz. By

one can verify , that the density (2.28) has really the Laplace transform (2.27). The second form is

IfIv(t) = -I t

j 'OO e-xv(pt /x)dx,

(2.29)

0

where

sin(vn)

v(~) = n [~v+~ -v+2cos(vn)]' It allows with easy to find asymptotical expressions for small and large time :

lfI(t) "-'

u" v -I y-t ,

t

--+

0,

(v)_v

vp - v- I { ql_v)t ,

t--+ oo •

The third form is given by the next Lemma proved in (Laskin, 2003):

Lemma . The complement cumulative distribution function

P(T > t) =

J OO IfIv(t')dt'

can be represented in the form (2.30)

68

Vladimir V. Uchaikin

2.4.4 nth arrival time distribution For the standard Poisson process, the pdf of the nth arrival time is given by Gamma distribution

I/I*"(t) = J1

( II t )" - 1

r-

(n - I)!

T (I1 )

= T, +...+ T"

(2.31)

«!" .

According to the central limit theorem

As numerical calculations show, p (l1 ) (r) practically reaches its limit form already by n = 10. In case of the fPp,

ET = 10'' ' I/Iv (t)tdt = 00 and the central limit theorem is not applicable. Applying the generalized limit theorem (Uchaikin and Zolotarev, 1999), we obtain :

2.4.5 Fractional Poisson distribution Now we consider another random variable: the number of events (pulses) N(t) arriving during the period t . According to the theory of renewal processes

PIl(t)=P(N(t)=n)=P

(

11 LTj >t ) -P

J=I

("+1LTj >t) ,

n =0, 1,2, .. .

J=I

and the following system of integral equations for PIl (r) takes place :

After the Laplace transform with respect to time, we obtain from here

The inversion yields :

69

2 Self-simil arity, Stoch asticity and Fractionality

oD~ PIl(t)

t- V

= J1 [PIl-l (z) - PIl(t)] + 1(1 _ v) 0 0, 0 < V :::; I. 11

(2.32)

This is a master equation system for the fractional Poisson processes. When v it becomes the well known system for the standard Poisson process :

dpll(t) = J1 [PIl-l (r) - PIl(t)] + O(t)OIlO' dt

--+

I

(2.33)

System (2.32) produces for the generating function 00

g(u,t) ==

L U"p"(t)

(2.34)

11=0

the following equation:

oD~ g(u,t) When v

--+

t- V

= J1(u - 1)g(u,t) + 1(1 _ v)

(2.35)

I it becomes the well known equation for the standard Poisson process :

dg(u,t) = J1(u - I )g(u,t) + O(t). dt

(2.36)

Comparing (2.32) with (2.33) and (2.35) with (2.36), one can observe that the equa tions for standard processes are generalized to the equations for correspondent fractional processes by means of replacement of the operator d/ dt with oD~ and of right side the term O(t) with t- V / 1( 1- v) . The solution to Eq. (2.35) is of the form

Applying the binomial formula to each term of the sum and interchanging the summations , one can rewrite it as the series

g(u,t) = I~U 00

11

[all (m+n)!(-a)lIl] n! n{;o m!1(v(mk+ n) + I) . 00

(2.37)

Comparing (2.37) with (2.34) yields a"

00

PIl(t) = n! n{;o

(m+n)! (-a)1Il v m! 1((m+n)v+ 1) ' a = J1t , 0< v :::; I.

This distribution becoming the Poisson one when v = I can be considered as its fractional generalization, called fractional Poisson distribution. The correspondent mean value and variance are given by

70

Vladimir V. Uchaikin

iu"

(N(t )) = r(v+ I) and

(J"2(t ) = (N(t )){ l + (N(t ))[2 1-2VvB(v , 1/ 2) - In .

Table 2.3 shows the properties of Fpp compared with those of the Poisson process .

Tab le 2.3 Properties of FPP compared with those of the Poisson process Poisson process (v

IJI(I )

u cr!"

P(n,l )

(J1t)" e -J1t n!

(N (I ))

pI

2

pI

(}N(t )

=

Fractiona l Poisson process (v

I)

< I)

plV-1 e; v(_ p IV) 00 (H Il) ! (_ J1 t v) k (J1t V ) " - Il! - L k=O - k! - n V(H Il)+ I )

J1 t V nv+ l ) V

J1 t r( v+1)

{

J1 t

V

[VII(V. I/ 2)

1+ r(V+I ) ~ - I

]}

2.5 Fractional compound Poisson process The last section joins both kinds of models and leads to bi-fractional (with respect to space and time variables) different ial equations. Their fundamental solutions form a new exten sion of Levy stable distributio ns - the class of fraction ally stable distributions.

2.5.1 Compound Poisson process The Poisson process admits a very simple but productive genera lization called compound Poisson process . The idea of this generalization is based on replacing unit jumps at random arrival times by jumps of random length X U), j = 1,2 ,3, ... at the same times. The random variable s are independent of each other and of arrival times. Consequently, instead of random function N(l )

N(t ) =

LI

j =l

for the Poisson process we have

2 Self-similarity, Stochasticity and Fractionality

71

N(t)

X(t) =

E x U)

j =l

for the compound Poisson process . Let N(t) be the Poisson process with the rate f1 and p(x) , - 0 0 < x < 00, denotes the probability density function for X U), then pdf f(x ,t) for X (r) is represented in the form :

f(x ,t) = e- I·U

00 (f1t)j . E - .,-p*J(x) , t > O. j =O

J.

This density obeys the integro-differential equation

d! = _ f1f(x ,t) + f1 ) '00 p(x - x')f(x' ,t )dx' = 0, ot

(2.38)

-00

with the initial condition

f(x ,O+) = 8(x)

(2.39)

or, equivalently, the equation

it =

-f1f(x,t) + f1

i:

p(x-x')f(x' ,t)dx' = 8(x)8(t)

(2.40)

with the condition

f(x ,O) = 8(x)8(t) .

(2.41)

The solution can be obtained by passing to characteristic functions :

d!~~,t)

= -f1[i - p(k)]!(k,t) = 8(t) ,

!(k,t) = exp{ -f1t [l - p(k)]}, f(x,t) = - 1 2n

Joo exp{ -ikx -

f1t [1 - p(k)}dk .

-00

2.5.2 Levy-Poisson motion Let us call the Levy-Poisson motion such a compound Poisson process which has a stable distribution of random jumps, that is

p(x) = g(x;a, B), It is readily seen than

and consequently,

ji(k) = exp {-Ikl a exp{ -i( Ban/2)sign k}}.

Vladimir V. Uchaikin

72

Therefore, the strict solution of Eq. (2.1) is represented by the series

Substituting here

a = I,

e = I and taking into account that g(x;

I, I) = 8(x-I) ,

we obtain

f(x,t; I , I) =

f:

f:

(j1.?j e- fl t r I8 (r ' x - l ) = (j1~)j e- fl t 8(x - j) . j=O J . j=O J .

After integrating this expression over small interval (n- £,n+£) , n = 0 , 1,2 , ... , 0 < e < I, we arrive at the ordinary Poisson distribution

From the other side, at the asymptotic of large

(j1t)j fl t L _.,_e... "-' 00

j=O J .

1

ut

00

dj8(j - j1t)· · ·

0

and we obtain the Levy motion :

f(x ,t ;a ,e) "-'rl"(x,t ;a , e) = p(x,t)

= (j1t)-I /ag((j1t)-I /ax;a , e) .

As shown above, this pdf obeys the equation

op(x,t ;a ,e) Ot

= L( a ,e )p (x.t: a ,e ), t > 0,

with the initial condition

p(x,O+) = 8(x), or, equivalently, the equation

op(x,t Ot;a,e)

= L( a , e) p (x .t; a , e) + u5:(x )5:() u t , t ~ 0,

with the initial condition

p(x,O-) = o.

(2.42)

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2 Self-similarity, Stochasticity and Fractionality

2.5.3 Fractional compound Poisson motion Let us come back to Eqs. (2.39)-(2.40) describing evolution of a jump-like Markov process started from the origin at t = 0. It can be rewritten in the form

a

at [f (x,t ) - l +(t)f(x,0)] = Kf(x,t) =

°

(2.43)

where

1+(t)={O, t :::;O, I,

t

> 0,

f(x,0)=8(x)

and

Kf(x,t) = -J1f(x,t) + J11: p(x-x')j(x',t)dx'. Replacing in Eq. (2.43) differential operator a/at by its fractional version aD;, we arrive at the correspondent generalization of fractional Poisson equation

aD; f(x,t) = Kf(x,t) + f(x,O)cI>v(t) where

(2.44)

t- V

cI>v(t) = oD,I+(t) = r(1 - v) Observe that the presence of function cI>v (z ) in the right-hand side of the equation guarantees against violating normalization. Indeed, because

1 : Kf(x ,t)dx = 1 : [-J1f(x,t) + J11: p(x-x')f(x' ,t)dx'] dx = - J11:f(x,t)dx+ J11:p(x)dxl :f(x' ,t)dx' =0, the condition

must be fulfilled. Following the same way as above we can arrive at the time-space bi-fractional differential equations for a model of fractional Levy-motion being alternative to that considered in Sect. 2.3.6. The correspondent master equation is of the form :

aD; p(x,t; a , e, v) = L( a , e)p(x,t; a , e , v) + 8(x)cI>v(t) , t

~

with the initialization condition

p(x,t) = 0, t < 0. This equation generalizes Eq. (2.42) and takes its form when v ----+ I.

0,

(2.45)

74

Vladimir V. Uchaikin

2.5.4 A link between solutions Let us dwell on a link between solutions of first-order and fraction al (v E (0, I )) which allows to avoid a special computational algorithms. One of the first applications of this approach can be find in the authors works (Ucha ikin and Zolotarev, 1999; Uchaikin, 1999). The space variable x doesn't participate in this transformations and will be tempor arily omitted. Consider equation s

and

df = Kf (t ) + 8(t ). dt Recall, that
t.J (A ) = Kj(A ) + I, ~

~

and solving them with respect to f v,w and f, we come to the interrelation:

The backward transformation yields

where

h

(t 'f) =

v ,W'

-1-1

2ni c

eAt -AV r A- wdA.

Direct calculations allow to verify that the solution of equation oD~ f v,o(t ) = Kfv ,o(t) + 8(t)

is expressed through the solution of equation

df = Kf(t ) + 8(t ) dt according to

whereas the solution of equations

2 Self-similarity, Stochasticity and Fractionality

75

t- W

oD~ ! W,I- W(t ) = K!w ,l-w (t) + r(1 _ (0 ) is connected with it by the interrel ation

2.5.5 Fractional generalization of the Levy motion Applying the above result to solving the fract ional equation of generalized Levy motion t- a

oD rP( x,t ;a ,9 )=L(a ,9)p(x,t ;a ,9) +8(x) (

rl-a

with the initial cond ition

p(X,O-) =

) ,t :2: 0,

°

and taking into account that equation

ap(X,tat;a, 9) = L( a ,9px,t ) ( ;a ,9 ) +uxut <;: ( ) <;:() ,t :2:0, with the same initial condition has the solution

we obtain

p(x,t;a ,v,9) =

10'' ' pt», (t/ rt

=

10- (t/ r) -V

;a, 9)g+Ct";v )dr

00

[ a g((t/ r )- Vlax;a , 9 )g+(r;

v)dr.

Introdu cing the notation

we repre sent the result in the form : ( -via x,a,v, 9) . - - v]« qt pX,t ( .,a,v, 9) -t

Solut ion of symmetrica l fract ional Levy mot ion equation i :"

oD~p(x,t ;a ,v,O) = - (- LiI)a I2 p(x,t;a ,v,0)+8(x) ( ) rl-a

76

Vladimir V. Uchaikin

is expressed as follows :

( - v]« x,a,v,. 0) 0) -t - - v/aqt px,t,a,v, ( . Recall that

g(x;a ,O)

= ~1°O e-ikx-Ikladk. 27r

-

00

The set of distributions q(x;a , v , e) (a E (0,2], v E (0 , 1]' e E [- ea , ea ] forms a new class of probability distributions appeared as univariate distributions (Botet and Ploszajczak, 2002) and multivariate distributions (Uchaikin and Zolotarev, 1999 ; Uchaikin, 1999), which were partially investigated in our works (EI-Wakil and Zahran, 1999; Kilbas et al., 2006) where it was named fractionally stable distributions . The reason of such term is that any random variable with distribution belonging to this class is equal in distribution to the ratio of two independent stable random variables (SRV's) :

X(a , v, e) ~ S(a , e) j [S(vW / a . A few words about a physical interpretation ofFSD. When a with subdiffusion generalization of the Fick law:

= 2 vi-I we deal

"() d Ol -v ( ) Ol-vdf(x,t) JX ,t =-KdXo t fx ,t =-Ko t dx· The fract ional derivative

Ol-vdf(x,t) = _I_.!!...- t' df(x,-r) d'r t dX r(v) dt Jo dX (t - -r)l -V

°

reflects here the memory effect on the diffusion packet evolution : the current den sity at time t is determined not only by the local density gradient at this point and at this time but rather by the evolution of of the density at this point during the entire period elapsed. This peculiarity can be explained by the presence of traps randomly distributed in the medium and trapping a tracer on a random time T with a powertype probability distribution P(T > r) oc t - V , t ----+ 00 . The subdiffusion with a = 2, OJ < 1 is of non-Markov but local nature. When a < 2 and v = I, then the super-diffusion takes place:

The fractional Laplacian -( _L1t)a/2 == -( -d 2 j dX2 )a /2 is a non-local operator. In this case, the rate of the density change depends not only on its gradient at this point but on its distribution over the whole volume at this moment. This super-diffusion, called also Levy-diffusion or Levy-flights, is a Markov but non-local process.

77

2 Self-similarity, Stochastici ty and Fractionality

When

a < 2 and v < I, the process is non-Markov and non-local.

Acknowledgments This work was partially supported by Russian Found ation of Basic Investigation (grant no 07-0 1-00517) and Deutsche Forschung sgemein shaft (SA 86 1/8-1) .

Appendix: Frac tional operators

• The Riemann-Liouville fractional integrals for J1 > 0 are 11

_

- 11

t'

1

_

f(~)d~

a1x f (x) = aDr f(x) - r(J1) Ja (x- ~)l -Il'

I

b 11 _ - 11 I f(~)d~ bf x1 (x) = XDb f(x) = r(J1) x (~ -x)l -Il '

(A.I) (A.2)

• The Riemann-Liouville fractional derivatives (0 < a < I) :

LX

(A.3)

I

(AA)

a I d f(~)d~ aDx f (x) = r(1 -a)dx a (x -~) a' b a I d f(~)d~ xDbf (x ) = - r( I _ a) dx x (~ -x)a '

• The Marchaud fractional derivatives (0

< a < I):

r

Da f( ) = a f(x) - f(x=f ~) d~ ± x r(1 -a)Jo ~l +a . • The Riesz potential for J1 > 0, J1

i- 1,3,5, ...:

- 2r(J1)cos(J11c/2)

a (

D f x)

-

00

f(~)d~

(A.6)

Ix- ~I l - Il '

< a < I):

_ (a -l ( I ) f x)

= =

1

00

I

• The Riesz derivative (0

(A.5)

=

a 2r( I -a)cos(an/2 )

1°Of(X) -f(X -~) -00

1 ~l l +a

j:

d..,

r2f(x) -f(x -~) -f(x+~)d~ ~l +a

a

2r( -a)cos(an/2) Jo

= [2 cos(a n/2)r

1

( D~f(x) + D ~ f(x) ) .

• The Feller potential (0 < J1 < I) :

(A.7)

78

Vladimir V. Uchaikin

I~,..f(x)

= u - oo lf f (x) + v x1~f(x) =

1

u+v+(u-v)sign(x-~)f(~)d~

00

-

Ix - ~ II - ,u

00

(AS)

,

where u2 + v2 #- 0, In particular, I~,II

where l,u is given by (A6). • The Feller derivative (0 < a

oa

f( ) = a 11,\' x 2Ar(I-a) =

<

1

I):

u+v+(u-v)sign(x-~)[f( )-f(j:)]

00

-

= 2ucos(,unj2)1,u,

Ix-~II+a

00

2Ar(~ _ a) .!aoo [(u + v)f(x) -

X

uf(x -

j:

~ d~

~) - vf(x+~)] ~ -l -a d~ , (A.9)

where

A = [(u + v)cos(an j2)f + [(u - v) sin(anj2)f In particular, O?,O = o ~,

og,1 = o~ , O~,II

= [2ucos(a nj2)r

I Oa ,

where o- is given by (A.7). • The n-dimensional Riesz integro-differentiation of order

a is

(-L1Il )- a/2 f(x)=_I- f f(~)d~ 'y,,( a) J~i' Ix - ~ I"-a'

(A.IO)

where

a >O, 'y,,(a) =

a#-n ,n+2,n+4, ...,

2an" /2r(a j2) jr((n -

a) j2)

and

i.

(_L1Il)a/ 2 f = _ I- f (_I)k (l)f(x dll,l(a) J~i' k=O k where

a >O, and

1= [a]+1

k~)I~ 1 -Il-ad~

(A.I I )

79

2 Self-similarity, Stochasticity and Fractionality

In particu lar,

y, (a) = 2r(a)cos(alr/2) , dl,l(a)

= - 2r(- a ) cos(a lr/ 2), a <

I

and the operators (A.12) and (A.13) coincide with (A.8) and (A.9) respectively. • Fourier transforms

.%{j(x) }(k) = .%{ - oo lf f(x) }(k) = (ik)-Il f(k) , .%{ _oo D~ f(x) }(k) = (-ik)a f(k),

r eik,xf(x)dx = f(k) .

JRn

.%{ x l~f(x) }(k) = (-ik) -Ilf(k) ,

(A. 12)

0 < j1 < 1.

.%{ x D~f(x) }(k) = (ik)af(k),

a > - 1.

.%{ olff(x)}(k) = Ikl- ll exp{ij1 (lr/ 2)signk }f( k), 0 < j1 < 1; .%{ o D~f(x)}(k) = Ikla exp{- ia (lr/ 2)signk}! (k), a 2 -I ;

(A.I3) (A. 14)

.%{ 1~,J(x) }(k) = [(u + v) cos (1l2]!" ) + i(u - v) sin ( Iln signk] Ikl- Il f(k) , 0 <j1 < 1,

.%{( _ L1n )a/2f(x) }(k) = Ikla f(k) .

(A.I5) (A.16)

In partic ular

• Laplace transform

1 00

2'{j(t)}(A) =

e- Atf(t) dt = ] (1. ) :

2'{oWf(t)}( A) = 1. - 11](1.) ,

(A. 18)

2'{oDf f (t )}(A) = Aa] (A).

(A.I9)

Referenc es Barnes J.A. and Allan D.W., 1996, A statistical mode l of flicker noise, Proc. IEEE, 54, 176- 178. Batchelor G.K., 1953, The Theory ofHomogeneous Turbulence, Cambridge University Press, Cambridge . Bertoin J., 1996, Levyprocesses, Cambridge University Press, Cambridge. Botet R. and Ploszajczak M., 2002 , Universal Fluctuations: The Phenomenology of Hadronic Matter, World Scientific, Singapore.

80

Vladimir V. Uchaikin

Bouchaud J.-P. and Georges A, 1990, Anomalous diffusion in disordered media : Statistical mechanics, models and physical applications, Physics Reports, 195, 127-293. EI-Wakil S.A. and Zahran M.A., 1999, Fractional integral representation of master equation, Chaos, Solitons and Fractals, 10, 1545-1548. Fa KS . and Lenzi E.K, 2003 , Power law diffusion coefficient and amomalous diffusion : Analysis of solutions and first passage time, Phys Rev E, 67, 0611105. Fa KS . and Lenzi E.K, 2005a , Anomalous diffusion, solutions , and first passage time: Influence of diffusion coefficient, Phys Rev E , 71,01210 I. Fa KS . and Lenzi E.K, 2005b, Exact solution of the Fokker-Planck equation for a broad class of diffusion coefficients, Phys Rev E, 72, 020101(R). Feller w., 1971, An Introduction to Probability Theory and its Applications. Vol. 2., Wiley, New York. Jumarie G., 2001 , Fractional master equation: non-standard analysis and LiouvilleRiemann derivative, Chaos, Solitons and Fractals, 12,2577-2587. Kilbas AA, Srivastava H.M. and Trujillo J.1., 2006 , Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam. Laskin N., 2003, Fractional Poisson process, Communications in Nonlinear Science and Numerical Simulation, 8, 201-213. Levy P., 1965, Processes stochastiques et mouvement brownien. 2nd edn, GauthierVillars, Paris. Lukacs E., 1960, Characteristic functions, Griffin, London. Magre O. and Guglielmi M., 1997, Modelling and analysis of fractional Brownian motions, Chaos, Solitons and Fractals, 8, 377-388. Mandelbrot B.B. and Van Ness J.W., 1968, Fractional Brownian motions , fractional noises and applications, The Siam Review , 10,422-437. Montroll E.W. and Schlesinger M.E, 1984, The wonderful world of random walks In: Nonequilibrium Phenomena II (Lebowitz J.L.,Montroll E. WEds), NorthHolland, Amsterdam, 61-175. Repin O.N. and Saichev A.I., 2000, Fractional Poisson law, Radiophysics and Quantum Electronics, 43, 738-741. Richardson L.E, 1926, Atmospheric diffusion shown on a distance-neighbour graph, Proc. Roy. Soc. London A, 110, 709-737 . Saichev AI. and Zaslavsky G.M., 1997, Fractional kinetic equations: solutions and applications, Chaos, 7, 753-764. Samko S.G., Kilbas A.A. and Marichev 0.1., 1993, Fractional Integrals and Derivatives - Theory and Applications, Gordon and Breach, New York. Samorodnitzky G. and Taqqu M.S., 1994, Stable Non-Gaussian Random Processes . Stochastic Models with Infinite Variance, Chapman and Hall, New York, London. Uchaikin V. v., 1999, Subdiffusion and stable laws, Journal of Experimental and Theoretical Physics, 88, 1155-1163. Uchaikin Y.Y. and Zolotarev Y.M., 1999, Chance and Stability. Stable Distributions and their Applications, Netherlands, Utrecht, VSP.

2 Self-similarity, Stochasticity and Fractionality

81

Van den Broeck, 1997, From Stratonovich calculus to noise induced phase transition . In Stochastic Dynamics , L.Schimansky-Geier and T. Poeschelieds), Springer, Berlin, Heidelberg. Wang X.-T. and Wen Z.-X., 2003, Poisson fractional processes, Chaos, Solitons and Fractals, 18, 169-177. Wang X.-T., Wen Z.-X. and Zhang S.- Y. , 2006, Fractional Poisson process (ll), Chaos, Solitons and Fractals, 28, 143-147. Zaslavsky G.M., 2005, Hamiltonian Chaos and Fractional Dynamics , Oxford University Press, Cambridge. Zolotarev Y.M., 1986, One-dimensional Stable Distributions , Amer. Math. Soc., Providence , Rhode Island.

Chapter 3

Long-range Interactions and Diluted Networks Antoni a Ciani, Duccio Fanelli and Stefano Ruffo

Abstract Long-range interact ions appear in gravitational and Coulomb systems, two-dimen sional hydrodynamics, plasmas, etc. These physical systems are studied by a variety of theoretic al and numerical method s, but their description in terms of statistical mechanics and kinetic theory remains an open challenge. Recently, there has been a burst of activity in this field, since it has been realized that some simplified model s can be solved exactly in different ensemble s (microc anonical, canonical, grand-canonical, etc.). Besides that, numeric al simulations and specific kinetic theory approaches have revealed the presence of out-of-equilibrium macrostates, called Quasi Stationary States (QSSs) , whose lifetime increases with a power of the number of particles . This discovery opens the interesting and intriguing possibility that the states observed in experiments where long-range interactions are involved are not Boltzmann-Gibbs equ ilibrium states. In this chapt er, after a brief review of recent results on systems with long-range interactions, we focus on the Hamiltonian Mean Field (HMF) model. We give a short presentation of its equilibrium properties and present the numerical evidence of the existence of QSSs. Then , we discuss an analytical approach to the characteriz ation of QSSs, pioneered by Lynden-Bell, that uses a maximum entropy principle. This approach captures some macroscopic features of QSSs and predicts the existence of phase transitions from homogeneous

Antoni a Ciani Dipartim ento di Fisica, Universita di Firenze, and INFN, Via Sansone 1,50019 Sesto Eno Firenze, Italy, e-mail: [email protected] Duccio Fanelli Dipartim ento di Energetica and CSDC, Universita di Firenze, and INFN, via S. Marta, 3, 50139 Firenze , Italy, e-mail: duccio. fanelli @unifi.it Stefano Ruffo Dipartim ento di Energetica and CSDC, Universita di Firenze, and INFN, via S. Marta, 3, 50139 Firenze , Italy, e-mail: [email protected]

A. C. J. Luo et al. (eds.), Long-range Interactions, Stochasticity and Fractional Dynamics © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010

84

Antonia Ciani , Duccio Fanelli and Stefano Ruffo

to inhomogeneous QSSs, which are then verified successfully in numerical experiments on the HMF model. The HMF model is defined on a lattice where all sites are coupled with equal strength. We here generalize the model to one where only a fraction of pairs of N sites are coupled, in such a way that the number of links scales as NL rv NY with I < r ::; 2. We present numerical evidence that QSSs exist in all this range of values of r and that their lifetime scales as N u (y-l ), with a = 1.5 for homogeneous QSSs and a = I for inhomogenous QSSs . We devote this paper to George W. Zaslavsky, who introduced long ago a model similar to the HMF in order to study structural transitions in crystals. George was also interested in twodimensional hydrodynamics, and in particular in the point vortex model, which also shows QSSs, and has more recently developed a theoretical approach to lattices with long-range interactions, for which the kinetic equations turn out to possess fractional derivatives .

3.1 Long-range interactions To provide an operative definition of long-range interactions we shall consider the two-body potential V(r) J

V(r) =
(3 .1)

r

where r is the modulus of the inter-particle distance. We here show that the energy U of a particle (excluding its self-energy) diverges if the potenti al does not decay sufficiently fast. Let us estimate this energy by considering a particle placed at the center of a sphere of radius R in a d-dimensional space where the other particles are homogeneously distributed (see Fig. 3.1). Being Q,d the angular volume in d dimensions (2n in d = 2, 4n in d = 3, etc .), we get

U

= foR Jp _

-

r\

"'''d

J

[r~ ] Q,drd-Idr

(3 .2)

R { [r'l- (J ] R is rv Rd-(J if d P fo r(d- I)- (Jd r ex. R is

[ln r]iS rv lnR

if d -

(J"* (J

0,

= 0,

(3 .3)

where we single out the dependence on R and we neglect the contribution of a small ball of radius 0 around the particle I . It is hence clear that the energy diverges with R if the exponent (J is smaller than the embedding dimension d , namely the dimension of the physical space where the interaction occurs. Inspired by this peculiar observation and following the customary paradigms (Dauxois et al., 2002), we will define an interaction to be long-range if (J ::; d (J ::; 3 in the Euclidean space). Remarkably, and according to the above definition, the surface contribution to the energy cannot be neglected when long-range couplings are at play. Each particle is I This requirem ent provide s an effective smoothing of the potential by eliminating problems that arise due to the singular behaviour of the interaction at short distance s.

85

3 Long-range Interactions and Diluted Networks

hence influenced by all other constituents, not just its local neighbours, as it happens for short-range interactions. When aiming at characterizing the main properties of long-range systems, one can make reference to two distinct interpretative levels: The first has to do with the associated properties of equilibrium, while the second deals with non-equilibrium features . As we shall be addressing in the following, inequivalence of statistical ensembles is being reported, an observation which motivates the search for a consistent thermodynamics description. On the other hand, the emergence of Quasi Stationary States (QSSs) points to the need of elaborating a comprehensive dynamical frame work stemming from the principles of kinetic theory. It is proved difficult despite progresses that investigating the physics of system subject to interactions decay ing with distance can show singularities at short ranges, which has been reported along these lines. For these reasons it is customary to resort to the so called meanfield approximation ((J' = 0) which turns out to be analytically tractable both in the canonical and the microcanonical ensemble. These systems will be object of our analysis . In this chapter we shall briefly review some of the phenomena that characterize long-range interactions, with reference to both toy-models and real physical systems (Dauxois et al., 2002; Campa et al., 2008; Dauxois et al., 2009).

3.1.1 Lack ofadditivity When attempting a statistical mechanics treatment of systems with long-range couplings, one faces the problem of lack of additivity, which can be exemplified as follows . Imagine to partition a given system into subsystems. Then, the total energy of the system does not correspond to the sum of the energies associated to each subsystems.

r

'-+----~R

Fig. 3.1 The energy in a three dimensional sphere of radius R diverges as R3is long-range, i.e. o ::; 3.

<J

if the interaction

86

Antonia Ciani, Duccio Fanelli and Stefano Ruffo

Let us tum to illustrate this concept, with reference to a simple mean-field Hamiltonian H

=

_~ .)2 2N (" L. 5 I

(3.4)

,

I

where the spins S, = ± 1, i = I, ... ,N , are all coupled. This is the celebrated CurieWeiss model. Notice that the rescaling prefactor 1/ N ensures the extensivity of this Hamilton ian (Kac et aI., 1963), since H oc N 2 • On the contrary, while being extensive , Hamiltonian (3.4) is not additive . To shed light onto this point , we divide the system into two equal parts, as schematically pictured in Fig. 3.2. Consider first the particular case when all spins in the left box are equal to I , while all spins in the right port ion are equal to -1 . It is clear that the (~) 2 = _ J~ • energy of the two different parts is the same, and reads E, = E2 = -

iN

iN -

At variance , the energy of the whole system is E = (~ ~) 2 = O. Hence, the energy of such a system is not additive, since E "* E, + E2. The fact that the system is non additive has strong con sequences in the construction ofthe canonical ensemble, i.e. follow ing the usual derivation which moves from the microcanonical ensemble. Consider a situation as depicted in Fig. 3.2, and imagine that system 1 is much smaller than system 2. The probability p(E,) that system I carries an energy which falls within the interval [E" E, + dE,], given that system 2 has an energy E2 (the conservation of total energy imposes that E = E, + E2), is

p( E I) =

J

Q2(E2) 8 (E,

= Q2(E -E,)

,

+ E2 -

E )dE2

(3.5)

(3.6)

where Q (E) is the density of states

2

+

Fig. 3.2 A microscopic configuration with total zero magnetization, illustrating the lack of additivity in the mean-field model (3.4).

2 Capital (resp. lower case) letter s will denote in the followin g extensive (resp. intensive) thermodynamic quantiti es, except for temperature, which will be denoted by the capit al letter T.

87

3 Long-range Interactions and Diluted Networks

£2(E) =

L O(H-E) .

(3.7)

{Si}

Using the entropy S(E) = In£2(E) (the Boltzmann constant ke being set to one) to express £22 and expanding the term S2(E - El) , one gets

p(Ed = exp [S2(E-EdJ

~ exp [S2(E) oc

where

£22(E)

~:: IE + ... J

El

e -{3EI ,

~ = f3 = T

(3.8)

dS

(3.9)

(3.10)

2 1

dE E

.

(3.11)

As an end result one obtains the usual canonical distribution. As it should be clear from the above, additivity is a crucial ingredient in the derivation , which in turn suggests that non additive systems may have peculiar behaviours when placed in contact with a thermal reservoir. The next sections are devoted to presenting a selection of important, both dynamical and equilibrium features which are ultimately related to the long-range nature of the interaction and the associated lack of additivity. We shall first focus on the equilibrium properties, to which the next section is entirely devoted . Then we will concentrate on dynamical aspects, so introducing the main topics addressed in this chapter.

3.1.2 Equilibrium anomalies: Ensemble inequivalence, negative specific heat and temperature jumps The lack of additivity is indirectly responsible for many unusual properties which have been detected for systems with long-range interactions (Dauxois et aI., 2002) . Among the most striking equilibrium anomalies it is worth including the inequivalence ofstatistical ensembles, negative specific heat in the microcanonical ensemble and possible temperature discontinuities at first order transitions. In the following, we elaborate on these concepts, by referring to specific examples (Campa et aI., 2009) . Consider yet another spin model with infinite range , mean-field like interactions. The model is defined on a lattice, where each lattice site i is occupied by a spin-I variable S, = 0, ± I . The Hamiltonian is given by

H=dtS;-~ ( LSi)2, 2N i=l

(3.12)

i=l

where J > 0 is a ferromagnetic coupling constant and d > 0 controls the energy difference between the magnetic (S, = ± I) and the non-magnetic (S, = 0) states. We set

Antonia Ciani , Duccio Fanelli and Stefano Ruffo

88

J = 1. This is a simpler version of the Blume-Emery-Griffiths (BEG) model (Blume et al., 1971), also known as the Blume-Capel model. As we will discuss later, the phase diagram of the model can be determined analytically, both within the canonical and the microcanonical ensemble, so allowing to shed light onto the inequivalence issue. A global parameter to monitor the evolution of the model is the average magnetization m, given by 1 (3.13) m=-ESi.

N

i

The canonical equilibrium state can be obtained by maximizing the free energy (see details in (Blume et aI., 1971» I

InZ

j(f3) = -- lim - , 13 N--->oo N

(3.14)

where Z is the partition function Z

=

E exp[-f3H] .

(3.15)

{5i}

In general, working in the microcanonical ensemble can prove rather cumbersome. However, with reference to this specific case study, the derivat ion of the equilibrium states in the microcanonical ensemble reduces to a simple counting problem (Barre et al., 200 I) . This is due to the fact that all the spins interact with equal strength, irrespectively of their mutual distance. A given macroscopic configuration is characterized by the quantities N+, N _, No respectively identifying up, down and zero spins . Clearly, the constraint N+ +N_ +No = N applies . The energy E of such a configuration is solely function of N+, N _ and No and reads E=I1Q-_I M 2 2N '

(3.16)

where Q = L~l S7 = N+ + N _ (the so called quadrupole moment) and M = L~l S, = N+ - N _ (the magnetization amount) play the role of the order parameters . The number of microscopic configurations Q compatible with the macroscopic occupation numbers N+, N _ and No is given by (3 .17)

Using Stirling 's approximation in the large N limit, the entropy, S = InQ can be cast in the explicit form S=

-N[(i -q)ln(l-q)+~(q+m)ln(q+m)+~(q-m)ln(q-m)-qln2]

,

(3.18)

89

3 Long-range Interactions and Diluted Networks

where q = Q/ Nand m = M / N are the quadrupole moment and the magnetization per site, respectively . We are hence in a position to compare the predictions in the microcanonical and canonical ensembles, and identify possible traces of inequivalence. In doing so, we shall mainly refer to the discussion in (Campa et al., 2009). We shall elaborate on this important phenomenon in the remaining part of this section . We build on the specific BEG model , while at the same time identifying the general, reference concepts. In particular, we will show, for this specific example, that the inequivalence of ensembles is associated to the existence of a convex region of the microcanonical entropy as a function of the energy. This latter conclusion holds in general for long range interacting systems. In Fig. 3.3 a schematic representation of the entropy and the rescaled free energy (defined in Eq. (3.14)) are depicted, being these observables related by a LegendreFenchel Transform (LFT)

(f3) = f3f(f3 ) = inf[f3e- s(e )] .

(3.19)

e

The entropy curve consists of two branches: the high energy branch is obtained for m = 0 (dotted line), while the low energy one refers to m "* 0 (full line). The two branches merge at energy value et where the left and right derivatives do not coincide ; hence microcanonical temperature (T (c) = ( aa~)) - I) is different on the two sides, yielding to a temperature jump (this is better shown in Fig. 3.5 as we will discuss later). In the low energy branch , a region where entropy is locally convex (thick line) arises, that corresponds to negative specific heat, according to the formula $;~ = - C"IT2 . In the same figure, the rescaled free energy (f3) is plotted. (f3) is a concave function and for this reason in the canonical ensemble the specific heat

.... , .. s(e)

,<.r"m =O

1> (/3)

..

/~

/'

slope =

/31/' ;

" ::

"'/;7: Cv

....

..

different slop,~e~s.-:.'--_ LFT

<0 :

e

/3

Fig. 3.3 On the left panel, sre) as a function of energy density e for the BEG model is plotted . Here, negative specific heat coexists with a temperature jump, the dash-dotted line is the concave envelope and the region with negative specific heat c" < 0 is explicitly indicated . On the right panel, the rescaled free energy 1jJ (f3 ) = f3f (f3 ) is represented. The first order phase transition is located in f31'

90

Antonia Ciani, Duccio Fanelli and Stefano Ruffo

is always positive . This is in turn the first stringent evidence that the two ensembles here considered, namely the micronanical and the canonical one, are inequivalent. The inequivalence is also appreciated when considering the inverse LFT transform. Mathematically, (3.20) s*(£) = inf [f3£ -1/>(13)] . f3 Since the inverse LFT of a concave function is always concave, one cannot recover the initial microcanonical entropy, which displays a "convex intruder". Hence s( £) i:s* (e) . Indeed, one gets the concave envelope of the entropy function as reported in the left side of Fig. 3.3 with the dashed-dotted line. Moreover we note that, at 13(, the left and right derivatives of 1/>(13) (given by £1 and £2 respectively) are different. This is the first order phase transition point in the canonical ensemble . The BEG model displays, indeed, phase transitions, both of first and second order, but the transition curves in the phase diagram obtained within the two ensemble are different, as clearly shown in Fig. 3.4. First, the position of the tricritical points, which connect the first order curve to the second order one, is not the same, thus implying that there is a region in which the canonical phase transition is first order, while the microcanonical one is second order. It is precisely in this region that the specific heat is negative. Again this is a general fact that we here learned with reference to a specific application: The region with negative specific heat develops in correspondence of a first order transition in the canonical ensemble. Back to the BEG model, in the microcanonical ensemble, beyond the tricritical point, the temperature experiences a jump at the transition energy. The two lines emerging on the right side from the microcanonical tricritical point (MTP) correspond to the two limiting temperatures, which are reached when approaching the transition energy from below and from above . Let us now turn to clarifying the ,

" ~ '<:

~

v,

CTP

~

tJ/J

Fig. 3.4 Schematic representation of the phase diagram , expanded around the canonical (CTP) and microcanonical (MTP) tricritical points. The dotted line refers to the common second order curve, while the solid line represents the canonical first order transition curve. The dashed lines correspond to the microcanonical ensemble (the bold one represents a continuous transition) .

3 Long-range Interactions and Diluted Networks

91

temperature-energy relation T (e) and the associated temperature jump as identified above . Also this curve displays two branches, as shown in Fig. 3.5: A high energy branch , corresponding to the zero magnetization state, and a low energy branch relative to the magnetized phase. Fig. 3.5a corresponds to the canonical tricritical point (CTP). Here, the lower branch of the curve has a zero slope at the intersection point. Thus the specific heat of the magnetized phase diverges at this point. This effect signals the canonical tricritical point, as it appears in the microcanonical ensemble. Increasing L1 up to a value in the region between the two tricritical points a negativ e spec ific heat in the microcanonical ensemble first arises (aT / ae < 0, see Fig. 3.5b) . At the microcanonical tricritical point the derivative sr/ ae of the lower branch diverges , yielding a vanishing specific heat (Fig. 3.5c). For larger values of L1 /1 a jump in the temperature appears at the trans ition energy (Fig. 3.5d). The lower temperature corre spond s to m = 0 solution and the upper one is the temperature of the magnetized (m "* 0) solution at the trans ition point. ,--~-----,--~---,---------,---n

0.336 :::,

0.334

0.3321

(b)

(a)

0.3320

iJ. /J =In(4)/3

f.."

~

0.3319 iJ. /J =0.462256

0.332 0.331 0.331

0.332

0.333

0.334

0.335

0.3310

0.3315

0.3320

(c)

0.331 0.330 iJ. /J =0.462407

0.330 0.328 0.27

0.329

iJ./J =0.462748

0.330

0.328

0.326

0.25 iJ. /J= 1/2

0.23 0.15

0.20

0.02

0.04

0.06

0.08

Fig. 3.5 Caloric curves T(e) for the Blume-Emery-G riffiths model. IJ. /J is the ratio between the local coupling term and the global ferromagnetic coupling.

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Antonia Ciani, Duccio Fanelli and Stefano Ruffo

3.1.3 Non-equilibrium dynamical properties As previously mentioned, long-range systems also present intriguing out-of-equilibrium features . When dealing with systems driven by short-range couplings (e.g. a gas of neutral particles) , out-of equilibrium initial conditions often lead to the same equilibrium state. After a short transient, the system converges (expone ntially fast) to thermal equilibrium, where the property of ergodicity is satisfied. At odd with this vision, interesting and unexpected dynamical behaviours can develop when con sidering long-range couplings. Simulations in the microcanonical ensemble can for instance reveal that the system is not ergodic, implying that only a limited portion of the accessible phase space is visited during the dynamical evolution. More importantly, when starting far from equilibrium, long-range systems tipically get trapped in the so-called Quasy Stationary States (QSSs), intermediate regimes which persist for a long time (diverging with the numbers of degrees of freedom) before the system relaxes to the dep uted thermal equilibrium. This interesti ng phenomenon is one of the topics addressed in this chapter. Before elaborating on the QSS concept, we introduce another interesting dynamica l phenomenon (Mukamel et al., 2005) . There, an Ising model on a onedimens ional lattice with both short and long-range interactions was analyzed. Such a model describes a system of spins mathematically specified by the follow ing Hamiltonian (3 .21)

where S, = ± I . Molecular dynamics simulations demo nstrated that the accessible region of extensive parameters (energy, magnetization, etc.) may be non convex.

1.5

(a)

m 0.5

[lZill -I

0

1

0 -0.5 1.5 (b)

[l3] -I

0

1

m 0.5

Fig. 3.6 Time evolution of the magnetization for K = - 0.4 (a) in the ergodic region (s = - 0.3 18) and (b) in the non-ergodic region (s = - 0.325) for model (3.21). The corresponding entropy curves are shown in the inset.

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3 Long-range Interactions and Diluted Networks

This implies that broken ergodicity can appear, due to the fact that the accessible magnetization states, at a given energy, can be disconnected. An example is illustrated in Fig. 3.6 which compares two different situations : on the one hand (panel a) different magnetized states (m = 0, m ~ ±0.5) are accessible in a microcanonical simulation. In the other case a magnetization gap forbids transitions between different states (panel b). Random dynamics is performed using the microcanonical Creutz algorithm (Creutz , 1983).

3.1.4 Quasi Stationary States As anticipated, long-range systems get frozen in Quasi Stationary States (QSSs), i.e. apparently stable states that persist for long time, followed then by a slow relaxation to thermal equilibrium. These regimes are for instance encountered within the framework of the celebrated Hamiltonian Mean Field (HMF) model (Antoni and Ruffo, 1995), a paradigmatic mean-field model to which we shall be making extensive reference in the following. Without entering into the details of its mathematical formulation, we shall simply mention that the HMF model describes the coupled evolution of N particles confined to move on a unitary ring. When it comes to characterizing the dynamics, it is customary to refer to the global magnetizat ion which provides a sensible measure for the degree of clustering of the particles on the ring. Starting from an out-of-equilibrium initial condition, after a violent relaxation, the systems gets stuck in a QSS, as confirmed by visual inspection of Fig. 3.7, where the magnetization amount is reported as function of time . Such picture is also particularly instructive to elucidate the dependence of the duration of QSSs on the system size. The relaxation to thermal equilibrium is driven by granularity and the 0.35

r--~--~-'--~-------,

BG

QSS

Fig. 3.7 Temporal evolution of the magnetization met) for e = 0.69 and for different particles numbers : 103 ( 102) , 2. 10 3 (8), 5. 103 (8), 104 (8) and 2.10 4 (4) from left to right (the number between brackets corresponding to the number of samples). The QSS state is indicated by the lower arrow, while the upper arrow denotes the Boltzmann-Gibbs (BG) equilibrium value of the magnetization.

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Antonia Ciani , Duccio Fanelli and Stefano Ruffo

QSSs lifetime toss diverges with the number of particles N . For a particular class of initial conditions, a power law toss 0<: N a is shown to accurately reproduce the lifetime scaling , the exponent being ex ~ 1.7. This rather unusual scaling was for the first time observed by Yamaguchi et al. in (Yamaguchi et al., 2004). 3 These latter observations extend to a vast realm of physical systems (Campa et al., 2009) and emphasizes the importance of deriving a consistent theoretical framework for the QSSs emergence. In real physical systems, in fact a huge number of particles is often involved, so making virtually infinite the time to equilibration. QSSs are hence the solely regime experimentally observable. In the last decades, several attempts have been proposed aiming at providing a systematic theoretical description of QSSs. The most successful interpretative scenario is inspired to a maximum entropy principle, based on a pioneering theoretical proposal of (Lynden-Bell et al., 1967). A statistical theory similar to Lynden-Bell 's was independently developed for the Euler equation in two dimensions by (Miller , 1990) and the deep analogy between the Vlasov and the Euler equation was later stressed in Ref. (Chavanis et al., 1996). It was recently proven in a series of papers (see (Antoniazzi et al., 2007 ; Chavanis, 2006) and Refs. therein) that Lynden-Bell theory enables one to predict with a good accuracy the relevant macroscopic observables associated to the QSSs for the HMF model. In Section 3.2.3.4 we shall review Lynden-Bell's theory, and present its main predictions with reference to the HMF case study. Despite this success , and the demonstrated predictive adequacy of the Lynden-Bell ansatz, many open problems still remain which make the puzzle of the QSSs ' origin and stability a fascinating one. As a physical example where QSS could playa relevant role, we quote the case of Single-Pass Free Electron Lasers (FEL) (Barre et al., 2004) . This is an experimental device which exploits the interaction of a relativistic electron beam with a magnetostatic field to produce powerful laser light, with tunable wave-length . Under particular assumptions (which are generally consistent with a typical experimental set up), the evolution of the system can be modelled by a one-dimensional Hamiltonian model (Bonifacio et al., 1990). After an initial violent relaxation, in which the intensity of laser light grows exponentially, an apparent stationary regime is attained , where small oscillations are displayed around a well defined plateau. Here, however, the average intensity is lower than the final level which is eventually attained at thermal equilibrium: Only after a finite time the system experiences a slow relaxation which takes the intensity towards the value predicted by Boltzmann-Gibbs statistics (Barre et al., 2004).

3.1.5 Physical examples A growing scientific community has recently begun to tackle the problem of longrange interactions with different viewpoints (Dauxois et al., 2002 ; Campa et al., All along this chapter, we shall elaborate on the correctne ss of the proposed scaling, providing novel numerical evidence .

3

3 Long-range Interactions and Diluted Networks

95

2008,2009). Long-range interactions are in fact ubiquitous and prove central in both fundamental applications and a large variety of physical systems and experiments, that are currently under development. In the following subsections we will briefly discuss the most relevant examples.

3.1.5.1 Gravitational systems The first important example of long-range interactions is given by the gravitational potential

V(r) =

G

-TIl'

(3.22)

In this case (J = I. The singularity at the origin makes the study of this interaction particularly hard, and generates phenomena like "gravitational collapse". To avoid this latter, it is customary to introduce a dedicated regularization. Clearly, the gravitational potential plays a central role in astrophysics and cosmology, especially as far as the problem of structure formation in the expanding universe is concerned (Peebles, 1980).

3.1.5.2 Plasmas Rarefied plasmas share many similarities with collision stellar systems . In particu lar the mean-field drives evolution and it is more important than the local fields of individual nearby particles. Here, again , the Coulomb force is of long-range type, and, as for the gravitational case, (J = 1. However, as opposed to the previous example, plasmas have both positive and negative charges : Hence they are neutral on large scales and can form static homogeneous equilibria. An important example in the context of plasma physics is played by the so-called beam-plasma instability: When a weak electron beam is injected into a thermal plasma, electrostatic modes at the plasma frequency (Langmuir modes) are destabilized (Elskens and Escande, 2002). The interaction of Langmuir waves and electrons constituting the beam is studied in the framework of a self-consistent one dimensional Hamiltonian, which bears an universality character being also found in other disciplinary contexts where wave-particles interactions are central (Elskens and Escande, 2002; Bonifacio et al., 1990).

3.1.5.3 Free Electron Lasers As already anticipated, the Free Electron Lasers (FELs) represent innovative laser sources that provide tunable powerful light, even at very small wave-lengths. The physical mechanism responsible for light emission and amplification is the interaction between a relativistic electron beam, a magnetostatic periodic field generated by an undulator and an optical wave co-propagating with the electrons. Under re-

96

Antonia Ciani, Duccio Fanelli and Stefano Ruffo

alistic assumptions, the longitudinal dynamics of the system can be described by a one dimensional model (Bonifacio et al., 1990): The particles do not interact directly with each other, but are indirectly coupled via the external field, which in turn provides the long-range connection (in this case (J = 0).

3.1.5.4 2D Hydrodynamics Two-dimensional dissipative-less incompressible hydrodynamics represents another important application where the interactions are of long-range nature (Eyink and Sreenivasan, 2006) . Indeed, the stream function 1jf is related to the modulus of the vorticity ro, via the Poisson equation -t.1jf = roo Using the Green 's function technique , one easily finds that the solution of Poisson's equation is

1jf(r) =

L

d2r' ro(r')G(r - r'),

(3.23)

where G(r - r') depends on the domain D, but G(r) rv _(21r) -lln [r], when r ----+ O. The kinetic energy being conserved by the Euler equation, it is straightforward to compute it on the domain D, with boundary so,

(3.24)

so.

This result confirms that logarithmic interactions are inassuming 1jf = 0 on volved . The long-range character becomes even more explicit if one approximates the vorticity field by point vortices ro(r) = L [j8(r - r.), located at r., with a given circulation Ii. The energy of the system reads now (3.25) The interaction between vortices has a logarithmic character, which corresponds to (J = O. Hamiltonian (3.25) has also been studied by George M. Zaslavsky in several important papers devoted to sticking of passive scalars around vortices (Leoncini et a1.,2004) .

3.1.5.5 Small systems In addition to large systems, where the interaction are truly long-range, one should consider small systems where the range of the interactions is of the order of sys-

3 Long-range Interactions and Diluted Networks

97

tem size. Also in this case the system is not additive, and many similarities with a pure long-range setting are displayed. Phase transitions are universal properties of interacting matter which have been widely studied in the thermodynamic limit of infinite systems. However, in many physical situations this limit is not attained and phase transitions should be considered from a more general perspective. This is the case for specific microscopic or mesoscopic systems : atomic clusters can melt, small drops of quantum fluids may undergo a Bose-Einstein condensation or a superfluid phase transition, dense hadronic matter is predicted to merge in a quark and gluon plasma phase, while nuclei are expected to exhibit a liquid-gas phase transition . Given the above , it is mandatory to develop a general understanding of phase transition for afinite systems .

3.1.6 General remarks and outlook Most of the theoretical approaches that have been proposed to explain the peculiar behaviour of systems with long-range interactions are currently under development. It is therefore important to define dedicated toy-models, suitable for theoretical analysis, which enable to address the rich phenomenology of long-range systems . A paradigmatic, though particular class of long-range system is characterized by mean-field models. An emblematic example, already mentioned above, is the Curie-Weiss model, where discrete spin variables are associated to each node of a fully coupled network, and mutually interact as prescribed by Hamiltonian (3.4) . This model is a long-range system with (J = 0, and constitutes an important entry to the wider class of non-zero (J systems . The model can be straightforwardly characterized in terms of its associated statistical equilibrium properties. It becomes however intriguingly complex when allowing for an effective dilution of the underlying network of connections, i.e. reducing the average number of links per node. Exploring the modification induced by the dilution to the equilibrium solution, as well as investigating the role of finite size corrections, has been the object of a recent publication (Barre et aI., 2009). Being also interested in the out-of-equilibrium diluted dynamics we will consider the presence of QSSs in an "ad hoc" modified version of the celebrated HMF model which includes dilution , as done for the Curie-Weiss model. After introducing the HMF model in Section 3.2, the role of dilution will be inspected via numerical simulations in Section 3.3.

3.2 Hamiltonian Mean Field model: equilibrium and out-ofequilibrium features The HMF model (Antoni and Ruffo, 1995) belongs to the class of toy-models, specifically designed to investigate the properties of long-range systems . Toy models allow one to capture the basic physical modalities of a class of systems under

Antonia Ciani , Duccio Fanelli and Stefano Ruffo

98

scrutiny, often reproducing with a satisfying degree of accuracy the correct experimental phenomenology. In doing so they enable for a drastic reduction in complexity, and return an ideal playground for theoreticians. We already recalled that the HMF model displays rather peculiar out-of-equilibrium features which are also shared by other physical systems (Elskens and Escande, 2002; Barre et al., 2004; Del Castillo-Negrete, 1998). As we will clarify in the following, the HMF admits a Hamiltonian formulation in terms of continuous variables . It is exactly solvable both in the canonical and microcanonical ensemble (Campa et aI., 2009), leading in this case to equivalent results, and displays a second order phase transition from homogeneous to inhomogeneous states. In the last years, several extensions of the original formulation have been proposed, so to account among the other for a spatial modulation of the interaction, higher dimensionality (particles lying on a torus or on a sphere rather than on a ring) and refined coupling mechanisms. In the following we shall however limit our discussion to the original HMF model. When performing numerical simulations starting out-of-equilibrium, the system is usually trapped in long-lived Quasi Stationary States (QSSs), before relaxing to the deputed equilibrium solution . The QSSs display a rich phenomenology that was alluded to in the preceding discussion, and that will be further developed below. We shall be in particular concerned with reviewing the literature devoted to the QSS and bring novel numerical insight on the relation between their lifetime and the system size. More specifically, this section is organized as follows . In the next Sects. 3.2.1 and 3.2.2, we will introduce the model and shortly discuss its equilibrium thermodynamics. Then, in Sect. 3.2.3, we will report on the out-of-equilibrium dynamics, with emphasis on the emergence of QSSs and on the discussion of the continuum limit of the discrete N-body picture . This yields the Vlasov equation, which plays a crucial role in understanding the properties of QSSs, as discussed in Sect. 3.2.3.4. Finally, in the last Sect. 3.2.3.5, we present new evidences relative to the lifetime of the QSS and discuss them with reference to the existing data in order to broaden and clarify the current picture.

3.2.1 The model The Hamiltonian Mean Field model (Antoni and Ruffo, 1995) describes the motion of N particles on a ring (see Fig. 3.8) and is characterized by the following Hamiltonian I N J N (3.26) H= - P] + [I - cos( ei- ej)] , 2

where

L

j =l

L

2N i ,j=l

ej E [0, 2n [ represents the position (angle) of the j-th particle on the ring and

P j stands for its conjugate momentum. Once two particles come to the same loca-

tion, we can either think that they cross each other or collide elastically, since they share the same mass. Such a model is nothing but a globally coupled XY model aug-

3 Long-range Interactions and Diluted Networks

99

mented with a kinetic term. Depending on the coupling constant J the interaction can be attractive (J > 0) or repulsive (J < 0). In this former (resp. latter) case, the HMF is obtained by retaining only the first harmonic of the one-dimensional selfgravitating (resp. Coulomb) potential V(x) ex: +Ixl (resp. V(x) ex: -Ixl) and assuming periodic boundary conditions. The HMF model was introduced independently by (Ruffo, 1994; Inagaki, 1993), who studied its equilibrium and out-of-equilibrium properties using quite different methods . However, an Hamiltonian of the kind of Eq. (3.26) appeared for the first time in a paper by George M. Zaslavskii (Zaslavskii et al., 1977) (see formula (2.16» . Indeed, Zaslavsky's Hamiltonian is closer to the one of the free electron laser (Barre et al., 2004) , but it can be studied in a similar way using the canonical ensemble formalism (Zaslavskii et al., 1977). The purpose of Zaslavsky was to describe structural transitions in crystals as a result of capture in a nonlinear resonance . The rescaling factor 1/ N , which appears in the potential term of Hamiltonian (3.26) is introduced to guarantee the energy extensiveness : This is the so-called Kac prescription (Kac et al., 1963). The equations of motion follow from (3.26) and read

0.5

o -0.5

-1

-1.5 '-----'-_-'----''------'--_-'----'_-'-_-'-----1._-'-_-'------' -1.5 1.5

Fig. 3.8 The Hamiltonian Mean Field model describes the coupled motion of a bunch of massive particles confined on a unit ring. The position of each particle is identified via the angle 8, as depicted in the cartoon.

100

Antonia Ciani , Duccio Fanelli and Stefano Ruffo

(3.27)

To monitor the evolution of the system, it is customary to introduce the local magnetization vector m, = (cos i , sin i ) and the average magnetization m, a global order parameter I N (3.28) m=- [mi .

e

e

N i=l

Let us introduce the norm m = [m] and the phase , cp , defined as tan(cp) = my.

(3.29)

mx

Making use of the above definitions, Eqs . (3.27) can be cast in the form (3.30) All along this chapter we shall concentrate on the ferromagnetic case (J > 0) , this latter being the interesting scenario when it comes to elucidating the emergent QSS dynamics.

3.2.2 Equilibrium statistical mechanics In the following we will discuss the equilibrium solution of the HMF in the canonical ensemble. It is obtained by applying the Hubbard-Stratonovich trick. We will treat only the ferromagnetic case J > 0 and, without loosing in generality, we will set J = 14 . The partition function of the HMF model reads

JIT N

Z

=

dpideiexp( -{3H) ,

(3.31)

i=l

where the integration is extended over all the phase space . Integrating over momenta, yields Z

2JT: ) N/2

= ( 73

{3N

exp( -2 )Zv,

(3.32)

with

4

In the antiferromagnetic case J < 0 the magnetization

In

remains zero for all tempe ratures.

101

3 Long-range Interactions and Diluted Networks

(3.33)

In order to evaluate this integral, one resorts to the Hubbard-Stratonovich transformation (3.34) x2] = ~ exp dyexp[-y2 + J2:Ux. y], 2 n

[1:

1

0 0 100

-00-00

where x and yare two-dimensional vectors . Eq. (3.33) hence becomes (3.35) with J1 = f3 / N. We can now exchange the order of the integrals in (3.35) and factorize the integration over the coordinates of the N particles . Introducing the rescaled variable y ----+ y jN / 2f3 , one obtains

Zv =

2~f3 i:L: dyexp [ -N (;~ -In(2nlo(y))) ] ,

(3.36)

where y is the modulus of y and 10 represents the modified Bessel function of order O. This latter integral can be evaluated with the saddle point technique in the meanfield limit (N ----+ 00). In this limit, the free energy f(f3) reads

f3f(f3)=-lim

~lnZ=-~ln(2f3n)+!!..+max(Yf32 -In(2nlo(y))) 2 2 y 2

N -> oo N

.

(3 .37)

The maximization of the last term in (3.37) yields the consistency equation Y

II (y)

f3

10(Y)"

(3.38)

For f3 < 2 Eq. (3.38) has a minimal free energy solution for Y = 0, which corre sponds to a homogeneous equilibrium distribution with zero magnetization. On the contrary, for f3 > 2 the minimal free energy solution is a non-vanishing f3-dependent value y that can be deduced numerically, so specifying the magnetized equilibrium solution. The value of m for such non-homogeneous states is given by the ratio of the Bessel functions II (y) m=-- . lo(y)

(3.39)

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Antonia Ciani , Duccio Fanelli and Stefano Ruffo

Hence, the system displays a phase transition at the critica l inverse temperature f3c = 2, which can be proven to be of the second order type with magnetization critica l exponent 1/ 25 . This prediction is consistent with numerical simulatio ns, as clearly displayed in Figs. 3.9 and 3.10. Here the equilibrium magnetization is plotted against the interna l energy e of the system. This latter follows from the free energy as

e = dljJ(f3)

(3.40)

df3 ' which yields

1 I -m 2 e =2f3+ -2- '

(3.41)

With reference to the microcanonical ensemble, the first numerical experiments, performed at constant energy, showed that for energies slightly below the phase transition energy, ec = (21,,) + ~, the system rapidly relaxes to an apparent equi librium different from the cano nica l one (Anto ni and Ruffo, 1995). This finding was initially thought to be the fingerprint of inequivalence between microcanonical and canonical ensem ble, considering the long-range nature of the interaction. However, it was later recognized that inequ ivalence only occurs when in presence of a first order canonical phase transition, which is not the case for the HMF. Indeed, it was then rigorously proved (Campa et aI., 2009) that ensembles are equivalent for the HMF model, the large deviation technique (Touchette, 2009) being applied to derive the exact HMF microcanonical solution . The aforementioned disagreement, as re-

1.0 0.8 m

0.6 0.4 0.2

0 0

0.0 - 0.2 0.0

0 0

0

0 0

0

0

0 0

1.6

e Fig. 3.9 Equilibrium magnetization In as a function of the energy per particle E = H / N . Symbo ls refer to direct N-body simulations for N = 102 and 10 3 , while the solid line is the theoretical predict ion. The vertical dashed line points to the critical energy, which is located at E" = 3/4. This means that the magnetization passes continuously from zero to a finite value, when decreasing the tempe rature (or increasing 13).

5

103

3 Long- range Interactions and Dilut ed Networks

vealed by microcanonical simulations, stems from the peculiar dynamica l evolution of the HMF model, which was subsequently fou nd to occur also in other context. The following section is devoted to discussing these important aspects into detai ls.

3.2.3 On the emergence of Quasi Stationary States: Non-equilibrium dynamics To investigate the HMF dynamics one has to perform direct (microcanonicai) numerica l simulations of Eqs. (3.27). A carefu l numerical analysis suggests that the evolution of the system is very sensitive to the choice of the initial condition. Depending on the specific traits of the initial particle distribution, the system can be froze n in long-lasting Quasi Stationary States (Antoni and Ruffo, 1995; Latora et al., 1998). In other words, there is no straight convergence to the Boltzmann distribu tion, and particles are appare ntly stuck in a intermediate regime, whose macroscopic charac teristics strong ly differ from the corresponding equi librium configuration. Yamaguc hi et al. (Yamaguchi et al., 2004) performed a comprehensive campaign of simulations employing a 4th-order and 6th-order symplectic schemes (Me Lachlan and Atela, 1992). The system was initialized in the so-called "water-bag" distribution. This corresponds to imposi ng a uniformly occupie d rectang le in phase space

!(e ,p ,0) = {fi =I /(4L1 eL1P) if -L1p :S;p
1.4

Ferromagnetic case

1.2 T

1.0

0 0

0.8

Theory N =IOO N= 1000

0.6 0.4 0.2 0.0 0.0

0.4

0.8

1.2

1.6

e

Fig. 3.10 Equilibrium temperatu re T vs. energy per particle e. Symbol s refer to N-body simulations for N

=

102 and 103 , while the solid line stands is the theoretical prediction .

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Antonia Ciani , Duccio Fanelli and Stefano Ruffo

where j( e,p, r) is the single particle reduced distribution function (see formula (3.53)) . This choice identifies a whole family of possible initial conditions param eterized via the quantities Lle and Ap, By tuning Lle and Ap we can adjust the specific characteristics of the selected profile in terms of the associated energy and initial magnetization. The spatial coordinates j are randomly chosen in the interval [-Lle ,Lle[. For the special setting Lle = n one obtains an initial magnetization of order 1/ ,jN, i.e. corresponding to a homogeneous initial condition in the large N limit. The momenta p j are also randomly chosen from the interval [-fJ.p, fJ.p[. With this choice, one can express the energy density e = H / N and the initial magnetization as functions of Ll and Ll p

e

e

mo=

sin(Lle) Lle '

(LlP?

1- (mo)2

6

2

(3.43)

£=--+-----'-------'--

To monitor the system dynamics one can track the global magnetization as a function of time. The initial evolution takes place on a timescale which is independent of the value of the number of simulated particles N . Such an early evolution corre sponds to the "violent relaxation" process (Lynden-Bell et al., 1967; Henon , 1964), which brings the system towards the QSS. The lifetime toss of the QSS, i.e. the time that is necessary to abandon the intermediate phase and eventually jump to equilibrium, is instead increasing with the system size N. Yamaguchi 's simulations, performed with reference to an initial homogeneous (mo = 0) water-bag, returned a power law divergence N a , with the rather surprising exponent ex ~ 1.7 (Yamaguchi et al., 2004), see Fig 3.11. To extract a sound estimate of toss. Yamaguchi et al. considered the magnetization time series, dropping the part relative to the violent relaxation. The sigmoid shape so obtained can be interpolated via the following ansatz 7,-----...,-------,,-----...,------,

~f

6

¥' / ,.,f '

5

,*'

//::~~11

4 3 -t/ /

2

3

4

5

loglON = log 10 toss as a function of loglON, as follows from direct N-body simulation for an initial water-bag distribution with m« = 0 and e = 0.69 .

Fig.3.n The logarithm of the timescale b(N)

3 Long-range Interaction s and Dilut ed Network s

105

t

(3.44) m(t) = [I + tanh(a(N )log1o())]c(N)+ d(N ) , toss where the best fit value for toss returns an estimate for the duration of the intermediate lethargic phase, which yield s to the scaling recalled above . The parameters c(N) and d(N) are also adjusted by a proper fitting procedure and respectively represent the half-width between the initial and the equilibrium levels of m(t) and the initial plateau of the magnetization. The parameter a(N) modulates the slope of the kink of the tanh function . The important conclusion is hence that the lifetime of the QSS diverge s when performing the mean-field limit N ----+ 00. In such a limit, the system is perm anentl y trapped in the out-of-equilibrium QSS and cannot eventu ally relax to the Boltzm ann equilibrium. Such an intriguing observation implie s that the limit for t ----+ 00 and N ----+ 00 do not commute. The system behaves in fact in a sensibly different fashion , depending on the order the two limits are taken. Importantly, and developing on this observation, QSSs can be seen as stable stationary equilibria of a continuous system , form ally recovered from the original discrete formulation , when sending N ----+ 00. As we shall prove in the forthcoming section, this procedure yield s the celebrated Vlasov equation , which is nowadays believed to return the correct interpretative framework for clarifying the puzzle of QSSs' emergence in presence of long-range interactions (Antoniazzi et al., 2007) .

3.2.3.1 The continuum limit: the VIasov equation As anticipated, we are here concerned with deriving the continuous counterpart of the HMF Hamiltonian, i.e. its equivalent description which hold s for diverging system size. To this end, one could invoke the rigorous result of Braun and Hepp (Braun and Hepp , 1977) who cast the problem within a solid mathematical perspective. At variance, and to preserve phy sical intuition, we will here follow a kinetic theor y argument which move s from the so called Klimontovich equation (Nicholson, 1983; Campa et al., 2009) . We will start by considering a general one dimensional Hamiltonian setting. The discrete N-body Hamiltonian hence reads N

H=

2

L ~ + U (8

(3.45)

i),

i =1

where U (8i ) refers to the potenti al which we express as a function of the individual particles coord inates 8i N

U (8 1, ... , 8N )

= L V(8i -

8j ) .

(3.46)

i<j

The HMF clearly falls within the above realm . The state of the N-body system is specified by the discrete, one-particle, time dependent density function ill given by

Antonia Ciani, Duccio Fanelli and Stefano Ruffo

106

I N

it(8,p,t) = -

L 0(8 -

8i(t))0(p - Pi),

(3.47)

Ni=1

where 0 stands for the Dirac function, (8, p) indicate the Eulerian coordinates in the phase space . (8i' Pi) are the Lagrangian coordinates of the i-th particle, whose dynamics is ruled by the following equations of motion

.

aU

(3.48)

Pi = - a8i' Differentiating with respect to time the one-particle density (3.47) and making use of Eqs . (3.48) one gets

(3.49) Recalling that ao(a - b) = bo(a - b) one can rewrite the previous equations as

(3.50) where

v(8 ,t) = N

J

d8'dP'V(8 - 8')it(8',p',t)

(3 .51)

is the self-consistent mean-field potential. This procedure eventually yields to the well-known Klimontovich equation (3.52) This equation is still exact, even for a finite number of particles N, and contains the information about the orbit of every single particle. Klimontovich equation is especially useful as a starting point for deriving approximate equations that enables one to describe the average properties of the system under scrutiny. One can in particular perform a perturbative development with respect to the system size, so to obtain an indication of the so called mean-field like approximation. More specifically, one can pose

I

1

it = (f,,(8,p,t) ) + V!J 0!(8 ,p,t) = !(8 ,p,t) + V!Jo!(8,p,t) ,

(3.53)

107

3 Long-range Interactions and Diluted Networks

where f( e,p, r) = (ht(e,p, t) ) represents the single particle reduced distribution function . The average is here taken over a large set of independent microscopic initial realizations, relative the same macroscopic state. Inserting (3.53) in (3.51) clearly implies I (3.54) v(e,t) = (v)(e ,t ) + VNov(e ,t) , where the first term reads

(v)(e ,t ) = N

J

de'dp'V(e - e')f(e' ,p' ,t) .

(3.55)

and represents the average potential. Inserting both expressions (3.53) and (3.54) into the Klimontovich equation one obtains

df

df

d (v) df de dp = __1_ (dO f + dof _ dv df _ d (v) df) VN dt P de de dp de dp

7it+ P de -

+ ~ dOv dof N de

(3.56)

dp '

Averaging the above equation over the initial realizations, finally yields (3.57) Equation (3.57) is still exact. Performing the limit for N ----+ 00, so neglecting finite size corrections, one ends up with the Vlasov equation, namely (3.58) This ultimately defines the correct framework to investigate the origin of QSSs . These are in fact intermediate out-of-equilibrium regimes of the discrete dynamics, which become stable solutions of the corresponding continuous picture . As opposed to the Klimontovich derivation, the Braun-Hepp theorem (Braun and Hepp, 1977) states that, for a mean-field microscopic two-body smooth potential, the distance between two initially close solutions of the Vlasov equation increase s at most exponentially in time. If we apply this result to a large N approximation of a continuous distribution, the error at t = 0 is typically of order 1/ VN , thus for any "small" e and any "large enough " particle number N, there is a time t up to which the dynamics of the original Hamiltoni an and its Vlasov description coincide, within error bounded by c. The theorem implies that such a time t increases at least as InN, and is linked with the fastest possible instability of the Vlasov dynamics. Since QSSs evolve on a time scale that diverges with N, the Braun-Hepp result suggests that QSSs might gain their stability from being "close " to some stable stationary states of Vlasov dynamics. This fact has been recently exploited to get mathem atical estimates of t that increase with a power of N (Caglioti and Rousset , 2008) . The time evolution could be further slowed down because the finite N system passes through

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Antonia Ciani , Duccio Fanelli and Stefano Ruffo

the numerous different stable stationary states of the Vlasov equation. These states might be "statistical equilibria" of the Vlasov equation. An interpretation of this process is given by Lynden-Bell's theoretical approach, which will be discussed in Sect. 3.2.3.4.

3.2.3 .2 On the properties of the Vlasov equation The discrete dynamics of Eqs. (3.48) conserves the energy and other selected quan tities depending on the symmetry properties of the potential V (e.g. the average momentum for translational invariant potentials, as it is the case for the HMF model) . The Vlasov equation (3.58) clearly conserves the very same constants of motion , which are to be properly expressed in term of the single particle reduced distribution function f( e ,p,t) . As an example, the energy density reads

h[f] =

II dPdef(e ,p,t)~2 II +

dpdef(e ,p ,t )(v(e ,t) ),

(3.59)

where (v( e,t) ) is the average potential. Similar relations are straightforwardly derived for the other quantities involved. Importantly, the total mass

J1 [f] =

II

dpdef(e ,p ,t)

(3.60)

is also conserved, which in turn corresponds to keeping constant the number of particles in the discrete scenario . In addition, the Vlasov dynamics (3.58) also preserves the following quantities

Cl1 [f] =

II

dpdef"(e ,p,t) ,

(3.61)

usually referred to as Casimirs . This is indeed a crucial property, which makes the Vlasov dynamics interestingly rich. As we shall demonstrate, the formal distinction between the aforementioned Lynden-Bell statistics (i.e. the statistical mechanics of the Vlasov picture) and the conventional Boltzmann-Gibbs description stems from these unique characteristics.

3.2.3.3 The Vlasov equation for the HMF model Making explicit reference to the case of the HMF model, the following version of the Vlasov equation (3.58) is readily recovered when taking the continuum limit

Jf Jf at + p Je -

. Jf {mx[f]sm(e) -my[f]cos(e)} Jp = 0,

(3.62)

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3 Long-range Interactions and Diluted Networks

with mx[f] = If cos( 8)d8dp and my[f] = Ifsin( 8)d8dp. System (3.62) conserves the energy

h[f] =

fJ

JJ

2

2

~2 f(B ,p,t)dBdp+ I - m~- my

(3.63)

and the average momentum

P[f] =

JJ

pf(8 ,p,t)d8dp

(3.64)

as well as the normalization of the distribution J1 [f ] of Eq. (3.60). The adequacy of such a scheme can be also tested via direct numerical simulations. We here present the computer simulations performed by Antoniazzi et at. aimed at confronting the N-body scenario to its corresponding continuum picture (Antoniazzi et al., 2007) . The algorithm employed to solve the Vlasov system in phase space is based on the so-called "splitting scheme", a common strategy in numerical fluid dynamics. The N-body simulations were performed for various choices of the number of simulated particles (10 3 to 106 ) . As it should be clear from the above, one expects that the macroscopic observable measured in the Hamiltonian discrete setting superpose to the homologous Vlasov quantities, over a time interval which gets progressively longer for increasing N . This conclusion is unambiguosly reached in (Antoniazzi et al., 2007) . We here solely present the comparison between the velocity probability distribution functions relative respectively to the continuum and N-body settings: The profiles are reported in Fig. 3.12 and display an excellent agreement. Curves refer to a large enough number of simulated particles (N = 106) , with respect to the selected time of integration. Motivated by such a nice correspondence, one wish to gain analytical insight into the QSS by resorting to the Vlasov scenario . This is eventually achieved by elaborating on the Lynden-Bell maximum entropy principle, to which next section is entirely devoted .

3.2.3.4 Lynden-Bell's theory The approach proposed by Lynden-Bell, and presented in his seminal paper (LyndenBell et al., 1967), is elaborated within the cosmological context and aims at explaining the observable stars distributions in elliptical galaxies. For an infinite number of interacting bodies, the system can in fact be described by a Vlasov equation, gravity being the governing force. Similarly to what observed above , galaxies are QSS which do correspond to "statistical equilibria" of the Vlasov dynamics. After an initial rapid evolution, the single particle reduced distribution function f becomes progressively more filamented and stirred at smaller and smaller scales, without attaining any final stable equilibrium. However, averaging over windows in phase space, which encompass the above filamentations, one obtains a coarse-grained function 1 that is smooth and likely to converge toward an equilibrium state. Thus, assuming the dynamics of each coarse-grained element of f to satisfy a mixing hypothesis during the violent relaxation process , one can naturally infer that

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Antonia Ciani , Duccio Fanelli and Stefano Ruffo

QSSs correspond to statistical equilibria of equation (3.58) . Like in the usual Boltzmann approach, one assumes that all the possible microstates are visited with the same probability, during the mixing dynamics. Hence the equilibrium is obtained as the most probable macrostate ]QSS, i.e. the one that maximizes the mixing entropy, consistent with all the constraints imposed by the dynamics. We shall here focus on the family of water-bag initial distributions, which correspond to a uniformly occupied rectangle in phase space (e,p) (3.42). As usual, the mixing entropy is the logarithm of the number of microscopic configurations associated with the same macroscopic state characterized by the probability density

](e,p)

SLB []] = In[W({f} )] .

(3.65)

To get the number W, the phase space is dispersed into a finite number of macrocells [q ,q + dq] x [p , p + dp], the volume dp x dq being much smaller than the whole phase space. Dividing these macrocells in V microcells with size h, these can be either occupied by the level fo or by the level 0, one excluding the other, similarly to Pauli principle. A combinatorial calculation detailed in (Lynden-Bell et al., 1967; Campa et al., 2009) yields to the final expression

0.3

J (P)

0.6

0.1 0.3

- 1.5

-I

- 0.5

0

0.5

- 1.5

1.5

-[

- 0.5

0

0.5

1.5

p

Fig. 3.12 Particle velocity distribution functions for e = 0.69 and different initial magneti zation : (a) m« = 0.3, (b) mt, = 0.5, (c) mt, = 0.7 . The solid lines refer to the solutions of the Vlasov equation (3.62), while the symbols correspond to the N-body simulations. The picture in (d) is the same as (a), but it has been plotted in logarithmic scale , in order to emphasize the agreement in the tails of the distributions.

III

3 Long-range Interactions and Diluted Networks

(3.66) where in particular the limit v ----+ 00 has been performed. Let us emphasize that this conclusion is fully justified from first principles and follows a direct counting process, as in the spirit of conventional statistical mechanics. The intimate nature of the Vlasov dynamics enters the picture via the exclusion principle, a direct consequence of the Casimirs invariance which makes the Lynden-Bell statistics very similar to the Fermi-Dirac one (it is a Fermi-D irac statistics for distinguishable "classical" objects). The analogy is purely formal since the contexts are different. Moreover, it is worth stressing that in the limit fa ----+ 00 the addit ional "fermionic" contribution can be neglected and the usual Boltzmann-Gibbs entropy is recovered. In order to compute the equilibrium distribution, one has to maximize expression (3.66), while imposing the conservation of the invariants . Consider the HMF model and refer to the water-bag initial conditions (3.42), the two levels being respectively fa = 1/ (4LleLlI') and O. Each element of the water-bag family is completely specified by the associated internal energy e and initial magnetization rna = (m.tO,myo) 6. The maximum entropy solution is derived from the constrained variational problem

SLB(e) =

mr

(SLB [!] I h(f) = e;

Jd8dp! = I).

(3.67)

The problem is solved by introducing two Lagrange multipliers f3 / fa and J1 / fa for, respectively, energy and mass. This leads, after some calculations, to the following analytical form of the distribution that maximizes Lynden-Bell entropy -

fQss(8,p)

= fa

I 2

. '

1+ exp(f3(p / 2 - mAf] cos 8) - mAf] Sill 8) + J1

(3.68)

Again we notice that this latter differs from the Boltzmann-Gibbs distribution because of the "fermionic" denominator which is originated by the form of the entropy. Besides that, it clearly manifests non-Gaussian features in the core, while the tails decay exponentially, as for usual Maxwellian profiles. The variational problem (3.68) results in a system of implicit equations that can be tackled numerically. Multiple stationary solutions are in principle possible. To identify the global maximum, which in turn corresponds to the equilibrium state, one has to punctually evaluate the entropy in correspondence of the selected sta-

J

tionary points . Depending on the predicted value of m[f] = mx[!F + my[f] 2 , we can ideally distinguish between two distinct regimes : The homogeneous case corresponds to mQSS = m[f] c:::: 0 (unmagnetized), while the inhomogeneous (magnetized) setting is found for moss "* 0 distributions. Indeed, the predictions of the LyndenBell equilibrium (3.68) are derived scanning the parameter plane (ma ,e), which, we In general, one should consider also the average momentum P = J dpd8 p f'o , but we here assume the distribution s to be centered around (8,p) = (0, 0), which clearly implies P = 0. This is not a restrictive choice: a non-zero average momentum results in the same evolution, with the superposition of a constant translational drift.

6

112

Antonia Ciani , Duccio Fanelli and Stefano Ruffo

recall, univocally identifies the initially selected water-bag distribut ion. The underlying scenario, as first recognized in (Antoniazzi et al., 2007 ; Chavanis , 2006) , is depicted in Fig. 3.13. When fixing the initial magnetization and decreasing the energy density, the system undergoes a phase transition, from a homogeneous to an inhomogeneou s QSS. The plane can be then formally divided into two zone s respectively associated to an ordered inhomogeneous phase , moss "* 0, (lower part of Fig. 3.13), and a disordered homogeneous state, moss = (upper part). These regions are separated by a transition line, collection of all the critical points (m eC ) , which can be segmented into two distinct parts . The dashed line corresponds to a second order phase transition, meaning that the magnetization is continuously modulated, from zero to positive values, when passing the curve from top to bottom . Conversely, the full line refers to a first order phase transition : Here, the magnetization experi ences a sudden jump when crossing the critical value (m e C ) . First and second lines merge toget her in a tricritica l point approximately located at (mo ,e) = (0.2,0.61). Such a rich out-of-equilibrium scenario results from a straightforward applic ation of the Lynden-Bell theory and proves extreme ly accurate versus direct simulations based on the discrete formul ation of Eq. (3.27). The existence of phase transition is in particu lar numerically confirmed. The values of the critical parameters ( mo,eC ) and the order of the transition are correctly predicted. Lynden-Bell's theory consti tutes indeed a powerfu l analytica l tool which enab les one to characterize the global properties of QSSs and unravels their unexpected richness . Having recognized the existence of at least two different classes of QSSs points out the need of further clarifying the issue of their time duration. Is the scaling NI.7 universal or, conversely, does it apply only under specific condition? In the following we settle down to further explore this important point.

°

o,

o,

--

0.8 ,-----.----,---.------,-----.---,---,--------,----,--------,

O .6 ~--·

-------

Fig.3.13 Theoretical phase diagram in the control parameter plane (mo,e) for a rectangul ar waterbag initial profile. The dashed line stands for the second order phase transition , while its continuation as a full line refers to the first order phase transition. The full dot is the tricritical point. The grey zone corresponds to a forbidden domain of the parameter space.

3 Long-range Interactions and Diluted Networks

113

3.2.3.5 On the lifetime of homogeneous and inhomogeneous QSS: Different scalings, numerical simulations and kinetic theory Let us start by reviewing the available results on the time duration of the QSS for the HMF model. As already mentioned in the introduction of Sect. 3.2.3, Yamaguchi et at. reported in (Yamaguchi et al., 2004) that the lifetime of the QSS diverges as N1.7 when increasing the system size. Their numerical simulations refer to initially homogeneous water-bags (rno = 0) and to a specific choice of the energy (e = 0.69), which lies in the portion of the parameter space yielding to (almost) unmagnetized QSS 'moss = 0). Later on, Moyano and Anteneodo performed an independent campaign of simulations (Moyano and Anteneodo, 2006) initializing the system as a water-bag specifed by the conditions rno = I, e = 0.69, which admits the equilibrium values M ;:::;; 0.31 and T ;:::;; 0.457 for magnetization and temperature respectively. When preparing the initial water-bag, Moyano and Anteneodo set 8i = 0 for i = I, ... ,N and assume regularly spaced values of momenta (instead of random ones as customarily done) with the addition of a small noise to allow for a statistical realization 7. In Fig 3.14 the temporal evolution of the temperature T(t) = L (p~ ) / N is plotted (the average is taken over different initial conditions). Then, Moyano and Anteneodo rescaled the time axis by a factor N1.7, clearly inspired to the work of Yamaguchi et al.. Different curves nicely collapse onto each other, see Fig. 3.15, suggesting that the lifetime of the QSSs at rno = 1 obeys a scaling law identical to that associated to the rno = 0 case. Two comments are mandatory at this point. On the one hand working at e = 0.69 for rno = I, one would expect to find a magnetized QSS, while Moyano and Anteneodo's data return a homogeneous configuration, as confirmed by visual inspection of Fig. 3.15. How can this observation reconcile with the existence of the transition line, as identified via the Lynden-Bell calculation? Also, having identified the same scaling for two extreme values of rno (resp.

T

Fig. 3.14 Time series of the temperature T for EO = 0.69 and different values of the simulated particle number N (N = 500 x 2 k , with k = 0, ... ,9) . Averages were taken over 2.56 x 105 / N realizations starting from a "isotropic" (almost regularly spaced momenta) water-bag initial configuration at t = O. Dotted lines correspond to temperatures at equilibrium, TEQ = 0.476 and at QSS, TQsS = 0.38 (this temperature corresponds to moss = 0 according to the formula EO = ~ + 1(l -m 2 ) ). The figure was kindly provided by the authors of Ref. (Moyano and Anteneodo, 2006) . No specific information is provided in (Moyano and Anteneodo, 2006) on the strength of the superposed stochastic contribution. As we shall argue in the following we do believe that the noise is practically too small to induce any sensible modification from the equally spaced configuration.

7

114

Antonia Ciani , Duccio Fanelli and Stefano Ruffo

set to 0 and I) and energy values respectively below and above the corresponding transition point , would naively suggest that the QSS lifetime is not sensitive to the initial water-bag characteristics. The whole picture is however far more compli cated, as revealed by an independent set of measurements performed by Pluchino , Latora and Rapisarda (Pluchino et al., 2004) . These latter authors assumed a waterbag initial condition, the particles positions being (Ji = 0 for all i, which corresponds to setting rno = 1. The energy was again E = 0.69, but, at odd with the procedure followed by Moyano and Anteneodo, the velocities were randomly assigned with uniform probability. Surprisingly, the QSS lifetime is now reported to increase linearly with the system size N, see Fig. 3.16. When it comes to the observed value of moss- refined analysis by Pluchino and Rapisarda themselves (Pluchino and Rapisarda, 2007) indicates that inhomogeneous configurations prevail, in agreement with the Lynden-Bell prediction. Even more astonishingly no comment is found in the literature concerning the apparent discrepancy of the two studies mentioned above . This puzzle finds its solution when considering the work of Campa , Giansanti and Morelli (Campa et al., 2007) . In this paper the authors focus on different classes of initial conditions, all relative to the energy density e = 0.69. In particular they numerically show that when starting from particles equally spaced both in momentum and positions, the system naturally evolves towards a homogeneous QSS configuration (even if the system is initiated in a region that Lynden-Bell theory baptises associated to inhomogeneous QSSs) . Conversely, if the particles are initially randomly distributed then the Lynden-Bell partition into magnetized and unmagnetized QSS is respected. Moyano and Anteneodo's simulations fall in the former class (assuming the stochastic perturbation weak enough) and the system rapidly moves towards a homogeneous QSS, similar to that faced by Yamaguchi et at. when dealing with the choice rno = 0 and e = 0.69. In this respect, following the intriguing dynam ical mechanisms identified by Campa, Giansanti and Morelli, which still calls for a sound theoretical interpretation, it is quite natural to expect an analogous depen dence with reference to the observed scaling properties. Completely different is the situation explored by Pluchino, Latora and Rapisarda, since the initial condition chosen in (Pluchino et al., 2004) (random distribution) returns preferentially magnetized QSS.

0.5

............................................................

................--f

~~

T

O.4.SltI:IlAk====:=~w.

. 10 2

10'

t/N1.7 Fig . 3.15 Time series of the temperature T as a function of t /N1.7 . Parameters are the same as in the caption of Fig. 3.14. The figure was kindly provided by the authors of Ref. (Moyano and Anteneodo, 2006).

3 Long-range Interactions and Diluted Networks

115

In conclusion looking with a critical eye to the results published in the literature one can argue that the QSS characteristics are non trivially affected by the initial distribution of the particles . The associated duration time gets equally influenced. Motivated by this complex, and apparently controversial, scenario , we hypothesize that two main scaling laws exist for the QSS lifetimes : Above the Lynden-Bell transition line we conjecture a N a scaling law, a closely resembling the value 1.7 detected by Yamaguchi et at. and by Moyano and Anteneodo, respectively. Conversely, below the Lynden-Bell transition line, we predict a N divergence for the QSS lifetime, when increasing the number of simulated particles . To shed light onto this issue and eventually verify the correctness of our interpretative picture we turn to numerical simulations. As we shall be discussing in the following we also argue that the exponent a is indeed 1.5. We have numerically integrated the equations of motion (3.27), by using a symplectic 4th-order integrator, the McLaghan-Atela's algorithm (Me Lachlan and Atela, 1992). The timestep dt = 0.7 was chosen as a trade-off between a good energy conservation (with a relative accuracy lilE / E I '" 10- 5 ) and the possibility to perform long enough simulations, for relatively large systems , in a reasonable amount of computation time. We consider a water-bag initial condition (3.42) where the spatial coordinates i randomly populate the interval [-lie , lie[, with a uniform distribution. Analogously, the momenta are assigned into the interval [-lip, lip[. The energy follows from formulas (3.43). We are interested in measuring the duration of the QSS as the system size is changed. Consider first e = 0.69 for three different values of rna respectively set to 0,0.2, 0.7. All cases fall above the Lynden-Bell transition line and are hence expected to yield to a demagnetized QSS, for which the scaling N a , a c::: 1.7 applies . In Fig. (3.17) the temporal evolution of the magnetization versus the rescaled time, t / NI.7 , for rna = 0 is plotted . Distinct curves relative to different choices of N col-

e

o

DWBI C

£:, WBI C

N

Fig. 3.16 Log-log plot of the QSSs' lifetime is reported as a function of the size N , for e = 0.69 and nu, = 1. The lifetime diverges linearly with N . The figure has been provided by the authors of Ref. (Latora et aI., 200 I). 8 In our study we consider the particles to be randomly distributed within the initial water-bag domain .

Antonia Ciani , Duccio Fanelli and Stefano Ruffo

116

lapse onto a universal profile, so pointing to the correctness of the proposed scaling . To further reinforce our conclusion, we also extract a direct estimate of the equilibration time following the fitting scheme introduced by Yamaguchi's et al. (Yamaguchi et aI., 2004) and revisited above, see Eq. (3.44). In Fig. 3.18 the best fitted value of toss is plotted versus N in a log-log scale. The three panels refer to the values of Ino here considered. The superimposed dotted line correspond to a power law scaling with a = 1.7 as suggested by Yamaguchi et al. investigations. The dashed line is instead calculated for a = 1.5, and, in our opinion, it also interpolates correctly the data. We shall return on this important point in the following. To complete our analysis we here report results of the simulations relative to the same choice of Ino but different values of c. In particular we set e = 0.58, i.e. a value that fall below the Lynden-Bell transition line for all selected values of InO. The associated QSS are hence expected to be of the magnetized type. Results of the simulation are presented in Fig. 3.19 : Here the time is rescaled by a factor N, and the curves show a clear tendence to accumulate over a universal profile, regardless of the specific N value to which they refer. This finding is in agreement with Pluchino , Latora and Rapisarda's simulations relative to the specific case Ino = I and more generally suggest that the lifetime of magnetized QSSs grows linearly with system size. In conclusion, and based on the above numerical evidence, we suggest that the QSSs' lifetime obeys to two distinct scalings depending on the initial parameters of the water-bag . The different scalings correlate with the out-of-equilibrium transition line as detected via the Lynden-Bell theory of violent relaxation. The suggested scenario is summarised in Fig. 3.20. Let us now concentrate on the possible origin of the different scaling. We shall be interested in finding a justification for the observed behaviours, which, we believe, should ultimately stem from kinetic theory. Consider the Klimontovich

0.4 ,....----r-.,.,..,.,.,.""----r--T"TTmnr'--...,....-,r-rT'!".,,,...--r-n

0.3

;:: 0.2

0.1

Fig. 3.17 The magneti zation for different values of N versus the rescaled time t /NI.7 is plotted . Here N = 2· 103 (8), 5 · 103 (8) and 2· 104 (4) from left to right (the number between brackets stands for the number of samples) are reported. Simulations refer to tnt, = 0 and e = 0.69.

11 7

3 Long-range Interact ions and Diluted Network s

6

4 3

4 4

3

4 3

3.5

4

10glO N Fig. 3.18 Logarithm of the timescale b(N) = logl o toss as a function of 10glO N for e = 0.69 and three different choices of mo. From top to the bottom , mt, = 0,0.2,0.7. The dotted and the dashed lines represent the laws toss ~ N I.7 and to ss ~ N I.5, respectively.

118

An tonia Ciani, Ducc io Fane lli and Stefano Ruffo

0.5 0.4

.,

0.3 0.2 0.1 0 0.0001

0.01

100

10000

1e+06

lIN 0.6 ,....-r~----.....~----.,-r~-----,

0.55 0.5

0.4 0.35 0 . 3 .~~

0.000 1

_ _----;-;~~_ _-, 10000 lIN

----!-~~

0.6 ,........,~-----.-~~---,.-n~----, 0.55 0.5

0.4 0.35 0.3 L.....u~--_---L~~---...J.....o........._ --' 10000 0.0001 lIN Fig.3.19 The mag netization for different values of N versus the rescaled time 1/N is plotted . Her e, N = 103( 102 ),5 . 10 3 (8) and 2 · 104 (4 ) (the number between bracke ts correspondi ng to the nu mber of samples) . Data are referre d to e = 0.58 and m« = 0, 0.2 , 0 .7 from top to bott om.

119

3 Long-range Interactions and Diluted Networks

equation (3.57) and imagine to develop around a homogeneous solution , as it is the case (after a fast transient) when working above the transition line of Fig. 3.20. Then, under this condition, it can be shown (Campa et al., 2009) that the first order correction to the Vlasov equation, i.e. the collisional term

~

/ aovaOf ) se ap

(3.69)

N \

vanishes . Generalizing the perturbative calculation so to account for higher order corrections, one can show that the next-to-leading correction to the mean-field Vlasov scenario scales as 1/ N1.5 (Campa et al., work in progress). This in turn suggests that the system migrates from the ideal Vlasov setting on time scales which get longer and longer when increasing the system size and in particular diverging as N 1.5, in quantitative agreement with the results of our simulations. In the opposite regime , when the QSS is magnetized, and so intrinsically non homogeneous, one cannot drop the collisional term (3.69), this latter is in principle present and contributes with a non-zero forcing which drives the relaxation to equilibrium . The equilibration time is hence expected to diverge linearly with N, as seen in our simulations. 0.75 r--~---.--~----.-~r--.--~--.---~----,

0.7

/

)storder - - 2ndord e r - tricritical point •

/ / / /

0.65

/

0.6~o

0.2

-0.4

N

0.6

0.8

Fig. 3.20 Two scaling laws for the QSS lifetime as a function of the number of simulated particles are detected: Above the transition line, as predicted by Lynden-Bell's theory of violent relaxation, the QSS lasts for a time that diverges as a power law of the system size, N a . a = 1.7 fits the data, but the alternative scaling a = 1.5 results equally appropriate. Below the transition line, when magnetized QSS are to be expected, one observes a linear dependence of the QSS duration versus N.

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Antonia Ciani , Duccio Fanelli and Stefano Ruffo

3.3 Introducing dilution in the Hamiltonian Mean Field model In this section we will discuss the effect of dilution for the HMF model. We will show that, in the thermodynamic limit , the diluted HMF model behaves, at equilibrium , as its fully connected analogue, independently on the degree of dilution. We shall then focus on the study of Quasi Stationary States. We will discuss the results of a dedicated campaign of numerical simulations which enabled us to demonstrate the robustness of QSS versus dilution . We shall in particular quantify the QSS duration versus the number of simulated rotors and shed light onto the role played in this respect by the amount of dilution. Long-range interactions on networks have been recently studied in (Barre and Goncalves, 2007). The ground state energy and the entropy of a diluted Ising ferromagnet have been obtained analytically in (Serva, 2009).

3.3.1 Hamiltonian Mean Field model on a diluted network A random graph of the Erdos-Renyi type (Erdos and Renyi , 1959) is constructed by taking an ensemble of N sites and, after choosing at random a pair of sites, a link joining the two sites is added with probability p. On each site i sits an XY spin variable m, = (cos ei, sin ei) ' Two randomly selected sites, say i and i . are then connected through a link to which we associate the coupling constant.Zj, The number of links NL is bounded from above by N = N(N - 1)/ 2, which corresponds to the fully connected case, since we are avoiding double edging of two sites and self wiring . Let us now introduce the dilution parameter y by scaling the number of links as (3 .70) where the normalization factor y! is introduced in order for the fully connected topology to be exactly reproduced when y --+ 2. We restrict our analysis to the interval I < y :::; 2, which corresponds to graph topologies that range from a number of links growing linearly with the size of the system (y = I) up to the fully connected case (y = 2). We refer to all these cases as diluted networks . The HMF Hamiltonian, in presence of dilution, can be cast in the form I NN

H

N

LPT + -4NL (b"Lj) Jij [I - COS(ei 2

=-

ej)] .

(3 .71)

i=l

where Jij stands for the coupling strength, that depend on the selected pair 0, j) . In fact, the couplings Jij are assumed to be randomly distributed according to the following law

121

3 Long-range Interactions and Diluted Networks

I with probability p

Jij = {

o with probability

= NL/N '" ~NY-2 ,

t

1- p = 1- _Ny-2 . y!

(3.72)

Hence, the prob ability distribution is written as

P(Jij) = p8(Jij - I) + (I - p )8(Jij).

(3.73)

We shall hereafter concentrate on the thermodynamic limit (N ----+ 00) and pro ve that the diluted system always converges to the corresponding fully coupled model for all values I < Y< 2.

3.3.2 On equilibrium solution of diluted Hamiltonian Mean Field We begin by con sidering the equilibrium properties of such a system and so write the canon ical partition function as

(3.74) We want to compute (In Z)J , where O J denote s the average over disorder. This is achieved via the celebrated replica trick , which is based on the ident ity (In Z)J

·

= I1m 11->0

(Z")J - I , n

(3.75)

where n is a assumed to be a real number. The central idea con sists in carrying out the computation for all integers n, extending the results for all n, and performin g in the end the limit for n ----+ O. The HMF replicated partition funct ion reads

[Jndeidpiexp-/3H (ei,Pi)]" N

ZIl

=

i= 1

. /3 /3N j nn detdp jexp [-"2 LL(pj)2 - 4N L L (I -cos(et - ej))] . 11

=

N

a= l i = l

N

a i= 1

L a (b.,,})

(3.76) Averaging over disorder eventu ally yie lds

122

Antonia Ciani, Duccio Fanelli and Stefano Ruffo

x

LP(Jij)exp [f3 N L L "

=

(I-cos(er -

4NL a (io'!' j)

Jij

f3

N

ej))]

N

I TITI derdpf exp [- 2 LL(Pf) 2] a= ll = 1

a 1=1

N

X

LP(Jij)TIexr[f3 L(I-cos(er-ej))] Jij

(3.77)

4NL a

ie]

Recalling Eq. (3.73) one then obtains (Z")J

=

I fIn

derdpfexp [-

a= ll= l

X

=

I

~ LE(pf)2] a 1= 1

N Y! 2 I I -2- -I- +---exp -TI i* j [ y! N 2-y y! N2- y 4Ny-1

[f3

"

f3

N

TITIderdPfexr[ -2 a= l l= 1

La (I-cos(e·a-e·a))]] 1

J

N

LL(Pf)2] a 1=1

f;; [I-~-I-+~-Iy! N2- y y! N2 - y

xexp ["In

x exp [

:;~!I ~ (I - co s( er - en) J]

l

(3.7 8)

To proceed further we series exp and the exponential and the logarithm and retain the lead ing order in N (Z")J

=

I fI nderdpf a= ll= 1

X

exr[L:L I*J a

So (ln Z)J

exp [ -

~ L E(pf)2] a 1=1

;~ (1- cos(er - en] = Z::,.

(3.79)

= InZ ", and one obtains the corresponding Hamiltoni an (3.80)

y:s:

In conclu sion , and as anticipated, for large enough N values and for I < 2 the diluted system is equivalent to Eq.( 3.26) . In other words, and irrespectively from the dilution amount, system (3.71) is expected to relax asymptotically to the equilibr ium configuration, which is eventually attained by the original, full y coupled, analogue.

3 Long-range Interactions and Diluted Networks

123

3.3.3 On Quasi Stationary States in presence of dilution We will here focus on investigating the emergence of Quasi Stationary States for the HMF model on a diluted network . We shall be in particular concerned with testing the robustness of QSS versus the dilution amount. Are the QSS still present when the average number of links per node is progressively reduced? And, in that case, how the duration time scales with the dilution parameter y? These are the questions that we plan to address in the following . Let us start by clarifying the numerical procedure that will be employed in the forthcoming characterization. The equation of motion of the diluted HMF model can be readily obtained and read

(3.81)

Our numerical implementation relies on the same symplectic 4th-order integrator used for the HMF model (Me Lachlan and Atela, 1992). The timestep here selected is dt = 0.5. The disorder is of the quenched type : The configuration of assigned links is fixed for every realization, without being further adjusted during the simulations. The quantities of interest are averaged over several configurations of the underlying network of contacts. To keep contact with the preceding discussion, we will limit our analysis to the water-bag initial condition specified by Eq. (3.42) . We recall again that, for this specific case the initial magnetization and the energy are obtained from the parameters of the water-bag as in Eqs .(3.43) . We begin our discussion by presenting the results relative to an initially homogeneous distribution (rna = 0). We focus on two different choices of the energy, E = 0.58 and e = 0.69, respectively below and above the transition line point f c = 7/12. Consider first the case e = 0.69, which in the fully connected scenario (y = 2) is shown to yield to an almost demagnetized QSS. In Fig. 3.21 we report on the temporal evolutions of the magnetization as recorded in our numerical simulations, for different values of the dilution , and by varying the system size (three choices of N in each panel, respectively N = SOD, 1000, 2000). Several observations can be made, as it follows from a straightforward qualitative inspection of the figures. On the one hand the QSSs do exist even in presence of dilution . The magnetization settles down into an intermediate characteristic plateau which is eventually maintained for long times (notice the logarithmic scale on the x-axis), displaying a sensible dependence on the number N of simulated particles, as we shall be commenting in the following . On the other hand, the value of the magnetization associated to the QSS regime of Fig. 3.21 is shown to decrease when increasing the system size. We hence argue that the QSS is of the homogeneous type, as it is indeed found for the fully connected reference case for the same choice of parameters (s > f c.). As a final comment, we also stress that the asymptotic value of the magnetization is independent on the specific choice of the dilution and compat-

124

An tonia Ciani, Ducc io Fane lli and Stefano Ruffo

y =1.7

0--0

-

S

e

N =500 N =1000 N =2000

0.2

0. 1

00

10

100

1000

10000

t

Y= 1.5

0.4 0--0

0.3

-

N =500 N =2000 N =5000

S 0.2

s

0. 1

Fig. 3.21 Temporal evol ution of the magnetiza tio n m(t ) for different particle numbers N = 500 , 1000, 2000, 5000. Different panels refer to disti nct choices of y. Horizontal solid lines represent the equi lib rium value meq c::: 0.3 . The energy of the system is always set to e = 0 .69 , and mo =O

3 Long-range Interactions and Diluted Networks

125

ible, within statistical fluctuations due to the finite number of realizations, with the equilibrium value calculated for the fully connected case (m eq '" 0.3, solid lines in the figures). This finding confirms in turn the adequacy of the theoretical argument developed in Sect. 3.3.2. In Fig. 3.22 we reorder the simulation outputs so to appreciate how the dilution r affects the QSS lifetime . The more the system is connected the longer the QSSs survive. We emphasize that, as the dilution takes the lowest value here considered (namely r = 1.5), the QSS lifetime gets reduced by two orders of magnitude, with respect to the corresponding fully connected case. While QSS are certainly present when dilution is accounted for, they tend to progressively reduce their duration as r approaches the limiting value I, where they formally fade off.

N =1000 0.4

0.3

S E

'-'y =1.5 - y= 1.6 - y =1.7 -y=1.8 -fully

0.2

0.1

0\

100 t

10000

N =2000 0.4

0.3

S E

.-.y = 1.5 - y =1.6 - y = 1.7 - y =1.8 -fully

0.2

0.1

0\

100

t

Fig. 3.22 Temporal evolution of the magnetization m(t) for different values of yand for N (upper panel) and N = 2000 (lower panel). The energy of the system is £ = 0.69.

=

1000

Antonia Ciani, Duccio Fanelli and Stefano Ruffo

126

In order to quantify our observation we turn to measuring the QSS lifetime TQSS via the very same fitting procedure as introduced in Sect. 3.2.3.5. We here recall that the sigmoid profile (3.44) can be numerically superposed to the simulated curves, by properly tuning the free parameters a(N),c(N) and d(N) . Result of the analysis are displayed in Fig. 3.23, where TQSS is plotted as a function of N in log-log scale, for different choices of y (symbols). The data follow a power-law trend , the exponent (slope of the linear profiles) being sensitive to the selected value of y. Starting from this observation, it would be extremely interesting to elaborate on a reasonable ansatz, physically motivated, which is capable of reproducing the scaling observed for y < 2, while converging to the well-known N1.7 solution as the y ----+ 2 limit is performed. We stress again that the 1.7 factor is being suggested to apply when homogeneous QSS are concerned, but our simulations as reported in Sect. 3.2.3.5 seem equally compatible with the more sound 1.5 choice . We introduce again a to label such a controversial exponent, regardless of its specific numerical value. A rather natural proposal would be to replace in the aforementioned relation the global number of degrees of freedom N, with the effective quantity Neff = N (y-l ), which quantifies the average number of links per node. Under this assumption a Na (y-l ) (382) TQsS 0<: Nef f = . which reduces to the correct functional dependence when yis set to 2. Also, working at fixed N, ansat; (3.82) predicts that the QSS lifetime would shrink as y ----+ I, in agreement with our numerical experiments. In Fig. 3.23 symbols referring to the simulation are compared to the proposed scaling law (3.82), for both a = 1.7 (black solid lines) and a = 1.5 (red solid lines). A convincing degree of correspondence can be clearly appreci ated, so pointing to the correctness of the propo sed scenario . Even more interestingly, our numerical data seem to be better interpolated by the

le +05

10000

1000

100

..-:'" ...

--

,..-

----

..........-

£ ~ :-:'

.--' -- ..

-r" .. .-... -.-. ~ --- ,..-' • .' --=..-.-+ .-=--., .............-.. _.,~.

--- .. , .......-=--

~ ...

--:.

~.'

Jo.

.'

1000

N

Fig. 3.23 toss is plotted versus N . Symbols refer to direct simulation, where the time duration of the QSS is estimated by resorting to the fitting procedure (3.44), as discussed in the text. Averages over several realizations are considered . The solid lines stand for ansat: (3.82) where ex is alternatively set to 1.7 (dotted) and 1.5 (dashed).

3 Long-range Interactions and Diluted Networks

127

a = 1.5 choice, this latter possib ly arising in the context of a standard kinetic theory treatme nt, see Sect. 3.2.3.5. Before procee ding with our discussion, a further com ment is mandatory. We emphasize in fact that, up to this point, we are solely dealing with homogeneous QSS, the selected energy amount being so far set above the out-of-equilibrium transition line. Though the analysis is developed with reference to rna = 0, the conclusions reached hold more genera lly and admit a natural extension to initially bunched (rna "* 0) water-bags, provided e > ec . The dual condition e < e; is addressed in the remaining part of this section, where e = 0.58 for rna = O. y =1.7 0.6 0.5 0.4

::

0.3 0.2

-

0.1

N=lOa N =I OOO

-

N =5aOO

-

N =IOOOO

0 0.01

1 tlN y - 1

100

y = I.8 0.6 0.5 0.4

::

0.3 0.2 0. 1 0

0.0 1

100 tlN y - 1

Fig. 3.24 The temporal evolution of the magnetization is plotted versus the rescaled time t / N y- l for e = 0.58 and for Y= 1.7 and Y= 1.8. Different curves refer to different system sizes. Horizontal solid lines represent the equilibrium value m eq ~ 0.5.

128

Antonia Cian i, Duc cio Fanelli and Stefano Ruffo

Indeed, when we analyze inhomogeneous QSSs for the diluted case, the time evolution of the magnetization shares many similarities with the corresponding fully coupled scenario . The system sets down into an intermedi ate QSS, which is now magnetized, and then drifts towards thermodynamic equilibrium. Expanding on the preced ing argument and recalling that for magnetized QSS a linear scaling with N is expected to apply, we are inclined to believe to the following hypothesis 'LQSS oc Neff

= N (y-

(3.83)

I) .

To assess the adequacy of such an ansatz, we now plot the time series collected in our numerical exper iments , as a function of the rescaled time t / N y-I , see Fig. (3.24). The curves nicely collapse onto each other, so validating the scenario of

y = 1.7

O':-:---.L------:~--...L---::__,O_=__-----'---_"J

0.55

0.6

0.65

0.7

E

Fig. 3.25 In the upp er panel the temporal evo lutio n of the magnetization is rep resent ed for N = 1000, Y = 1.7 and different energy value s. Every curve is averaged over 10 reali zation s of the sys tem. In the lower panel the magnetization in the QS Ss (mQss ) is displayed versus the energy amo unt.

3 Long-range Interactions and Diluted Networks

129

Eq . (3.83) . Also in this case, such conclusions are shown to apply also for the more general setting where initial bunched water-bags are selected. In conclusion, we have here proven that QSSs do exist in diluted HMF dynamics. Their lifetime diverges with the system size, following however a different scaling depending on the specificity of the selected initial condition. The effect of dilution translates into a modification of the scaling exponent, which is correctly guessed on the basis of an intuitive ansatz:

3.3.4 Phase transition In the preceding discussion we have concluded that the high energy (unmagnetized) state and the low-energy (magnetized) state preserve their identity, when the dilution is accommodated for. What can we say about the phase transition separating these two reg imes ? While it is evident that a transition still occurs, can one elaborate on the specific role of y? Is it affecting the critical value of the energy at which the transition takes place for a given ma amount? And what about the order of the transition ? We shall here provide a preliminary answer to these questions for the specific case ma = 0, which, we recall , yields a first order out-of-equilibrium phase transition, as evidenced by the Lynden-Bell variational problem. In Fig. 3.25 (upper panel) the temporal evolution of the magnetization is reported for y = 1.7. Different curves refer to distinct e values, scanning the interval from 0.58 to 0 .62 . The corresponding transition is represented in lower panel of Fig . 3.25 . The transition seems to occur for an energy value larger than 7/12, i.e. the fully coupled reference value . Additional simulations are however necessary to shed light onto this issue, clarifying the role of finite N corrections, particularly crucial for diluted graphs. It can be moreover argued that the dilution mechanism mutates first order into second order transitions.

3.4 Conclusions After introducing the definition of long-range interactions and commenting on the main features of the statistical mechanics of these systems, notably ensemble inequivalence (Campa et al., 2009), we have focused on the equilibrium and out-ofequilibrium properties of the Hamiltonian Mean Field (HMF) model. The main purpose of our work was to test how the out-of-equilibrium dynamics of the HMF model is affected by dilution. The model is then defined on a random graph of the Erdos-Renyi type with a scaling of the number of links as N Y, N being the number of sites of the lattice, with I < y ::; 2. We have analytically proven, by using replica trick, that the equilibrium statistical mechanics is the same as for the fully connected y = 2 graph.

130

Antonia Ciani, Duccio Fanelli and Stefano Ruffo

It is well known that the HMF model displays Quasi Stationary States (QSSs) when starting from non equilibrium (e.g. water-bag) initial conditions. The system relaxes on a time scale of 0(1) on such QSSs, and then perform relaxation to Boltzmann-Gibbs equilibrium on a much longer time scale, which depends on system size N . In a seminal work by Yamaguchi et al. (Yamaguchi et al., 2004) it was argued that, for homogeneous QSS , the relaxation time diverges with a power law toss rv N a with ex = 1.7. Moreover, it was found, by using a theory introduced by Lynden-Bell, that both homogeneous or inhomogeneous QSSs can be emerge depending on the features of the (water-bag) initial condition. The separation between these two types of QSSs is indeed an out-of-equilibrium phase transition (Antoniazzi et al., 2007) . We here argue that the relaxation to equilibrium of inhomogeneous QSSs scales with the exponent ex = I . This is also based on previous numerical results present in the literature (Moyano and Anteneodo, 2006; Latora et al., 2001). Besides that, we show that the lifetimes of QSSs for the diluted HMF model are well fitted by the law toss rv Na (y-l ), with ex = 1.5 for homogeneous QSS and ex = I for inhomogeneous QSS . This would indicate and exponent ex = 1.5 for the fully connected HMF model, at variance with what previously found in the literature (Moyano and Anteneodo, 2006; Yamaguchi et al., 2004). The puzzle remains to be solved, although a theoretical argument, based on a kinetic theory that we are currently elaborating, would suggest this exponent. Finally, much remains to be done for the characterization of the out-of-equilibrium phase transition for the diluted model. A first numerical analysis seems to suggest that dilution may change the phase transition from first to second order. Acknowledgements We acknowledge collaboration on this project with Julien Barre and Franco Bagnoli. This research is funded by the PRIN07 research project on Statistical physics of strongly correlated systems at and out of equilibrium and by the Galileo project of the Italo-French University: Study and control ofmodels with a large number of interacting particles.

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Barre r., Dauxois T., De Ninno G., Fanelli D. and Ruffo S., 2004, Statistical theory of high-gain free-electron laser saturation, Physical Review E, Rapid Communication, 69, 045501 (R). Barre J. and Goncalves B., 2007, Ensemble inequivalence in random graphs, Physica A, 386, 212-218 Barre L , Mukamel D. and Ruffo S., 200 I, lnequivalence of ensembles in a system with long-range interactions, Physical Review Letters, 87, 030601. Blume M ., Emery Vol. and Griffiths R.B. , 1971, Ising model for the A transition and phase separation in He 3 -He 4 mixtures, Physical Review A, 4, 1071-1077. Bonifacio R., Casagrande F., Cerchioni G., De Salvo Souza L., Pierini P. and Piovella N., 1990, Physics of the high-gain FEL and superradiance, Rivista del Nuovo Cimento, 13, 1-69. Braun W. and Hepp K., 1997, The Vlasov dynamics and its fluctuations in the 1/ N limit of interacting classical particles, Communications in Mathematical Physics 56 , 101-1l3. Caglioti E . and Rousset F., 2008, Long time estimates in the mean field limit, Archive for Rational Mechanics and Analysis, 190,517-547. Campa A., Dauxois T, Fanelli D. and Ruffo S., work in progress. Campa A., Dauxois T and Ruffo S., 2009, Statistical mechanics and dynamics of solvable models with long-range interactions, Physics Reports, 480, 57-159. Campa A., Giansanti A. and Morelli G., 2007, Long-time behavior of quasistationary states of the Hamiltonian mean-field model, Physical Review E, 76, 041117. Campa A., Giansanti A., Morigi G. and Sylos Labini F., 2008, Dynamics and TherModynamics of Systems with Long-range Interactions: Theory and experiment, ALP Conference Proceedings, New York. Chavanis P.H., 2006, Lynden-Bell and Tsallis distributions for the HMF model, European Physical Journal B, 53, 487-50 I. Chavanis P.H., Sommeria J. and Robert R., 1996 , Statistical mechanics of two dimensional vortices and collisionless Stellar systems, The Astrophysical Jou rnal, 471, 385-399. Creutz M ., 1983 , Microcanonical Monte Carlo simulation, Physical Review Letters, 50,1411-1414. Dauxois T, Ruffo S., Arimondo E . and Wilkens M . (Eds.), 2002, Dynamics and Thermodynamics of Systems with Long Range Interactions, Springer, New York. Dauxois T., Ruffo S. and Cugliandolo L.F. (Eds.), 2009, Long-range Interacting Systems, Oxford University Press, Oxford. Del Castillo-Negrete D., 1998, Nonlinear evolution of perturbations in marginally stable plasmas, Physics Letters A, 241, 99-104. Elskens Y. and Escande D., 2002, Microscopic Dynamics of Plasmas and Chaos, lOP Publishing, Bristol. Erdos P. and Renyi A., 1959, On random graphs, Publicationes Mathematicae Debrencen, 6, 290 . Eyink G.L. and Sreenivasan K.R., 2006, Onsager and the theory of hydrodynamic turbulence, Rev. Mod. Phy s., 78, 87-135.

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Henon M., 1964, L'evolution initiale d'un amas spherique, Annales d 'Astrophysique , 27, 83-91. lnagaki S., 1993, thermodynamic stability of modified Konishi-Kaneko system, Progress in Theoretical Physics, 90, 577-584. Kac M., Uhlenbeck G.E. and Hemmer P.c., 1963, On the van der Waals theory of the vapor-liquid equilibrium. 1. discussion of a one-dimensional model , Journal ofMathematical Phys ics, 4, 216-228. Latora Y., Rapisarda A. and Ruffo S., 1998, Lyapunov instability and finite size effects in a system with long-range forces, Physical Review Letters, 80,692-695 . Latora Y., Rapisarda A. and Tsallis C; 2001 , Non-Gaussian equilibrium in a longrange Hamiltonian system, Physical Review E, 64, 056134. Leoncini X., Kusnetsov L. and Zaslavsky G.M ., 2004, Evidence of fractional transport in point vortex flow, Chaos, Solitons and Fractals, 19, 259-273. Lynden-Bell D., 1967, Statistical mechanics of violent relaxation in stellar systems, Monthly Notices of the Royal Astronomical Society, 136, 101-121 . Me Lachlan R.1. and Atela P., 1992, The accuracy of symplectic integrators, Nonlinearity, 5, 541-562. Miller J., 1990, Statistical mechanics of Euler equations in two dimensions, Physical Review Letters, 65, 2137-2140. Moyano L.G. and Anteneodo C, 2006, Diffusive anomalies in a long-range Hamiltonian system, Phys ical Review E, 74, 021118. Mukamel D., Ruffo S. and Schreiber N., 2005, Breaking of Ergodicity and Long Relaxation Times in Systems with Long-Range Interactions, Physical Review Letters, 95, 240604 . Nicholson D.R., 1983, Introduction to Plasma Theory, John Wiley, New York. P.1. E. Peebles, 1980, The Large-scale Structure ofthe Universe, Princeton University Press, Princeton PluchinoA., Latora Y. and Rapisarda A., 2004, Glassy phase in the Hamiltonian mean-field model, Phys ical Review E, 69, 056113. Pluchino A. and Rapisarda A., 2007, Anomalous diffusion and quasistationarity in the HMF model, AlP Con! Proc., 965, 129-136. Ruffo S., 1994, Hamiltonian dynamics and phase transitions, Marseille Conference on Chaos, Transport and Plasma Physi cs, edited by S. Benkadda et al., 114-119. Serva M., 2009, Magnetization densities as replica parameters: the dilute ferromagnet, submitted to Physica A . H. Touchette, 2009, The large deviation approach to statistical mechanics, Physics Reports, 478, 1-69. Yamaguchi Y. Y., Barre J., Bouchet F., Dauxois T. and Ruffo S., 2004 , Stability criteria of the Vlasov equation and quasi-stationary states of the HMF model , Physica A, 337, 36-66. Zaslavskii G.M., Shabanov Y.F., Aleksandrov K.S. and Aleksandrova J.P. , 1977, A model for a phase trans ition due to nonlinear resonance of lattice vibrations, Soviet Physics JETP, 45, 315.

Chapter 4

Metastability and Transients in Brain Dynamics: Problems and Rigorous Results Valentin S. Afraimovich, Mehmet K. Muezzinoglu and Mikhail I , Rabinovich

Abstract Experimental neuroscience is often based on the implicit premise that the neural mechanisms underlying perception, emotion and cognition are well approximated by steady-state measurements of neuron activity or snapshot of images . We will unfold a new paradigm in the study of brain mental dynamics departing from the stable transient activity neural networks , as supported by experiments. Transients have two main features: (I) they are resistant to noise, and reliable even in the face of small variations in initial condition, (2) the transients are input-specific, and thus convey information about what caused them in the first place . This new dynamical view manifests a rigorous explanation of how perception, cognition, emotion, and other mental processes evolve as a sequence of metastable states in the brain and suggests the new approaches to the diagnostics of mental diseases. The mathematical image of robust and sensitive transients is a stable heteroclinic channel that is possibly the only dynamical object that satisfies all required conditions. We discuss the ideas that lead to the creation of a quantitative theory of mental human activity. For the convenience of the reader we put all mathematical details into Appendices.

Valentin S. Afraimovich Instituto de Investigacion en Comunicacion Optica, Universidad Autonoma de San Luis Potosi, Karakorum 1470, Lomas 4a 78220 , San Luis Potosi , S.L.P., Mexico , e-mail : [email protected] .uaslp.mx Mehmet K. Muezzinoglu BioCircuit Institute, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 920930328, USA, e-mail : [email protected] Mikhail 1. Rabinovich BioCircuit Institute , University of Californi a, San Diego, 9500 Gilman Drive, La Jolla, CA 920930328, USA, e-mail : mmuezzin @ucsd.edu

A. C. J. Luo et al. (eds.), Long-range Interactions, Stochasticity and Fractional Dynamics © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010

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4.1 Introduction: what we discuss and why now The imaging experiments with human brain provided the researchers with the amazing possibility of revealing the relation between mental functions and corresponding spatio-temporal patterns .

4.1.1 Dynamical modeling of cognition The tradition for understanding thought based on dynamical systems theory has its roots in the cybernetics era of the I940s . It was a time when information theory, dynamics, and computation were brought together for studying the brain (Ashby, 1954). However, with the dominance of symbolic artificial intellect and "information-processing psychology" and the absence of a good experimental technology in the I960s and I970s, dynamical-systems-based approaches were not extensively pursued . More recently, the idea that dynamics is a relevant framework for understanding cognition has become popular again . For example, Thelen and Smith (1994) described the development of kicking and reaching in infants in terms of dynamical notions such as the stability of attractors in a phase space that is defined by the body and environmental parameters. Movements to new stages in development are explained in terms of bifurcations to new attractors as a result of the change in order parameters, e.g., infant weight , body length, etc., as the infant grows . Thelen and Smith believe that "higher cognition" is ultimately rooted in these types of spatial skills learned in infancy, and thus that higher cognition will itself be best understood dynamically. They contrast their account with traditional "information processing" theories of development, in which new developmental stages are caused by brain maturation and the increasing ability of maturing infants to reason logically. The work (Port and Gelder , 1995) formul ates the general idea of continuity, namely the cognition should be characterized as a continual coupling among brain, body, and environment that unfolds in real time, as opposed to the discrete time steps of the artificial intellect. This is said to contrast with computations focus on "internal structure," i.e., its concern with the static organization of information processing and representational structure in a cognitive system . A dynamical approach means that the organization of brain structures is also dynamical and functional , i.e., not only anatomical. Thus the dynamical approach to cognition is a confederation of research efforts bound together by the idea that natural cognition is a dynamical phenomenon and best understood in dynamical terms . This contrasts with the "law of qualitative structure" (Newell and Simon, 1976) governing orthodox or "classical" cognitive science, which holds that cognition is a form of digital computation . The temporal characteristics of information flow in the brain depend on both the neural network architecture and the type of connections. In particular, inhibitory connections supported by interneurons are responsible from the spatio-temporal transient activity. On the other hand, excitatory cell assemblies, and associated connections, ensure that information "goes to the right place at the right time" (Buzsaki ,

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2006). One can say that any coarse-grain cognitive pattern (mode or representation) observed in experiments can inhibit alternative activity patterns among workspace neurons (Dehaene et aI., 2003) . At the same time, these modes sequentially inhibit each other, because only one representation is possible at any given time . As a result, mental activities, i.e., cognition, emotion, and consciousness arise in the form of sequences of representations. A dynamical system for the modeling of emotion, cognition and their interactions is a set of quantitative variables changing continually, concurrently, and interdependently over quantitative time in accordance with dynamical principles, which are embodied in a set of differential equations. Dynamics in this sense includes a very successful experience of dynamical modeling that scientists use to understand natural phenomena via nonlinear dynamical models . This experience includes a set of concepts, proofs, and tools for understanding the behavior of systems in general. An important insight of dynamical systems theory is that behavior can be understood geometrically in some projection of the state (phase) space. The behavior can then be described in terms of attractors, trans ients, stability, coupling, bifurcations and chaos , etc. Although classical cognitive science has interpreted cognition in principle as something that happens over time, the dynamical approach sees cognition as being in time, i.e., as an essentially temporal phenomenon. For example, when a dynamical model of the information (sensory) coding is created, time is included in the coding space (Rabinovich et al., 2006b) . Details of timing (durations, rates, synchronies, etc.) matter (Buzsaki and Draguhn, 2004).

4.1.2 Brain imaging Recent work in brain imaging has revealed many fundamental properties, and, in particular, the functional organization of the brain systems (see, for example (Haynes et al., 2006)) . Most of these results have been expressed in the form of averaged-in-time spatial patterns indicating the brain areas that are simultaneously activated in various emotional and cognitive states. These findings create an impression that we can get a clear fingerprint, i.e., a portrait of the specific emotion or the execution of the specific cognitive function . However, neither emotions nor cognition are frozen functional patterns . The underlying neuro-dynamics, i.e., the temporal evolution of emotion and cognition and their reciprocal link, can be extracted only by detection of the sequential brain activity in representing and translating a sequence (Keele et aI., 2003). Some brain imaging methods of today provide very high temporal resolution . First of all, Positron Emission Tomography (PET). Based on this technique, it is possible to get time-resolution about 5-10 nanosecond (Bailey et al., 2005) . Also, high temporal resolution characterizes the electroencephalography (EEG) , whose resolution is about 2 milliseconds (Rowan and Tolunky, 2003) . Unfortunately, the EEG method has a very blur spatial resolution.

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Blood-oxygen-Ievel dependent (BOLD) imaging , a Magnetic Resonance Imaging (MRI) technique, can provide whole brain coverage approximating a one-second temporal and one-millimeter spatial resolution, hence making it a strong candidate for the application of the nonlinear dynamical systems theory on the emotion , the cognition, and their interaction in the brain .

4.1.3 Dynamics of emotions The quest for the dynamical origin of emotions goes back many decades . (Franz , 1935) devoted attention to the dynamics of emotions by describing emotional sequences together with their content. He emphasized that only the balance between incorporation, elimination and retention represents the fundamental dynamics of the biological process called life. Recently, (Zautra, 2003) has developed a twodimensional approach in which both positive and negative emotions are conceptualized and measured as co-occurring simultaneous dynamical processes. Whereas behaviorism dominated the psychological and psychiatric sciences during the first half of the 20th century, cognitive science has become a central paradigm of the latter half. This new line of interest was fostered by the promise of the dynamics of cognition as an integrated and fertile approach to understanding the mind (Bruner et aI., 1990). 'The cognitive revolution" has started to inform us about the dynamics of emotion as well. Due to the great progress in the measurement and imaging technologies within the last two decades, we now have a deeper understanding of the neural substrate s of emotion and cognition. Although further progress, especially in the temporal resolution, is still needed to delineate many important details, it is clear from the current evidence that human mental life is governed by a complex nonlinear dynamical system in a non-stationary (i.e., transient) regime. The experimental insight into the matter corresponds to a period when the mature theory of dynamical systems was itself in a change. In particular, the recent shift in the interest towards complex systems has been establishing a valuable array of tools and an important motivation for physicists and engineers to tackle mental phenomena - an ancient problem that still lacks a rigorous formulation . The extension of dynamical systems research towards the complex systems analysis (Bar-Yam, 2003) promises resolutions on these qualitative aspects, that are required to apply the rich theory of dynamical systems on brain-like complex systems. It is reasonable to emphasize here three approaches that are closely connected to each other : (i) structural approach, which focuses on the specific architecture of connections (e.g., "small-world" type (Watts and Strogatz, 1998» ; (ii) informational approach that describes the dynamics of principal information flows (Maass et al., 2002 ; Majda and Harlim, 2007) ; and, (iii) approach considering functional modes (Friston et aI., 2000 ; Rabinovich et aI., 2008b) . In this work we mostly follow the last one.

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4.2 Mental modes To build a dynamical model of mental activities, one needs to choose convenient dynamical variables that should be consistent with experimental data.

4.2.1 State space As the two main players of human mental life, emotion and cognition have been under the spotlight for researchers for a long time both individually and jointly. Given the necessity that any useful analysis must be based on an accurate quantification of the investigated phenomenon, numerous attempts have been made to assess cognition and emotion. Being directly related to the processing of auxiliary information, cognition has attracted relatively more attention in these efforts, particularly in the form of task development assessing decision making tasks(Schraagen and Chipman, 2000; Hollnagel, 2003). Although, tests aiming solely at emotional quantities also exists in(see , for example, (Price et al., 1985)), the assessment of emotions or their effects have often been attempted in a cognitive theme, involving, for example, appraisal (Scherer, 1993; Thagard and Aubie, 2008), decision making (Pessoa and Padmala, 2005), or memory (Lee, 1999; Fales et al., 2008a) . The critical constructive question to tackle the problem is: What is the best medium to describe the evolution of such emotional and cognitive modes while capturing the functional complexity? To answer this question we can use the experience of the investigation of the complex systems in nature , in particular, turbulent flows (Landau and Lifshitz, 1959). It tells us that we need to know just the equations for the coarse grain liquid particles, i.e., the micro details related to molecular dynamics are not directly effective for the macroscopic description. However, such details are consequential for the parameters of a macroscopic model. Although the situation with the cortex dynamics is much more complex, the analogy suggests some insight and direction. A similar approach, a neural mass model, has been suggested in (David and Friston , 2003; Zavaglia et al., 2006). We also use this analogy , but in a different perspective (see below).

4.2.2 Functional networks The dynamical system perspective and the models of both cognitive functions and emotions is based on the assumption that the brain is a complex neural network of many dynamical sub-networks (neural clusters) working in coherence within a sequential time structure (see Fig . 4.1). The understanding of this structure underlying certain emotional and cognitive functions and realizing it on a reasonable model can be helpful in evaluating and predicting specific features in a psychiatric disorder.

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Fig.4.1 Schematic representation of three different functional modes in a complex neural netwo rk. The nodes are different brain centers . The connections between them can be activated or inhibited depending on informational inputs. Figure modified from (Young et al., 1994).

New results in the brain imaging, particularly functional MRI (fMRI) data, have revealed some fundamental properties and the functional organization of the brain systems that correlate with emotion and cognitive functions (Phan et aI., 2002 ; Dolcos and McCarthy, 2006) . Each of the brain centers that form the functional emotional subcircuit or mode is itself a very complex dynamical system with several characteristic time scales (see Fig. 4.2). These systems are open to an enormous range of neural stimulations from a wide range of brain areas . The spatiotemporal pattern of brain activity underlying an emotion is typically very sensitive to the external or internal stimuli. Amygdala, for example, receives information from both cortical and subcortical structures . These include highly processed information from the visual system , the auditory cortex , the olfactory and gustatory neocortex and the somatosensory cortex. In short, it is directly informed about each of the five senses. The amygdala also receives projections from association cortex, from the thalamus (relaying basic, unprocessed sensory signals) , the hippocampus (high level information about the relationship between objects and events in the external world), and from a range of structures that represent internal bodily states, such as hunger and thirst.

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It is important to take into account that emotions and cognition are active processes that result in specific changing of the brain organization in time and dynam-

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Cognitive

Emotional

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Fig. 4.3 An illustration of the emotion-cognition tandem including the self. All three mental activities develop in time in parallel and continuously exchange information.

ical brain's response to environmental information and representation of the self. These processes are determined by the functional (not necessary anatomical) connections between brain areas or neural circuits that participate in the execution of cognitive functions and generation of emotions. At different segments or steps of temporal emotional or cognitive process, these networks form different temporal sequences that execute and represent different emotions and cognitive functions in the brain . Dynamical variables describing emotion, cognition, and their mutual interaction form a joint workspace (or state space) . To understand how to choose such variables, we have to know some details of the emot ional and cognitive modes organ ization. Let us consider emotion. As we already pointed out, an emotion is a result of the coordinated dynamical activity of many brain areas (sub-networks). The examples of such areas are the posterior orbitofrontal cortex, anterior temporal sensory association areas and the amygdala. These areas are not all inclusive, but they may have a key role in temporal emotional processing. They sequentially interact in evaluating the sensory and emotional aspects ofthe environment for decision and action of complex behavioral coping response (see Fig . 4.3).

4.2.3 Emotion-cognition tandem As the two main players of human mental life, emotion and cognition have been under the spotlight for researchers for a long time both individually and jointly. Most theories view the emotion-cognition interaction cross-sectionally, disease-specific and based on a particular plain of knowledge (i.e. neuroanatomical, pharmacological, etc.). To illustrate our point let us refer to the current theories of Panic Disorder. The neuroanatomical theory of panic, as proposed by (Gorman et al., 2000), describes a functional relationship between different anatomical parts of the brain .

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However, the theory of anxiety and panic disorders must be multi-dimensional. In particular, the theory must handle "alarm" responses (amygdala and central gray nuclei), abnormal cognitions (striatal circuits) , and control-ling pathological behaviors (involving executive cortices.). Under stress, people lose the ability to maintain positive feelings because they have become inversely linked, causally, with negative states. This state of affect simplification has the effect of reducing informationprocessing capacity and therefore reducing emotional clarity, i.e. it is one of the key components of emotional cognition. The mental process in the joint emotion-cognition workspace is indeed a competition for resources that are needed to carry out each process . Two basic types of such finite resources are the energy (e.g., oxygen and glucose) and the information (attention and memory) . Emotional appraisal and cognitive-emotional dynamics interplay continuously in time. This interplay could have an extremely important (perhaps central) role in diagnosis. Nevertheless, little attention has been given to the temporal dimension of emotion and emotion-cognition interaction processes (Damasio, 1994; Scherer, 1999; Williams and Gordon, 2007). By focusing on nonlinear dynamical interaction between cognition and emotions, (Lewis, 2005) provides a valuable platform for integrating psychological and neural perspectives on the emotion-cognition interface . He discussed a wide-ranging and timely theoretical formulation of emotion cognition relations and, in particular, emphasized: (a) bidirectional interactions between appraisal and emotion ; (b) sub-cortical psychological and neural constituents underlying the emergence of emotion-appraisal processes; and (c) large-scale functional coupling through oscillatory neurophysiological mechanisms. The joint emotio-cognitive behavior has its basis in the dynamical coordination of many brain centers , which often participate in both emotional and cognition activity (Pessoa, 2008 ; Fales et al., 2008b ; Lewis, 2005). Due to this overlap, emotion and cognition are integrated in the sense of being partly separable (Gray et al., 2002; Raichle and Gusnard, 2002) . (See Fig. 4.3). Emotional and cognitive modes in the brain interfere and exchange information reciprocally. As we have already emphasized, this relation is driven by a competition for energy and informational resources (Raichle and Gusnard, 2002 ; Kelso, 1995; Ganguli et al., 2008; Keightley et aI., 2003) . On neurobiological level, the cognitive control of emotion follows from the direct inhibition of negative emotional modes by the centers that underlie the correct behavior (Keightley et al., 2003) . On this basis, a lot is known about the interaction of emotion and cognition. However, to the best of our knowledge, the developing theory of emotion still lacks a mathematical model that accounts for emotion-cognition interaction in time based on observable principles.

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4.2.4 Dynamical model ofconsciousness Consciousness underlines main aspects of cognitive human behavior. Understanding the neurobiological and dynamical mechanisms of consciousness has proved to be one ofthe most mysterious problems for neuroscientists. Progress in recent years, however, allows one to develop some theories of consciousness through integration of evidence from physiological, behavioral and modeling studies . Any kind of cognition (perception, image recognition, self, awareness, etc.) can be conscious. For example, a conscious awareness is the process in which external or internal stimuli are perceived, recognized and can be intentionally acted on. Experiments using simple sensory stimuli suggest that even the primary sensory areas in the brain may be involved in the process of conscious awareness (Meador et al., 2002) . Once conscious awareness is established, it is fed back in time to process the primary input (Libet et al., 1979) . It is accepted from the Williams James time that consciousness is a continuous and transient process (James, 1890). After half a century, (Gurwitsch, 1943) has put this pioneering idea on a time axis : transients link the current mental state to what came before and what is to come after. According to him, every mental state has a (finite) duration and comply with a temporal order as imposed by the continuously flowing stream of consciousness. In the context of our work, mental states correspond to conscious metastable states, and temporal order to a sequence of metastable states. Due to the provocative nature of this topic for physicists, we would like to clarify the dynamical modeling of consciousness. What are the main differences between conscious and unconscious cognitive functions? It is the ability of a human being to predict the future based on the past experience and knowledge, and to use it in the present in order to create the future. Dynamical model of any conscious cognition has to include two sub-systems, namely, the part that represents the cognitive process in the present (it can be a basic model that we discuss below), and second part that reconstructs a possible future based on the analyses of the past and using the present cognitive metastable state as initial conditions. In contrast with the first sub-system the second one resides in an imaginary universe where the time is compressed. The fast prediction together with the best possible version of future (as a result of decision making) is used for the correction of the present. Because the conscious activity is able to hold only a limited number of items at once , it only gives us a fragmentary picture of the future . This is one of the reasons why we are mistaken so often . From the neurophysiological point of view, the consciousness is a temporal process associated with low-amplitude irregular high-frequency EEG activity (20-70 Hz). In contrast unconscious states like deep sleep, coma, general anesthesia, and epileptic states show a predominance of slow rhythms, high-amplitude and more regular spectrum at less than 4 Hz (Baars , 1988). As fMRI investigations indicate the consciousness involves widespread, low-amplitude interactions of many brain centers.

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The fact that time is a critical attribute of the consciousness is clearly illustrated by recent experiments (Arzy et al., 2008) , which show that human consc ious experience is the ability to not only exper ience the present moment but also to recall the past and predict the future , a facility that is called "mental time travel." To build the dynamical model of the con sciou sness it is nece ssary to formalize the main consciousness properties (see, for example, (Laureys , 2005) .) These are: (i) Consciousness is a simulator (or game generator) that enables the brain to process experiences that are not actually occurring. Such a simulator of experience can attach to or detach from perception and physical action as nece ssary, but may be overruled by emotion and perceptions (for example , pain) . During the representation , details of perception are substituted by simulation , causing various illusions and false memories . (ii) The simulator detaches during dreaming, remembering and imagining - detached simulation of an event causes the same cortical neurons to fire that actually event does. (iii) Simulation allows complex situations to be explored many times for more efficient learning (e.g., thinking about future lines of play in a chess game, rather than actually playing each line). (iv) Consciousness is a sequential dynamical process: The constantly evolving sequence of conscious content simulate s cause and effect in a sequence of world event s. Conscious content may evolve due to an interruption of attent ion to simulate a different sequence of events . Its availabil ity depends on quality of representation, where quality is defined by stability in time, strength, reproducibility, sensitivity, and metastab ility.

The quality of simulat ion depends on the stability against diverse perturbations, on the reproducibility that is necessary for the repeated analysis of the same event s, and on the life time of metastabl e states - too short life time may not be sufficient for storing the results in memory. One of the most popular approaches to the consciousness model ing is based on the global workspace theory (Dehaene et al., 2003 ; Shanahan , 2005) . The main idea of this theory is as follow s. Informational inputs from inside the brain or outside of it (external stimul i) activate excitatory neurons with long-range axons, leading to the emergenc e of a global pattern among workspace neuron s. Accord ing to the global workspace theory, transient links among speciali zed brain processing modules and generate new information depending on incoming signals (Wallace, 2005). However, this theory cannot explain the underlying dynamical mechan ism. To model the main dynamical properties of the con sciousness we suggest here to merge the global workspace theory with the theory of the transient dynam ics of complex networks based on the winnerless competition principle, see below, and also (Rabinovich et al., 200 I, 2006b, 2008a). The most difficult problem here is to find convenience for the measurements and sufficient for the dynamical description time dependent variables . We suggest use as dynamical variables the strength s of cognitive modes that is discussed below. Nonlinear differenti al equations that describe the interaction of these variable s is a dynamical model that we are looking for.

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PCClRsp

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Fig. 4.4 The default network, as explored so far, compri ses six brain regions: Ventral medial prefrontal cortex (vMPFC), posterior cingulate / retosplenial cortex (PCC/Rsp ), inferior parietal lobule (lPL) , lateral temporal cortex (LTC), dorsal medial prefrontal cortex (dMPFC), and hippocampal formation (HF+). The approximate locations of these core areas are marked in the figure (adapted from Buckner et aI., 2008).

We would like to mention here one more problem that is directly related with con sciousness, namely the self. Despite being treated continuously as a fundamental que stion since the earliest era of psychiatry (James, 1890), the concept of "se lf" is a relatively new and very active field of study in the neuro-psychiatry. Among a few establishments in this line, there is growing evidenc e that the self-reference is a stand-alone proce ss in human brain (Kelly et a\., 2002; Wicker et a\., 2003 ; Northoff et a\., 2006 ; Broyd et a\., 2009) . It is also believed to be in close relation with a distinct brain system, called the default network. The search for a basel ine condition in brain imaging studie s has revealed the default network (see Fig . 4.4) , whose activity was not initially interpreted beyond assuming it as a "re st state." The pioneering studie s (Gusnard and Raichle, 200 I ; Raichle et a\., 200 I ; Gusnard et a\., 200 I) enlightened the neural substrates of this activity, its distinct connecti vity among other brain systems, and its cruci al role in self-referencing in both active and passive cogn itive states.

4.3 Competition -

robustness and sensitivity

In this section we discuss the main principles of mental dynamics.

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4.3.1 Transients versus attractors in brain Traditional efforts in modeling dynamical phenomena are predominantly based on the fact that dynamical systems tend to converge to stable fixed points or dynamical states (limit cycles or strange attractors) where the density of all flows (matter, energy, or information) are balanced. A dissipative complex network of agents (i.e., neurons , brain centers , etc.) with symmetric interactions gives rise to a convergent behavior involving multiple attractors (Hopfield , 1982; Cohen and Grossberg, 1983). The basins of attraction can be arranged to partition the state space to satisfy certain needs . When applicable, a Lyapunov function can be a handy tool in the analysis and design of such systems, since it translates the fixed points as extrema on an energy landscape while providing a clear image of the basins (see Fig. 4.2) . There may be certain neural phenomena, such as an associative memory (Wills et al., 2005), that would enjoy a dynamical model operated in this mode , i.e., as a map from initial conditions to the attractors. However, computing with attractors makes a very limited use of complex dynamical networks in general : once the attractor (or its vicinity) is reached, the "dynamical" nature of the system becomes irrelevant. Since attractors mark the terminal states of the process, this perspective assigns merely a quantizer role to such network that could be formulated equally and effectively by an algebraic transform. Furthermore, it overlooks the qualities of the path from the initial condition to the attractor, a phase where nonlinear systems could exploit their remarkable repertoire of behaviors. Therefore, confining dynamical models to symmetry assumption is not only unrealistic, but also rules out a continuum of opportunities within the modeling capacity of dynamical systems . Let us discuss here a contrary paradigm. A fundamental responsibility of the nervous system that shapes the dynamical processes in the brain is the internal regulation of the organism so that the available resources are distributed properly among emotions, thoughts, and actions. This constitutes the origin of the time dependence in the functional brain organization underlying the emotional or the cognitive processes: at different time steps of these processes, the participating brain centers (or even the networks of centers) can be different and the activity on anyone of them alone might not be sufficient to identify the ongoing emotion or cognitive function . A spatiotemporal competition of multiple networks of the brain centers , on the other hand, can only achieve this. In fact, the dynamic interaction of modes forms a reproducible temporal pattern that is specific to the ongoing emotion or cognitive function. Technically, such temporal aspect of encoding is the only possible scheme for the brain in order to be capable of executing a continuum of emotions and virtually infinitely many cognitive functions while possessing a finite number of elements (i.e., centers) . Thus, emotions and cognition are sequential dynamic processes resulting from the interactions of different brain subsystems (modes) and their coordination and synchronization in time.

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Fig. 4.5 Energy landscapes are illustrative tools in attractor-oriented analysis and design . The top row shows three landscapes accommodating different types of attractors: (a) a global attractor, (b) distinct local attractors , and (c) a continuous attractor. The surfaces on the bottom row are associated to two multi-stable systems where the basins of attraction are color-coded. In simple systems , one can obtain clear-cut borders as in panel (d). As the system complexity increases, the borders can become fractal (see panel (e)), which makes the computation with attractors questionable . (Figure courtesy of T. Nowotny)

4.3.2 Cognitive variables The critical constructive question to tackle the problem is the choice of the best medium to describe the evolution of such emotional and cognitive modes while capturing the functional complexity. We suppose that the specific cognitive activities can be described by the interac tion of the finite number of cognitive modes . We can describe in this way the fMRI series of snapshots taken at consecutive time instants (i.e., an fMRI movie), when a subject is busy with the execution of some cognitive job , for example, decision making . As experiments showed, the distinct networks of coherently working brain centers underlie this activity and they constitute an image of a spatial pattern in each frame . In spite of the vast diversity of these patterns across the frames, each of

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them can be considered as a sequential activity of the finite number N of the functional modes. The overall movie, then, is a reproducible spatio-temporal sequence illustrating the time-varying ordered contribution of each (observable) mode in the process. There are several ways to extract these modes from the experimental data, in particular it can be a main principal components that describe the brain activity in time (Friston et al., 2000; Koenig et al., 200 I). Thus, a snapshot of the brain activity at time t captures a combination of the cognitive variables in the form '[~ Ai(t)U (i), where U(i) is the fixed spatial form (e.g., coordinates) of i-th cognitive mode, and Ai(t) is its level of activity, e.g., the average intensity of voxels covering U (i) in the BOLD image at time . The number N of these modes depends on the level of details that we wish to describe. These modes in the modeling may include appraisal, execution of a strategy (policy), or even as broad as decision making, or default network and self-reflection activities . Despite being coarse-grain partitioning of the cognitive universe, these functions can still be distinguished by the active brain subnetworks underlying them . We denote these cognitive modes by the nonnegative time-varying variables Ai, i = I, ..., N, indicating the average activity of the corresponding sub-networks. As a mental activity, a cognitive process requires both energy and informational resources (e.g., attention and working memory) to proliferate. In the ecological model that we suggest below, we encapsulate all these resources in a real variable RA within the interval [0, I], which denotes the ratio of the resources supplied to the total demand from the cognitive process .

4.3.3 Emotional variables It is known , and is an active field of research , that distinct emotional modes have certain mental fingerprints characterized by the specific connections of brain centers involved in it. A well-explored example is the sadness network in human brain (Lewis et al., 2008). Following the same reasoning presented for the cognitive variables above, the emotion-related functional can be read-out in the form Bi(t)V(i) , where B, are the emotion modes and V(i) are their temporal and spatial activities . Since the spatial functions and are assumed to be known for all modes and fixed for all times, the temporal evolution of emotion modes are captured by Bi(t). Another critical observation is that the activity patterns of emotions are unlikely to coexist for a long time, yet overlap during smooth transitions. This observation is consistent with the general ecological principle that we expand below and adopt throughout the modeling effort. Analogous to RA , which represents the resources allocated to cognition, we introduce the positive quantity RB denoting the supplied rate of resource demand from the emotional process . The manner in which individuals receive, process, and interpret information in time is the key to understanding of the emotional representation, execution of the

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cognitive functions and the biobehavioral organization. For healthy people , these processes are sequentially organized in time following a stable (i.e., robust) onedirectional information flow. Although the execution of multiple mental processes seem feasible in healthy human brain, the critical resource of attention is available to just one of them at any given time. Therefore, a progress (in the form of a meta-stable state transition) can actually occur in only one process (cognitive or emotional) at an arbitrary time instant.

4.3.4 Metastability and dynamical principles We suppose that the reader has got some knowledge about emotion and cognition, thus it is the right time to introduce the principles that form the basis of their dynamical modeling . These principles are: (i) existence of metastable states representing the modes in the unified emotioncognition working space, (ii) structural stability of the transients that are formed by the switching of the system (brain) among these states, (iii) ecological, i.e., competition principle governing such switchings. The first item, i.e., metastability, is a general nonlinear dynamics concept, which describes states of delicate equilibrium. Metastability in the brain is a phenomenon, which is being studied in neuroscience to elucidate how the human mind process information and recognizes patterns . There are semi-transient signals in the brain, which persist for a while and are different than the usual equilibrium state (Abeles et al., 1995). The metastable activity of the cortex can also be inferred from the behavior (Bressler and Kelso, 2001) . Metastability is a principle that describes the brain 's ability to make sense out of seemingly random environmental cues (Oullier and Kelso, 2006 ; Werner, 2007) . The metastability is supported by the flexibility of coupling among diverse brain centers or neuron groups (Friston, 1997,2000; Ito et al., 2007 ; Sasaki et al., 2007; Fingelkurts and Fingelkurts, 2006) . The temporal order of the metastable states is determined by the functional connectivity of the underlying networks and their causality structure (Chen et al., 2009) . The mathematical image of a metastable state is a saddle in the state space of the system . The image of the transition between these saddles is the unstable separatrix connecting them (see Fig. 4.6). Such construction is named a heteroclinic chain .

4.3.5 Winnerless competition - structural stability oftransients Competition without a winner (or, continuously changing winners), is a widelyknown phenomenon in systems involving more than two interacting agents that sat-

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isfy a relationship similar to the popular game rock-paper-scissors or the voting paradox (Borda, 1781; Saari, 1995). The participants of such a process can become winners periodically, or, especially when the number of participants is more than three, the process can be non-cyclic and even can be terminated following a stable sequence of transients and one participant becomes the ultimate winner. (Busse and Heikes, 1980) modeled the thermo-convection in a horizontal liquid layer rotating around the vertical axis. Different convective patterns with rolls orientation 0°, 120°, and 240° periodically switch among each other (Rabinovich et aI., 2000). As a generic dynamical phenomenon, which is rare in simple systems yet common in complex ones, the sequential switching among saddles can provide concise and constructive formulations in a variety of real-world problems (Afraimovich et aI., 2008). Prototype dynamical models that are widely accepted in computational neuroscience (Wilson and Cowan, 1973), and ecology (Lotka, 1925) have been shown to exhibit a transitive winnerless competition for a fairly broad range of parameters (Rabinovich et aI., 2006a; Huerta and Rabinovich, 2004; Afraimovich et aI., 2008) . For mathematical description, see Appendix I. Since the time spent within a saddle vicinity is inversely proportional with the (logarithm of the) noise level (variance) (Kifer, 1981; Stone and Holmes, 1990), the characteristic time of such a transient varies in a wide range . In a stable heteroclinic sequence, the order of temporal winners is fixed and the noise is able to accelerate the process. Thus, the noise must be large enough to maintain the switching behavior at the desired rate (on average) and small enough to keep the heteroclinic nature of system on track, i.e. to maintain stability. Embedding a structurally stable heteroclinic skeleton in the phase space (see Fig. 4.6) results in a channel which routes the volume around it along the imposed sequence . Within this volume, the system behavior is reproducible with finite accuracy. Since the location of the saddles conveys input-specific information, which is activated the corresponding metastable states and their sequential order by the strength and the topology of the connections, then the system becomes both noiserejecting and input-sensitive (due to following the stimulus-specific channel) simultaneously. The key mechanism underlying the winnerless competition in the Brain

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is inhibition, which is known to exist in neural systems at micro- and macroscopic levels (Aron, 2007 ; Kelly et al., 2008 ; Jaffard et al., 2008; Buzsaki et al., 2007). There is substantial experimental support (Abeles et al., 1995; Jones et al., 2007 ; Rabinovich et al., 2008a) (also outlined below) that metastability and transient dynamics are the key phenomena that can contribute to the modeling of cortical processes and thus yield a better understanding of a dynamical brain.

4.3.6 Examples: competitive dynamics in sensory systems The way sensory signals are processed in animals is through the activation of specific groups of neurons, which are determined by both the quality and the quantity of the stimulus . The intrinsic dynamics of neural networks produces firing pattern s that encode the informational input and relays them to further processing centers upstream . In general, this code is spatio-temporal and sequential. Transitive winnerless competition is a sound mathematical object to explain these reproducible transient phenomena. The competitive nature of neuronal processing in general provides another clue that a winnerless competition may be underlying a sequential activity. When characterizing the dynamics of biological sensory systems, the variability observed in the recordings from the neural activity appears as a noise, but is not actually a noise component. It should be viewed as fragments of the competitive sequent ial activity. Such phenomenon has been observed recently in experiments with olfactory and gustatory sensory systems (Jones et al., 2007; Mazor and Laurent, 2005). An analysis of the response in the rat's gustatory cortex to prototype tastes reveals that a reproducible taste-specific switching pattern is triggered shortly after the stimulus is presented (Jones et al., 2007) (see Fig. 4.7). Experimental observations in the olfactory systems of locust (Stopfer et al., 2003) and zebrafish (Friedrich and Laurent, 200 I) reveal odor- and concentration-specific, reproducible and transient patterns of activity in principal neurons . Here the odor representations are spatio-temporal successions of states, or trajectories, each corresponding to one odor identity and one concentration (Stopfer et al., 2003) (also see Fig. 4.8). The rules governing the WLC resemble the competition of different species for the environ-mental resources (Afraimovich et al., 2008) . This analogy links the WLC concept with brain functions via a popular standpoint in brain research . This ecological perspective to the brain organization suggests that it is the strict competition for finite resources among brain networks that maintains the collective dynamics. For example, a competitive activity of multiple brain areas (Fox et al., 2005, 2007) are fundamentally important for thinking, in particular, for sentence comprehension (Just and Varma, 2007) . These results provide an indirect support for the argument that the WLC is widespread in the nervous system . The WLC opens a new alley in modeling and analysis of large scale cortical processes with temporal resolution.

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4.3.7 Stable heteroclinic channels The mathematical object that represents a reproducible transient activity is the Stable Heteroclinic Channel (SHe), consisting of saddle sets, their vicinities, and the pieces of trajectories connecting them. A SHC is characterize d by two properties: (i) the conditions on the structural stability of the SHC, and (ii) the relatively long (but finite) passage (or exit) time that the system spends in the vicinity of a saddle in the presence of moderate noise.

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Let us consider a channel that consists of saddles each having one-dimensional unstable manifolds, i.e., a separatrix leading to the next saddle . To obtain the condition on the channel's stability, we must consider elementary phase volume in the neighborhood of each saddle that is compressed along the stable separatrices and stretched along the unstable separatrix . Let us order the eigenvalues of saddle i as

Ai > 0 > Re { Ad}

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(4.1)

4 Met astability and Transients in Brain Dynamics

153

The number (4.2) is called the saddle value. If Vi >

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then the compression along the stable manifolds dominates the stretching along the unstable manifold, and the saddle is called as a dissipative saddle . If all saddles in the heteroclinic chain are dissipative, then the trajectories in their vicinity cannot escape from the chain, providing stability. In the absence of noise, the state vector approaching to a saddle along a stable manifold is confined to the neighborhood of the saddle indefinitely. The exit from a saddle 's neighborhood is possible only due to a strong enough perturbation . The dependence of the exit time on the noise level was studied in (Kifer, 1981) and (Stone and Holmes , 1990). A local stability analysis around a saddle fixed point results in the relation

.= Ai1 (1) T11f '

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where '[;i is the mean time spent in the neighborhood of saddle i (provided that initial points belong to the stable manifold), i.e., the life time of a metastable state, and 111 1is the level of noise. In the framework of a specific model , one can derive the inequalities on the model parameters using the conditions (4.2) and (4.3) that guarantee the stability of heteroclinic channel.

4.4 Basic ecological model 4.4.1 The Lotka-Volterra system The competition within and between cognitive and emotional modes can be described by the Generalized Lotka- Voterra (GLV) model (Lotka, 1925), given by (4.5) Here X i ;::: 0 is the i-th competing agent , E is the input that captures all (known) external effects on the competition, '[; is the time-constant, pi's are the increments that represent the resources available to the competitor i to prosper, qJij is the competition matrix , and 11 (z ) is a multiplicative noise perturbing the system . The system (4.5) has many remarkable features, see Appendix 2. Depending on the control parameters' ratio, this model can describe a vast diversity of behaviors. When connections are nearly symmetric, i.e., qJij ;::::: qJji, two or

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154

more stable states can co-ex ist, yielding a multi-stable behavior - the initial condition determines the final state. When the connections are strongly non-symmetric, a heteroclinic contours or limit cycles in their vicinities can emerge. Dynamical chao s can be observed in this case (Muezzinoglu et al., to appear) . A specific kind of dynamical chaos , where the order of the switching is deterministic, but the life-time of the metastable states is random , can also be observed (Varona et al., 2004) . We think that such reproducibility of metastable states ' order, despite the irregularity in timing, can be interesting for the processing of observed data. For a given model , the values of the control parameters that ensure the stability of the transients can be obtained from inequalities to be derived from (4.1) and (4.2). In (Afraimovich et al., 20 I0) such conditions have been generalized in the case of weakly-interacting subsystems like (4.5). As we already discussed , cognition and emotion are strongly connected. Nevertheless, it is reasonable to suppose that the modes within one family are more strongly connected than the modes between these two families . That means, one can consider that, one family does not "destroy" the dynamics of the other family, but modulates it. In particular, cognition support emotional equilibrium, whereas emotion excites or inhibits cognition. Therefore, it is natural to describe their interaction based on coupled subsystems of type (4.5). Taking also into account the dynamics of resources, we should write a third set of equations, describing the resource modes (i.e., attention, memory, and energy) . It is important to emphasize the special role of attention in this interaction: it selects the sensory cues that are critical for current decision making process . Based on experimental evidence, the dynamics of attention can also be described by a competition among informational entities . For the sake of simplicity, let us consider that these entities are total emotional i3 = L~l BJt) and total cognitive A: = L~, Ai(t) activities . Finally, we write the model in the following form .

"fA ' :tAi(t) = Ai(t) [(}i(I,B,D) ·RA-

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(4 .8)

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(4 .9)

The nonnegative variables Ai and Bi, as described above, correspond to the cognitive and emotional modes , the union of which are denoted by A and B respectively . The proposed model is merely a formul ation of the competition within and among these two sets of modes . Both of these modes are open to the external world through the quantities / and S, denoting the cognitive load and the stressor, respectively, and

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D is the control parameter characterizing the medication. The coupled processes evolve on time scales determined by the parameters 'LA and 'LB. Both processes are open to the brain noise, which appears as the multiplicative perturbation 11 (t) in the equations. The variables RA and RB characterize the resource dynamics, where rf!A and rf!B control the level of competition for resources . The competition within cognitive and emotional modes are regulated by the selfexcitations (J and S, and by the competition matrices p and c, and by the time constants 'LA and 'LB, respectively . Note that the GLV equations modeling the two coupled processes has a rich repertoire of dynamical behaviors. Therefore, the choice of the triples ((J , p, 'LA) and (S, ~ , 'LB) determines not only the quantitative attribute s (i.e., time scales, transient and/or steady-state characteristics), but also the qualitative nature of each behavior. In fact, the cognitive and the emotional brain processes have different qualities : the former is usually characterized as a sequentially ordered brain activity advancing on a regular pace, whereas the latter is a highly variable, fast, and sometimes unpredictable activity. Based on these observations, the suitable operating regime for a (healthy) cognitive process is the stable heteroclinic chain. There is no particular constraint posed at this point on the quality of emotional dynamics ; it can follow also a heteroclinic sequence with a short switching period, a recurrence behavior or a strange attractor. Cognitive and emotional brain processes have different qualities : the former is usually characterized as a sequentially ordered brain activity advancing on a regular pace , whereas the latter is a highly variable, fast, and sometimes unpredictable activity. Based on these observations, the suitable operating regime for a (healthy) cognitive process is the stable heteroclinic chain . There is no particular constraint posed at this point on the quality of emot ional dynamics; it can follow also a heteroclinic sequence with a short switching period , a recurrence behaviour or a strange attractor.

4.4.2 Stress and hysteresis In this example, as adopted from (Rabinovich and Muezzinoglu, 2009), an auxiliary stressor S triggers these emotions, which in turn disrupts an ongoing cognitive sequence. Thus, the simulation demonstrates the feed-forward chain of events : S =} negative emotions =} cognitive disruption. Let us consider N = 5 cognitive modes and M = 5 emotional components. The multiplicative perturbation 11 (r) is a white noise with variance 10- 8 and 10- 3 for the cognitive and the emotional dynamics, respectively, and the time constants are 'LA = 'LB = 20. Without loss of generality, we prescribed the finite heteroclinic sequence of saddles el ----+ e2 ----+ ••• ----+ es for the emotional modes. The mode es is a stable attractor (i.e., without any unstable manifold so that the system is confined to the vicinity

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of es once it enters its domain of attraction). This state marks the terminal cognitive mode, such as the execution of a certain coping strategy, whereas the preceding modes denote the cogn itive tasks that lead to this resulting activity. They could be named, for instance, as perception, appraisal, evaluation, and selection, in their order of appearance in the sequence. The feasible values of Pij that can establish the desired heteroclinic skeleton in the A network constitute a broad continuum in the parameter space. A set of sufficient conditions that determine a part of this region in the form of simple inequalities on o, and Pi) can be found in (Afraimovich et al., 2004b) . Following these conditions, we set Pii = 1.0 for i E {I, ... , 5}, Pi-l ,i = 1.5 for i E {2, ... , 5}, Pi,i+l = 0.5 for i E {1, ... ,4 }, and Pij = Pj -l ,j+2 for j E {2,3,4} and i rf:- j -I ,j,j+ I. In this illustration, the five emotional modes were organized as a heteroclinic sequence, yet as a cyclic one by introducing es ---+ el transition. We note that we do not necessarily name the emotional components individually, but interpret their mean activity as the degree of anxiety, a negative emotional state. In this respect , the precise dynamical quality of the emotional network is not of primary consideration in our design; for the sake of our illustrations, the emotional behavior could have been realized simply as a limit cycle , or as a strange attractor. The ~ij was evaluated as done above for Pij , yet taking into account the es ---+ el transition, which results in ~s ,s = 0.5. We disregard any transient drug effect in the simulations, thus assume that D, thus both matrices P and ~ are fixed. These matrices configure the competition within the cognitive and the emotional modes . The interaction between them are regulated by the choice of the increment functions a and S, as well as through the resource competition (4.8) and (4.9) . All five increments o, in the cognitive process were modeled as I - L=l Bi(t), i.e., inversely proportional to the total (negative) emotional activity. The increments Si for the five emotional modes were considered as independent of the cognitive activity in this example; they were all equal to the externally applied stressor quantity S, which was assumed to be non-negative. The resource competition RA vs. RB is regulated by Eqs. (4.8) and (4.9) with parameters CPA = CPB = 0.3 and random initial conditions. With the selected parameters, the integration of the ordinary differential equa tions were performed by the Milstein approximation. The results shown in Fig. 4.9 were obtained. The figure illustrates the suppression of and delay in the cognition due to the emotional activity, which is induced by an external stressor. An interesting prediction that can be derived from the model is the contrast in the switching regimes of the total activity in the cognitive and the emotional network during the rising and decay periods of S. This can be better observed in Fig. 4.10 .

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4 Metastabi lity and Transients in Brain Dynamics

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Fig. 4.9 Simulation of the stresso r-induced emotion-cognition interaction generated by the proposed ecological model. The bottom curve is the tempora l profile of the stressor, which triggers the emotional activity depicted on the second row. Arousal of these emotional modes affect the ongoing cognitive activity negatively, as seen on the first row. This effect is due to two couplings between the cognitive and emotional processes : (i) the direct interaction encoded in the cogni tive increme nts (J (see text), and (ii) through the resource competition, whose trace is shown on the third row.

4.4.3 Mood and cognition Such interaction in the absence of the stress depe nds on the psychiatric profile of the individuals. The individuality is fixed by the value of parameters in the framework of our mode l. In fact all we need to know is j ust a ratio of charac teristic timesca les, the level of excitation, and the degree of the non-sy mmetry of the inhib itory connections. Let us first consider an average (i.e., healthy) person. Suppose that a cog nitive activity, in the form of a transient process, is to be finished in the solution of an internally-formulated problem. We represe nt this cognitive process by five modes, the mood (in the absence of the stress) by three modes, and an emotional memory by two modes. The mode l parameters are indicated in the caption of Fig . 4.9. The initia l conditions are as follows : At t = 0 all but one cognitive modes are equal to zero . One emotional modes, namely the one representing a negative mood, is not equal zero, whereas the other is set to zero . The memory mode, reminiscent of a positive image, is slightly larger than the intensity of the memory mode , which we assume to be representing a negative image . The phase portrait of the considered dynamical process is given in Fig. 4.11, and corre-

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4 Met astability and Transients in Brain Dynamics

159

sponding time series in Fig. 4.12 . One can see that, at the first stage of the process , the positive reminiscences suppress the negative emotions and support a cognitive activity. Then, the negative remin iscences push out the positive ones and negative emotions become prevalent (see also (Varona et al., 2004)). As a result, the cognitive activity goes down. However, the cognition still controls emotion partially, and eventually negative reminiscences vanish, yielding the success of the cognitive job. Of course, for another psychiatric profile, the interaction between the mood and cognition could be different. For example, the reminiscences could be exchanging with each other chaotically as well as the emotion. In some region of the control parameters, negative reminiscences can generate a depression lasting for a very long time . The discussed dynamical model is a viable mathematical description of the mental brain dynamics delineating the crucial elements of emotion, cognition, and attention memory. The analysis and simulation of this self-perturbating system can reveal different characteristics and interaction schemes of the two processes. This should be a scenario-based approach, where the exact emotional and cognitive modes , as well as all known (or investigated) interactions are encoded in the model.

Jo:lffi I ~ ~ ~ = ~'l 0

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time Fig. 4.12 A self-induced emotion-cognition interaction as generated by the proposed ecolog ical model. The interpretations are as in the previous figure. Here a certain cognitive mode, the A4 denoted by the green curve , triggers the emotional activity, which suppresses the cognitive activity in return. The emotional activity is time-limited as encoded in ~ (see text); the cognitive process returns to back its track after this period .

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What aspects of the proposed dynamical model of the emotion-cognition interaction are specific to an individual or to any given psychiatric disorder? We can think of two answers to this important question : (i) the coupling schemes connecting emotion +-* cognition +-* memory processes; and (ii) the time constants setting pace for each process . The former determines the scheduling of the components in the course of the mental activity. For instance, the appraisal is likely to trigger a coping strategy, which is shaped by the current emotional state. The co-occurrence or a particular sequential order of these processes in time forms a non-coincident pattern that may be unique for an individual and/or may be indicative of a disorder. The timing of these episodes is another mental characteristic. For instance , prematurely terminated appraisal may result in an improper coping strategy, or a coping may not arise at all if the appraisal gets stuck, occupying the stage indefinitely .

4.4.4 Intermittent heteroclinic channel A simple SHS does not suffice in describing some psychiatric disorders, because the cognitive chain of metastable states is sequentially interrupted by stereotyped emotional behavior. An example of such disorders is a well known obsessive-compulsive disorder (OCD). It is a type of anxiety disorder that traps the patient in endless cycle s of repetitive feelings , unwanted thoughts, and unwanted repetitive acts, which the sufferer actually realizes as being foolish, but is (s)he is unable to resist compulsive rituals (Huppert and Franklin, 2005 ; Hollander et al., 2007) . Compulsive rituals are performed in an attempt to prevent the obsess ive thoughts or make them go away. Although the ritual may make the anxiety disappear temporarily, the person must perform the ritual again when the obsessive thoughts return . This OCD cycle can progress to the point of taking up hours of the person 's day and significantly interfere with normal activities . People with OCD may be aware that their obsessions and compulsions are senseless or unrealistic , but they cannot stop themselves. Here we suggest a mathematical image of cognitive-emotion interaction that corresponds to OCD. It is an intermittent heteroclinic channel. Here we wish to notice that, from the dynamical point of view, normal emotion - cognitive activities are happened when the corresponding control parameters (see (Afraimovich et al., 20 I0)) are placed in a narrow corridor between pathological stable states, like deep depression, coma and so on, and edge of chaos (strange attractor) i.e. such normally mental activities are close to the boundaries of different instabilities. It is important for the understanding, diagnostics and treatment that different disorders are characterize by different levels and features of dynamical instabilities . Quantitative description of such instabilities can be done in terms Lyapunov exponents (Afraimovich et al., 20 I0). According to our computations the OCD is described by very specific dynamical object i.e. Intermittent Heteroclinic Channel (lHC) . In a IHC a chain of metastable cognitive modes sequentially interrupted by OCD metastable state that we named ritual for the sake of simplicity. The model suggested here, i.e., a system of three

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4 Met astability and Transients in Brain Dynamics

ritual

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sets of GLV type equations, is able to describe most manifested anxiety form s as illustrated in Fig. 4.13 . Interaction withinlbetween cognitive and emotional modes and related dynamics are regulated by the ratios of self-excitations, the strength of competition, and external inputs (Afraimovich et al., 20 I 0). Depending on the psychiatric application, one must choose combination of parameters to be adjusted . This choice can change not only the quantitative performance of the model (i.e., time scales, transient and/or steady-state characteristics), but also the qualitative nature of solutions (e.g., steady-state vs. transient, chaos vs. regularity).

4.5 Conclusion First of all, we have to say that, we do not know the origin of a thought. However, we hope that, we delivered to the reader our vision how to build a dynamical theory of the cognition and emotion, and how to develop corresponding models . Our focus here was on the specific mental phenomenon: emotion-cognition interaction. Principles : Because of the brain's exclusive complexity and necessity to get a successful description of a continuum of different cognitive and behavioral brain functions, it is possible to build such a theory just in case if we can find a reasonable level of abstraction. We think that this abstraction should not be at the level of neuronal groups , not even of brain centers, but of transient cognitive modes and intermediate entities , particularly the metastable states. The main principles that have to be a

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basis of such theory are: (i) robustness of brain dynamics against noise, (ii) reproducibility of mental activities in similar environment and personal shape, and (iii) ecological principle - competition of mental modes for the energy and information recourses. In the framework of such theory normal mental activities can be viewed as emergent properties of the dynamics of complex functional networks, whereas mental disorders can be viewed as distortions of these dynamical networks . Quantification of mental processes based on types of dynamical distortion is a new direction towards diagnostics, modeling, and tackling mental disorders. Analyses of transient patterns of mood have emerged as a new field in psychiatry. The creation of this new field can be called a "Project for a Dynamic Psychiatry" as a paraphrase of Freud's "Project for a Scientific Psychology" (Freud, 1895). Dynamical/mages: Under basal resting conditions, most healthy neural sub-systems demonstrate irregular complex dynamics that represent weakly interacting multiple mental processes that operate over multiple time scales. These processes prime the brain for an adaptive response , making it ready and able to react to new cognitive information or internal and external psychological perturbations. This reaction in a normal situation leads to a robust and reliable condition. The dynamical principles that we have discussed above provide us an understanding of the origin of the robustness and the reliability of mental behavior. This behavior is a result of temporal brain activity that is open complex nonequilibrium system with finite energy and informational recources. We have shown that the competition between different modes, each functionally depending on incoming information, solve the fundamental contradiction between robustness and sensitivity to weak informational signal. The dynamical image of such activity in state space of the corresponding dynamical model is a Stable Heteroclinic Channel (SHC) that is a sequence of metastable states, whose vicinities are connected by unstable trajectories, i.e., separatrices. We have analyzed here the simplest variant of heteroclinic channel i.e. with one dimensional unstable separatrices. As our preliminary computer experiments indicate a heteroclinic channel that consist of saddles with many dimensional unstable separatrices can be nevertheless relatively stable. Everything depends on the values of the positive Lyapunov exponents: if one of them is clearly dominant, it decisively determines the departing direction, consolidating the robustness of transient behavior. When there are several positive Lyapunov exponents of the same order, the functionally-oriented cognitive behavior can be distorted (Afraimovich et aI., 2008). From this point of view, certain psychiatric disorders involving emotion and cognition can be distinguished by the dynamical parameters quantifying these exponents. Dynamical Characteristics in Clinic: Recent clinical observations have shown that a mental disorder (as a mental health) cannot be described by analysis of the mood in short period of time. Such mental disorders are dynamical. For example, the authors of a recent paper, (Katerndahl et aI., 2007) have asked the basic questions about the levels of mood variability between healthy and disordered people . They analyze

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and compare the dynamic patterns of hourly mood variation among newly diagnosed primary care patients with either major depressive disorder or panic disorder compared with patterns in patients without either disorder. Their premise is that, in "normal" persons, mood states might vary over time in a dynamical pattern similar to that seen for heart rate. Heart rate variability in normal persons has been shown to have some level of chaoticity. Normal controls displayed a circadian mood pattern with chaotic dynamics. Depressed subjects did not show a circadian pattern of mood variation. Panic disorder subjects had variable patterns of mood dynamics but generally did not match the combination of circadian pattern and dynamical chaos seen in controls. Taken together, these results suggested that healthy individuals (i.e. without a disorder) might experience a normal circadian rhythm in mood with superimposed mood changes as the chaotic response to multiple social or biological stressors during a day, while either the circadian rhythm or the responsiveness to stressors is impaired in those with mood or anxiety disorders. Our efforts that focus on new dynamical models of emotion and cognition in fact suggest to clinicians new approaches for recording and analyzing data, and, furthermore, for diagnosis. Now is a time that we can use functional brain imaging to identify patterns of brain activity in response to selected stimuli, and gene mapping to identify genetic features associated with specific mental disorders. If we can add transient brain dynamics to psychiatry toolbox, the ability to identify and classify mental and behavioral disorders will be greatly enhanced. Acknowledgements The authors are grateful to Alex Bystritsky, Ramon Huerta, Alan Simmons, and Irina Strigo for multiple constructive discussions. This work has been supported by U.S. Office of Naval Research through the grant ONR-NOOOI4-07-I-0741. V. A. was partially supported by PROMEP grant UASLp·CA21. The authors thank Thomas Nowotny for providing Fig . 4.5

Appendix 1 Here we present a mathematical background of the ideas and results discussed above, (see also (Rabinovich et al., 2008b)).

Stable heteroclinic sequence We consider a system of ordinary differential equations (4.10)

where the vector field X is C2-smooth . We assume that the System (4.10) has N equilibria Q I , Q2, ... , QN, such that each Qi is a hyperbolic point of saddle type with one-dimensional unstable manifold WQi, that consists of Qi and two "separatrices",

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the connected components of WQi \ Qi which we denote by r; + and also that r; + C WQi+ l , the stable manifold of Qi+l .

r;- . We assume

Definition At. The set r := U~ I Qi U~11 r; + is called the heteroclinic sequence.

Denote by A? ), ... , A2 ) the eigenvalues of the matrix ~X I Qi' By the assumption above one of them is positive and the others have negative real parts . Without loss of generality one can assume that they are ordered in such a way that A(i) > 0 I

> Re::l'''2(i) -> ReA(i) > ... -> ReA(i) 3 d

We will use the saddle value -ReAdi) Vi

=

A(i)

(4.11)

I

The saddle Qi is dissipative if Vi > I . It means that a displacement from the stable manifold of Qi becomes much smaller after going through a vicinity of Qi (Shilnikov et al., 1998) and (Shilnikov et al., 200 I). Definition A2. The heteroclinic sequence T is called the stable heteroclinic sequence (SHS) if (4.12) Vi > I, ... ,N. It was shown in (Afraimovich et al., 2004a,b) thatthe conditions (4.11) and (4.12) imply stability of F , in the sense that every trajectory started at a point in a vicinity of QI remains in a neighborhood of T until it comes to a neighborhood of QN. In fact, the motion along this trajectory can be treated as a sequence of switching between the equilibria Qi, 1,2, ... , N . Of course , the condition r; + C WQi+ 1 indicates the fact that the System (4.10) is not structurally stable and can be only occurred either for exceptional values of parameters or for systems of a special form . As an example of such a system one may consider the generalized Lotka- Volterra model (4.5) (see (Afraimovich et al., 2004a,b». In the space of the generalized Lotka- Volterra models, the occurrence of heteroclinic connections is a structurally stable event.

Stable heteroclinic channel We consider now another system, say,

x = Y(x),

x

E jRd ,

(4 .13)

that also has N equilibria of saddle type QI , Q2, ... , QN with one-dimensional unstable manifold WQi = r; + Ur; - U Qi, and with Vi > I, i = 1, ... , N . Denote by Vi a small open ball of radius e centered at Qi (one may consider, of course, any small

4 Metastability and Transients in Br ain Dynamics

165

neighborhood of Qi) that does not contain invariant sets but Qi. The stable manifold WQi+ ! divides Vi onto two parts: V/ containing a piece of Ij + and another one, V i-.

Assume that Ij + n ir;

I

:f. 0 , i

= 1,2, ... ,N -

I , and denote by Ij ;+I the connected

r;\ Ujf-iVt containing Qi. We assume that Ij ;+1 coincides with the connected component of r;\ Viti containing Qi and that Ij ;+1 n tr = 0 if

component of

j:f. i, i + I . Denote by 0 0 (Ij ; +I ) the 8-neighborhood of Ij ;+1 in jRd.

uI:,1

)UJ=I

v:

Definition A3. Let V(c , 8) = 0 0 (Ij ;+1 We say thatthe System (4 .13) has a stable heteroclinic channel in V (e, 8) if there exists an open set V \ V~ of initial points such that for every Xo E V there exists T > 0 for which the solution x(t ,xo), 0 :::; t :::; T, of (4 .13) satisfies the following conditions: i) x(O,xo) = xo, ii)for each 0 :::; t :::; T, x(t ,xo) E V(c, 8), iii)for each I :::; i :::; N there exists t, < T such that X(ti'XO) E

vt

Thus, if E and 8 are small enough then the motion on the trajectory corresponding to x(t ,xo) again can be treated as a sequence of switching along the pieces Ij ; +1 of unstable separatrices between the saddles Qi, i = I, ... ,N. It follows that the property to possess a SHC is structurally stable: if a System (4 .13) has a SHC then a C l - close to (4 .13) system also has it. We prove this fact here under additional conditions. Denote by Ij t c the inter-

vt

section r i i+1 n It is a segment for which one end point is Qi while the other one , say Pi, belongs to oV/, the boundary. Let W/ foc := WQi n Vi, the piece of the

stable manifold of Qi and Vi(Y) := 0y(W/foJ n V/ ' Y < e, where Oy(B) is the yneighborhood of a set B in jRd. The boundary OVi(y) consists of W ; loc: a (d - 1)dimensional ball , Bi, "parallel to" W ; foc and a "cylinder" homeomorphic to Sd-2 x I where Sd-I is the (d - 2)-dimensional sphere and I is the interval [0, 1]. Denote by Ci( y) this cylinder. The proof of the following lemma is rather standard and can be performed by using a local technique in a neighborhood of a saddle equilibrium (see (Shilnikov et al. , 1998,200 I; Afraimovich and Hsu, 2003)).

Lemma At. There is 0 < Co < I such that for any e < Co and any I :::; i :::; N there exist Ci < E and I < J1i < Vi for which the following statement holds: if fi :::; Ci Xo E Clfi) then (4 .14) where "dist' is the distance in jRd, t' > 0 is the time and x( 1:i,XO) is the point of exit of the solution of (4.13), going through xo,from

vt

rt.;

A segment has two end points: one of which is ~ and another one, say Ri+1 E oVitl ' Fix C < Co.

Lemma A2. There exist members 11 < )1, then : i) there is t;

x, >

> 0 such that x(t; ,xo)

I and v;

E OVitl'

> 0 such that ifxo E

Oy;(~) , 0

<

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Valentin S. Afraimovich ct al.

ii) the following inequality holds (4.15)

iii) every point x(t ,xo). 0 ::; t ::; ti belongs to the (KiYi)-neighborhood of

r;1+

I.

The lemma is a direct corollary of the theorem of continuous dependence of a solution of ODE on initial conditions on a finite interval of time. Now, fix the numbers }1i , fi satisfying Lemma AI. We impose a collection of assumptions that will guarantee the existence of SHe.

Assumption A N. The point RN E CN(fN) \ (BN U W~!oJ. The lemma A2 implies that there exists YN -I < YN-I such that X(tN-I ,xo) E CN(fN) for every Xo E 0 YN_I (PN-I) . Fix a number £N-I < fN-I such that (4.16)

Assumptlon Av .i] . The point RN-I E CN-I (£N-I) \ (BN-I UW~-lIoJ· Again , there exists 0 < YN -2 < YN-2 such thatx(tN-2 ,xo) E CN-I (£N-I) for every Xo E 0 YN_2 (PN-2) . Fix a number £N-2 < fN-2 such that (4.17)

Continuing we come to

Assumption Ai. (i = I, ... , N - 2) The point Ri; I E Ci; I (£N-I ) \ (Bi+ I U WJ+l!oJ . After that we choose It < fi such that (4.18) where

Yi is fixed in such a way that x(ti ,xo)

E Ct+1 (£i+l) provided that Xo E 0:n(Pi) .

The following theorem is a direct corollary of Lemmas A I, A2, Assumptions A N-A2 and the choice of the numbers £i, Yi.

Theorem AI. Under the assumptions above the System (4.13) has a SHC in V(f , 8) where

ut

8 = maxKiYi and the set of initial points (see Definition A2) U = 0Y1 (PI) n

Corollary AI. (Rabinovich et a1., 2008b) There exists a > 0 such that every system x=Y(x)+Z(x)

167

4 Met astability and Transients in Brain Dynamics

where IIZllcl < (J also has a SHC in V(s, 8), maybe with a smaller open set U of initial points. The proof of the corollary is based: i) on the fact that the local stable and unstable manifolds of a saddle point for an original and a perturbed system are CI-close to each other; ii) on the theorem of smooth dependence of a solution of ODE on parameters and iii) on the open nature of all assumptions of Theorem A I . The conditions R, E C, (et+ I) with ei « I looks rather restrictive, generally. Nevertheless, for an open set of perturbations of a system possessing a SHS, they certainlyoccur. Theorem A2. If a System (4.13) has a SHS then there is an open set 'f,f in the Banach space of vector fields with the C r-norm such that the system

i=X(x)+Z(x) has a SHC, for every Z E 'f,f . See the proof in (Rabinovich et al., 2008b). So, an orbit stays in a SHe until it goes out of a vicinity of QN. Then it could go to an "inner part" of the phase space and after some time come again to the same or other SHe. Some numerical simulations show that such a behavior indeed occurs in GLYM. GLYM. It is even more observable for the case of intermittent heteroclinic channels. Qualitatively it is very similar to the behavior of a trajectory of a Hamiltonian system processing sticky sets (Zaslavsky, 2005 ; Afraimovich et al., 2004a) . It is an interesting problem to study the similarity and difference of statistical features for both cases.

Appendix 2 We are going to investigate a transient multispecies competition in the framework of the following form of GLYM: (4.19) Here each ai(t) ?: 0 represents an instantaneous density of the i-th specie 's, ?: 0 is the interaction strength between species i and i . (Ji(E) is the growth rate for species i that depends on the environmental parameter E «(JJ Pii) is the overall carrying capacity of species i in the absence of the other species ; T/i is environmental Pij

noise. The product a, [(Ji(E) + T/i(t )] determines the interaction of the species i with the environment. We will consider a non-symmetrical species interaction,

Pij

-I- Pji .

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A Generalized Lotka- Volterra model in the absence of noise

°

The system (4.19) provided that 11i(t) == is a simplest lattice dynamical system with quadratic inhibitory couplings. It has many remarkable features . Let us list some of them (Afraimovich et aI., 2004b) . (i) It is a dissipative system that is simple to see fixing the ball of dissipation 1l

La] = R 2 , where R is large enough . j

(ii) It has n invariant hyperplanes a, = 0, i = I, ... .n, and many invariant linear subspace formed by the intersections of some of them. (iii) Because of that, it is simple to calculate the eigenvalues of the matrix of the linearized at the equilibrium S, = (0 0 · · · ()j 0 · · · ) system (we set ()j(E) = ()i). They equal to - o, and ()j - Pji()i, j = I, ... , n, j i- i. Thus , if all of them are negative then S, is the stable node, and if ()jl - Pj;i()i > for some ji then S, is the saddle point. (iv) Assume that it is true and consider a restriction of the system onto the invariant plane ~ji = {aj = O}.lt has the form

°

n

f hl ,i

The point (()i, 0) is the equilibrium saddle point , and the system has no other equilibria in the positive quadrant provided that I - Pijl P N i- 0. Therefore, the unstable separatrix has no choice but to go to the equilibrium (0, ()jl) that is the stable node, and the phase portrait in the positive quadrant is very simple : all trajectories except the stable separatrices of the saddle (()i , 0) and the unstable node in the origin go to the node (0, ()jl ) as t ----+ +00.

Stable heteroclinic sequence Selection of saddles. We look for the conditions under which the system (4.19) has a SHS consisting of saddles Sk = (0, ... ,0, ()ik ' 0, ... , 0) linked by heteroclinic trajectories, k = I , ... ,N ~ n. The saddles Sk have the following increments (eigenvalues of the linearized system at Sk): ()j - PPk ()ik' j i- ik, and -()ik · The saddles Sk = (0, ... ,0, ()ik' 0, ... ,0), k = 2, ... ,N are selected in such a way that : there is one positive eigenvalue, and the rest, are negative . Then the following inequalities are verified (4.20) and the other eigenvalues are negative .

4 Metastability and Transients in Brain Dynamic s

169

Heteroclinic connections. To assure that there is a heteroclinic orbit fik-1 ik joining Sk-I and Sb as it was said, the following condition has to be satisfied (4.21) This orbit belongs to the plane Pik-l ik = n j""ik_1,ik{a j = O}, where the point Sk has a I-dim strongl y unstable direction (determ ined by ik+ d. Leading directions. Under the following conditions (4.22) and (4.23) the separatrix fik-1 ik come s to Sk follow ing a leading direction, transversal to the on the plane ~k-l ik (Afraimovich et al., 2004b).

a ik -axis

Dissipativity of saddles. The saddle value (4.24) is defined for every saddle Sk . We assum e that Vik >

I, k= l , oo . ,N.

(4.25)

It means that every saddle Sk is "dissipative" . It was shown in (Afraimovich et al., 2004a,b) that if all saddles have onedimensional unstable manifolds, then under the conditions above, the SHS consi sting of the saddles Sk and joining them separatrices fik - I ik is stable in the following sense: if one choo ses a positive initial cond ition in a small neighborhood of So, the trajectory going through it will follow the sequence {fik - 1i k } , staying in a small vicinity of them until it come s to a neighborhood of the last saddle SN. In anoth er words the system possess a SHC in a vicinity of this SHS.

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Phan K.L., Wager T., Taylor S.P. and Liberzon I., 2002, Functional neuroanatomy of emotion: a meta-analysis of emotion activation studies in PET and fMRI, Neurolmage,16,331-348 . Port R. and Gelder T.J. van., 1995, Mind as Motion : explorations in the Dynamics of Cognition. MIT Press, Cambridge. Price D.D., Barreil lE. and Barrell r.J ., 1985, A quantitative-experimental analysis of human emotions, Motivation and Emotion, 9, 19-38. Raichle M.E . and Gusnard D.A, 2002, Appraising the brain's energy budget, Proc Natl Acad Sci USA, 99, 10237-10239. Rabinovich M.l. and Muezzinoglu M.K., 2009, Mutual emotion-cognition dynamics, arXiv :0909.1144. Rabinovich M.I., Ezersky A.B. and Wiedman P., 2000, The Dynamics of Patterns, World Scientific, Singapore. Rabinovich M.l., Volkovskii A, Lecanda P., Huerta R., Abarbanel H.D.l. and Laurent G., 2001 , Dynamical encoding by networks of competing neuron groups: Winnerless competition, Phys. Rev. Lett., 87, 068102. Rabinovich M.I., Huerta R. and Afraimovich v., 2006a,Dynamics of sequential decision making, Phys. Rev. Lett., 97, 188103. Rabinovich M.I., Varona P., Selverston A.1.and Abarbanel H.D.I., 2006b, Dynamical principles in neuroscience, Reviews ofModern Physics, 78, 1213-1265. Rabinovich M.l., Huerta R. and Laurent G., 2008a,Transient dynamics for neural processing, Science, 321, 48-50. Rabinovich M.I., Huerta R., Varona P. and Afraimovich V.S., 2008b, Transient cognitive dynamics, metastability, and decision making, PLoS Computational Biology , 4, e I000072. Raichle M.E., MacLeod AM., Snyder AZ., Powers W.J., Gusnard D.A and Shulman G.L., 200 I, A default mode of brain function, Proc Natl Acad Sci USA, 98, 676-682. Rowan AJ. and Tolunky E., 2003, Primer ofEEG : With a Mini -Atlas, ButterworthHeinemann. Saari G., 1995, Basic geometry of voting, Springer, Berlin. Sasaki T., Matsuki N. and Ikegaya Y., 2007, Metastability of active CA3 networks, J Neurosci, 27, 517-528. Scherer K.R., 1993, Studying the emotion-antecedent appraisal process : an expert system approach, Cognition, 7, 325-355. Scherer K.R., 1999, Appraisal theory. In Handbook of Emotion and Cognition . Chichester, Wiley, England. Schraagen lM. and Chipman S.P., 2000, Cognitive Task Analysis, Lawrence Erlbaum Associates. Shanahan M., 2005 , Global access, embodiment, and the conscious subject, Journal of Consciousness Studies, 12, 46-66. Shilnikov L., Shilnikov A., Turaev D. and Chua L., 1998, Methods of Qualitative Theory in Nonlinear Dynamics Vol I, World Scientific, Singapore. Shilnikov L., Shilnikov A., Turaev D. and Chua L., 2001, Methods of Qualitative theory in Nonlinear Dynamics Vol II, World Scientific, Singapore.

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Stone E. and Holmes P., 1990, Random perturbations of heteroclinic attractors, SIAM J. Appl. Math., 50, 726-743. Stopfer M., Jayaraman V. and Laurent G., 2003, Intensity versus identity coding in an olfactory system, Neuron, 39 , 991-1004. Thagard P. and Aubie B., 2008, Emotional consciousness: a neural model of how cognitive appraisal and somatic perception interact to produce qualitative experience, Consciousness and Cognition, 17, 811-834. Thelen E. and Smith L., 1994, A Dynamic System s Approach to the Development of Cognition and Action, MIT Press, Cambridge. Varona P., Levy R., Arshavsky YI. , Rabinovich M.I. and Selverston A.I., 2004, Competing sensory neurons and motor rhythm coordination, Neurocomputing, 58 , 549-554. Wallace R., 2005 , Consciousness: A mathematical Treatment of the Global Neuronal Workspace Model, Springer, New York. Watts D. J. and Strogatz S. H., 1998, Collective dynamics of 'small-world ' networks. Nature , 393, 440-442. Werner G., 2007 , Metastability, criticality and phase transitions in brain and its models . Biosystems, 90 , 496-508. Wicker B., Ruby P., Royet 1. and Fonlupt P., 2003, A relation between rest and the self in the brain, Brain Res Rev, 43, 224-230. Williams L.W. and Gordon E., 2007, Dynamic organization of the emotional brain: Responsivity, stability, and instability, The Neuroscientist , 13, 349-370. Wills T.J., Lever C; Cacucci E , Burgess N. and O'Keefe J., 2005, Attractor dynamics in the hippocampal representation of the local environment, Science, 308, 873. Wilson H.R. and Cowan J.D ., 1973, A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue, Kybernetik, 13 , 55-80. Zaslavsky G., 2005, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, New York . Zaslavsky G., 2007, The Physics ofChaos in Hamilton ian Systems, Imperial College Pr, London. Zautra A. , 2003, Emotions, Stress, and Health , Oxford University Press, New York. Zavaglia M., Astolfi L., Bablioni E and Ursino M., 2006, A neural mass model for the simulation of cortical activity estimated from high resolution EEG during cognitive or motor tasks, Journal ofNeuroscience Methods, 157, 317-329. Zeeman E., 1977, Catastrophe theory: Selected papers , Addison-Wesley, Reading, MA.

Chapter 5

Dynamics of Soliton Chains: From Simple to Complex and Chaotic Motions Konstantin A. Gorshkov , Lev A. Ostrovsky and Yury A. Stepanyants

Abstract A brief review of soliton dynamics constituting one-dimensional periodic chains is presented. It is shown that depending on the governing equation, solitons may have either exponential or oscillatory-exponential decaying tails. Under certain conditions, solitons interaction can be considered within the framework of Newtonian equations describing the dynamics of classical particles . Collective behaviour of such particles forming a one-dimensional chain may be simple or complex and even chaotic. Specific features of soliton motions are presented for some popular models of nonlinear waves (Korteweg-de Vries, Toda, Benjamin-Ono, KadomtsevPetviashvili, and others) .

5.1 Introduction One of the key topics of George Zaslavsky's research has been transition from regular motions to chaotic ones and the chaotic behavior of dynamic systems . He developed this approach from relatively simple nonlinear oscillators (starting to work with B.Y. Chirikov) to quantum chaos and ray chaos in acoustics . The brief review below deals with the complex dynamics of "wave particles " - solitons which can form one-d imensional "chains" and two-dimensional "lattices" and chaotic ensemK.A. Gorshkov Institute of Applied physics of the Russian Academy of Sciences, Nizhny Novgorod, Russia, e-mail : Gorshkov @hydro.appl.scin-nov.ru L.A. Ostrov sky ZelTech/NOAA ETL, Boulder, USA and Institute of Applied physics of the Russian Academ y of Sciences, Nizhny Novgorod , Russia, e-mail : Lev.A.Ostrovsky @noaa.gov Y.A. Stepanyants Department of Mathematics and Computmg, faculty of SCIences, University 01 Southern Queensland, Toowoornba, Australia, e-mail : Yury.Stepany ants@usq .edu.au

A. C. J. Luo et al. (eds.), Long-range Interactions, Stochasticity and Fractional Dynamics © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010

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bles and still preserve their identity upon interactions. In general this topic is extremely broad . Here we limit ourselves by using an asymptotic approach based on the assumption that interacting solitons are sufficiently strongly separated in space to be overlapped only by their asymptotic "tails" so that the effect of their interaction and the effect of the latter reveals itself at time intervals significantly exceeding the characteristic duration of each soliton . This allows application of the direct perturbation method developed in (Gorshkov and Ostrovsky, 1981; Ostrovsky and Gorshkov, 2000) . As a result, a number of non-trivial features of soliton ensembles can be assessed. It turns out, for example, that soliton chains in the Kortewegde Vries equation behave as Toda solitons , that the wave fronts-kinks can form a double-chain system, that solitons in resonators can behave chaotically, and solitons in non-integrable systems can exist as "multi-hump" solitons with the peaks distributed both regularly and chaotically. In spite of such a plethora of effects, we, as already mentioned, address only one kind of approach to soliton interaction and do not consider other, also effective approximate methods such as the Green function method (Keener and McLaughlin, 1977) or the perturbation method based on the application of the inverse scattering method to systems close to integrable ones (Karpman and Maslov, 1977). In a number of cases the solutions described here are confirmed by exact solutions and/or numerical calculations. Complex dynamics of nonlinear wave systems can often be described in terms of the interaction of compact coherent structures such as solitons or kinks. The solitonic concept is especially successful when the governing model equations belong to the class of completely integrable systems or are close to such class. In this brief review it will be demonstrated that even in the case of completely integrable equations, e.g., the Korteweg-de Vries (KdV) equation, the behaviour of the soliton ensembles can be rather complex and interesting . In the case of non-integrable equations that are close to integrable ones, complex dynamics can be revealed even within few interacting solitons . For instance, even three oscillatory-tail solitons in a circular resonator can demonstrate very complex and, apparently, even stochastic behaviour. In what follows the KdV equation plays a role of a basic model equation and we will refer to it many times. In the reference frame moving with the speed of long linear perturbations it can be presented in the form

u, + txuu, + f3u xxx = 0,

(5.1)

where ex and f3 are some constant coefficients, which are determined by a specific physical problem . As is well known, Eq. (5.1) possesses periodic and solitary solutions . The former solutions are known as cnoidal waves, whereas the latter ones are known as solitons . Solitons playa fundamental role in the dynamics of localized initial perturbations due to their stability and persistence in their interactions with each other and even more general perturbations. The solitary solution for Eq. (5.1) can be presented in the form

u(x,t) == UKdV(~ =x- Vt) =A sech 2 [(x - Vt) /il]'

(5.2)

5 Dynamics of Soliton Chains

179

where V = aA /3 is the soliton speed and ~ = (12f3/aA) 1/2 is the characteristic soliton width, whereas A is its amplitude (maximum deviation from the zero level). KdV solitons as well as solitary solutions for other nonlinear partial differential equations (PDEs) will be treated as interacting classical particles whose dynamics is determined by Newtonian equations of motion (ODEs) . The interaction potential between solitons is completely determined by the asymptotics of fields of individual solitons. Such a concept was introduced for the first time in (Gorshkov et a1., 1976) and then was developed in many papers [see the reviews (Gorshkov and Ostrovsky, 1981; Kivshar and Malomed, 1989; Ostrovsky and Gorshkov, 2000) and references therein] .

5.2 Stable soliton lattices and a hierarchy of envelope solitons In many cases stationary nonlinear waves in dispersive media can be presented as a periodic sequence of solitary waves. In several cases such representation has been rigorously proven (Toda, 1989; Zaitsev, 1983). In particular, a well-known periodic cnoidal wave solution to the KdV equation can be presented as (Toda, 1983):

cn2 (2Kx ) =

~k2 {(~)2 2K'

f. sech/ [nKK' (x- i)] + ~K + .s: -I} + I 2KK' ,

(5.3)

i= - oo

where cn (x,k) is the elliptic function of the argument x and modulus k, E(k) and K(k) are complete elliptic integrals with the modulus k [see, e.g., (Weisstein, 2003)], K'(k) = K(k'), and k' =~. Important question then arises regarding the stability of nonlinear periodic waves and chains of solitons . To a certain extent, the stability problem is related to the problem of the evolution of an initially perturbed periodic wave. One of the widely adopted approaches to this problem is the application of one of the versions of the averaging method (Whitham, 1965, 1974; Ostrovsky and Pelinovsky, 1972; Karpman and Maslov 1977; Keener and McLaughlin, 1977; Kaup and Newell 1978; Grimshaw, 1979). However, this approach has a drawback because the governing set of equations for the parameters of the perturbed periodic wave (amplitude, frequency, wavenumber) contain in the first approximation hydrodynamic-type nonlinearities which are not balanced by dispersion or dissipation. As a result, nonphysical discontinuities appear for modulated waves; they can be removed , however, in the high-order approximations. Another evident limitation of the averaging method is in its inapplicability to the description of perturbations whose period is comparable with the period of a carrier wave. The situation becomes much simpler when a nonlinear wave can be treated as a sequence of well-separated solitons . In this case a modulated nonlinear wave can be considered as a perturbed chain of particles - solitons interacting with each other. As has been shown in (Gorshkov and Ostrovsky, 1981; Ostrovsky and Gorshkov, 2000), if soliton velocities (as well as, other parameters such as widths, amplitudes)

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K.A . Gorshkov, L.A . Ostrovsky and Y.A. Stepanyants

differ only slightly from some mean value Vo (i.e. Wi - Vol « Vo) , then equations describing coordinates of their centers (maximums) take the form of Newtonian equations for interacting particles : 2----+ d S if3 m yf3 ~ t

(----+ = L ----+ Fy Sj

-

----+)

Si

(5.4)

,

j#i

----+

where S i are spatial coordinates of solitons , m yf3 are elements of the "tensor of mass" which is determined via the derivative of the y-component of the soliton's mo----+ ----+ mentum overthe l3-component of the mean soliton velocity Vo : m yf3 = a p y/ a V f3 '

Is

Sil] than their characteristic widths, ISi- Sjl > ~ (the difference between soliton

and F y '" exp [-A,(Vo) j are forces which are determined by the asymptoties of individual solitons . Distances between solitons may be arbitrary but greater widths ~i and ~j is insignificant in this approximation). For the Lagrangian systems a condition of reciprocity takes place: (5.5a) and the force

F y (s)

can be determined via the "potential function" V

(s): (5.5b)

Equations (5.4) adequately describe soliton interactions when their collisions are elastic; in this case the number of solitons before and after the interaction is unchanged, and the nonsolitonic wave field (radiation) is absent. In the onedimensional case, Equations (5.4) are the equations describing a set of particles moving and interacting on a straight line. If the wave field of an individual soliton diminishes sufficiently fast in space (in many cases it decreases exponentially with distance from the center), then one can consider only the interaction of an n-th soliton with its nearest neighbours having numbers n - I and n + I. Equations (5.4) reduce in this case to a chain of coupled nonlinear oscillators (Gorshkov and Papko, 1977): aPd

2S 11

, (

)'(

)

aV dt2 =V SI1- SI1 -1 -V SI1+I- S11

,

(5.6)

where the prime sign stands for the differentiation V' (x) = dV / dx. The simplest solution to this equation is SI1 = nAo which represents an equidistant set of "particles" on a line or a periodic nonlinear wave in terms of original field variables with the spatial period Ao (see, e.g., Eq. (5.3) when k ----+ I). By linearizing Eq. (5.6) around stationary state SI1 = nAo, one can show that the chain is stable if V'(Ao)ap /av > 0, otherwise, when V'(Ao)ap/av < 0, it is unstable. Physical interpretation of this result is straightforward: the chain is stable when solitons repel each other, and unstable, when they attract each other. In the

5 Dynamics of Soliton Chains

181

long-wave limit in the stable case, when the differe nces in the right-hand side of Eq. (5.6) can be replaced by the differential operator Vi (Ao)d 2 / dn2 , the corresponding POE is of a hyperbolic type, while in the unstable case the POE is of an elliptic type (Whitham, 1965, 1974). Meanwhile, due to the discrete nature of Eq. (5.6), the dispersio n is essential in that equation, especially for the small-scale perturbations. The dispersion stabilises the process of singularity creation and provides the existence of smooth solutions and stationary waves. In many cases soliton fields possess exponential asymptotics, so that Vi (S) ----+ aexp( - AS) when S ----+ 00 , where a and A are some constants. In a stable case of repelling solitons with such exponential asymptotics, Equations (5.6) represe nt the well-known model of the Toda chain (Toda, 1989). This model has an exact solution in the form of a periodic stationary wave:

where dn (x,k) is another elliptic function linked with the en-function [see Eq. (5.3)] by the formu la dn2(x ,k) = k2 [cn2 (x, k) - I] + I [see, e.g., (Weisstein, 2003)], and V and k are related by the following dispersion relation:

0.5 u (x)

I

o

(a)

50

100

ISO

200

1.5 ,

< ,

-.

-,

-- -

"

-

-,

'.

"

-

250 x

---

,-

300

"

-

350

400

, '.

450

,

.-- - -

-- -

500

"

"-

u (x)

0.5

(b)

o

100

200

300

400

500 x

600

700

800

900

1000

Fig.5.1 (a) Unperturbed chain of KdV solitons and (b) modulated chain of the same solitons (note that the horizontal scales are different).

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K.A. Gorshkov, L.A . Ostrovsky and Y.A. Stepanyants

2VK(k)

cn 2 (2kK )

E(k)

I - cn2 (2kK)

+ K(k) =

(5.8)

I.

In the limiting case of an infinite period, solution (5.7) reduces to the solution for solitary wave - the Toda soliton:

n - Vt ) ] , 5,,-5,,-1 =Ao+~ln [ I +f3 2 cosh _2 ( -~-

(5.9)

with the following relationships between the parameters: f3 = sinh (I / ~),v = sinh (1 / A). With respect to the original sequence of solitons, Equations (5.7) and (5.9) can be treated as stationary waves of envelopes. Solutions (5.7)-(5 .9) are illustrated in Fig. 5.1. One of the important properties of the Toda chain model is its complete integrability (Manakov, 1974). An outcome of this property is that the interaction of solitons (in our case - the envelope solitons) conserves their number and individual parameters (only phase shifts appear). Arbitrary localized perturbation (of the appropriate polarity - this will be specified further) evolves into the diverging sequence of solitons described by Eq. (5.9) . Similar properties of repelling chains of solitons were revealed from the exact solution for the KdV equation (Kuznetsov and Mikhailov, 1974). The existence of periodic envelope waves (5.7)-(5.9) provides a way to constructing a wide class of solutions . Indeed, periodic envelope waves (5.7)-(5.9) can be treated again as chains of equidistant envelope solitons (5.9) [see Eq. (5.3)], if the period of the wave is not too short in comparison with the characteristic soliton 's width . Inasmuch as solitons (5.9) possess exponential asymptotics and behave as repelling particles, their sequence can be treated again as the Toda chain. This procedure can be repeated again and again , one can construct multi hierarchical envelope waves of different orders . An envelope wave at each order represents a perturbation of the soliton chain of the previous order. An envelope wave of the N-th order can be presented in the analytical form (Gorshkov and Papko, 1977): ~

(x,t) =

L

(0 )[x- Vot-5j(t)],

(5 .10)

j = - oo

~ [ f32 - 2 nm- 10"tm- 5"1/1+1 (tm+ I ) ] 5"1/1 -5"1/1 -1 = Am-I +~m-I ~ In 1+ mcosh ~ llm - - OO

tm =

[H!E PT

~exp

m-I

-u-

(Am-I )] tm-I, m-I

tn

m= 1,2, · · ·,N.

Here (x,t) is the initial wave field, <1>(0) (~ = x - Vat) is soliton solution for the initial wave field, 10" and Am are the average speed and mean distance between solitons in the m-th order chain, correspondingly, aPT is the coefficient of interaction

5 Dynamics of Soliton Chains

183

for Toda solitons. The quantities 13m and I::!.m can be presented in terms of Vm + dSIlIII + 1 / dt in each order using the relationships presented after Eq. (5.9). Solution (5.10) represents a N-periodic function with aliquant periods in general. Note that for the integrable models there are so-called N-zone exact solutions which represent periodic and conditionally periodic analogs of multisoliton solutions . Such solutions are described in (Zakharov et aI., 1980; Ablowitz and Segur, 1981) for the KdV equation, and in (Dubrovin et aI., 1976) for the Toda equation and some other equations. A detailed comparison of exact solutions against the approximate ones (5.10) has not been done thus far. It is obvious that the solutions (5.10) correspond to only a part of the general exact solutions . Indeed, as follows from the condition of applicability of Eq. (5.6) in each order of the hierarchy, V,1l » dSIlIII + 1 / dr, when all N periods tend to infinity, Equations (5.10) will be reduced to the equation describing multisolitons with close velocities only, whereas the exact solutions in the same limit describe multisolitons with arbitrary velocities. The Toda equation can be generalized for waves of modulation in the chain s consisting of envelope solitons of the initial wave field (Gerdjikov et al., 1996). In this case the real values SIl(t) should be replaced by the complex ones ZIl(t) = SIl(t) + i f{J1l (r), where SIl(t) represent the coordinates of the centers of envelope solitons, and f{J1l (r) are the phases of the carrier wave of the initial wave field. The Toda equation in the complex form preserves the integrability property and the form of the solution with the replacement of real parameters characterizing the solution by their complex counterparts. The twofold increase of the number of parameters results in the richer family of solutions of the Toda equation. Such solutions were investigated in details in (Gerdjikov et al., 1996) for the groups of almost equid istant envelope solitons within the framework of the nonlinear Shrodinger (NLS) equation. For slightly modulated nonlinear waves close to the periodic sequences of solitons, the approximate results described above are more general as they are not related to the integrability of the initial nonlinear equation and provide clear physical interpretation in terms of interacting Newtonian particles . One may say that the equation of Toda chain plays a similar role to that which the NLS equation plays for quasi-harmonic waves. These two equations, to a certain extent, are complementary to each other. Indeed, if the modulated quasi-harmonic wave can be described by the NLS equation and the envelope wave is a periodic cnoidal wave of a large period , it can be presented as an infinite chain of solitons with exponential tails (Toda, 1989; Gerdjikov et al., 1996). Hence , the perturbation of such solitons can be described by the Toda chain equation. On the other hand, if the initial sequence solitons is only slightly modulated in amplitude, the envelope quasi-harmonic wave can be described by the NLS equation. And to complete this scheme, one should mention the case when the envelope of slightly modulated quasi-harmonic wave is described by the NLS equation and represents again quasi-harmonic wave, which in turn can be described by the NLS equation, and so on, and so on. The relationship between the NLS and Toda equations can be schematically illustrated in Fig. 5.2.

184

K.A. Gorshkov, L.A . Ostrovsky and Y.A. Stepanyants

Where a) Toda Eq.

----+

Toda Eq.

----+

Toda Eq.

----+ •• •

b) Toda Eq. ----+ Toda Eq. ----+ NLS Eq. ----+

•••

Toda Eq. ----+ NLS Eq. ----+

•••

c) NLS Eq. ----+ Toda Eq. ----+ Toda Eq. ----+

•••

NLS Eq. ----+ Toda Eq. ----+

•• •

d) NLS Eq.

•• •

----+

NLS Eq. ----+ NLS Eq. ----+

Fig . 5.2 Schematic presentation of various possibilities for modulated periodic waves: a) pure Toda equation hierarchy ; b) and c) random intermittent hierarchy of Toda and NLS equations when the original wave is describ ed either by Toda equation (case b) or by NLS equation (case c), d) pure NLS equation hierarchy.

It should be noted, however, that in such a hierarchical structure amplitudes of the tallest solitons must still be relatively small to be in consistent with the weak nonlinearity assumption under which the basic equation was derived . At each level of the hierarchical structure, envelope solitons are smaller, wider, and the time of interaction between them increases. One more interesting example of application of this theory is a periodic sequence of slightly modulated Benjamin-Ono (BO) solitons (Ablowitz and Segur, 1981). These solitons posses algebraically, rather than exponentially, decaying tails and are described by the function:

(5.11 ) where the soliton half-width ~ is related to the amplitude, ~ = 4/ A. It is interesting to note that more general stationary solutions to the BO equation in the form of periodic waves can be presented similarly to Eq. (5.3) as an infinite sequence of BO solitons (5.11) (Zaitsev, 1983) (such representation is rather general ; as shown in the cited paper, it is valid for a wide class of "soli tonic" equations, includ ing KdV, BO, Toda equation, Kadomtsev-Petviashviliy equation, and others). Then, the function determining the force exerting on the nearest soliton by its neighbor is (see Eq. (5.6» : U'(S) = (2A /~)(NS? For small soliton deviations S from their equilibrium distances Ao, the forces in the right-hand side of Eq. (5.6) can be approximately presented as

U'(S) ;:;: -3(2A /~)(NAo? [S/Ao-4(S /Ao?] .

(5.12)

Such presentation, which is simply a Taylor's expansion of the force, is rather general ; it may work for solitons with non-exponential asymptotics. The first nonlinear term in the Taylor series (5.12) is usually quadratic, but it can be, in principle, cubic or even of higher-order nonlinearity. Chains of quadratically and cubically interacting particles were studied in the Report by Fermi , Pasta and Ulam (1955) which gave rise to what is known as the FPU-problem - anomalously slow stochas-

185

5 Dynamics of Soliton Chai ns

tizatio n in chains of nonlinear oscillators (see also the paper by (Dauxois, 2008) where the significant contribution of Mary Tsingou to the numerical comp utations on that problem is described). As is well-known (Toda 1989), a discrete chain of quadratically interacting particles can be reduced to the KdV equation in the longwave limit; similarly, a chain of cubically interacting particles can be reduced to the modified KdV (mKdV) equation in the same limit. The n, KdV equation has periodic stationary solutions which can be presented by infinite sequences of solitons with exponential tails on the basis of which the Toda chain hierarchy can be constructed as described above (the same is true for the mKd V equation). From this consideration it follows that in the first row of the scheme presented in Fig. 5.2, the entry equation may be any nonlinear equation possessing soliton solutions with any kind of asymptotic behavior at infinity. To concl ude this section, let us discuss briefly the case of strong modulation of soliton lattices . Notice first that Eq. (5.6) is able to describe the interact ion of solitons not only of almost equal speeds, but solitons whose speeds are significantly different. In the latter case, two solitons may overlap in the process of interaction despite the repulsion force which may act between them - this case is known as the "overtaking" interaction in contrast to the "exchange" interaction taking place for solitons of almost equa l amp litudes and speeds [see, e.g., (Scott et aI., 1973)]. These two cases of soliton interactions are illustrated in Fig. 5.3. u

3

1.0

'I'\ 2 1\ ,\I'

'I I I I ,

(a)

I I I ,

0.5

I , I I

,/

0.00

(b)

/

20

u 1.0

I

I

I I

1\

I I , I

I I I I

\ \

I J

\ \

' I

\/

I \ I I

I

I I I

\,

,

-,

50 x

30

3

0.5

0.00

50 x

Fig . 5.3 Exc hange (a) and overtaking (b) interactio n of solitons. In the former case, energy from the tallest soliton transmits into the smallest soliton, which gradually grows, accelera tes and moves ahead. In the latter case, the tallest solito n simply covers the smallest soliton and forms jointly with it a single-crest pulse, which then disi ntegrates then into the same two solitons. (In Fig . 5.3a curves 2 and 3 are shifted back artificially to visualize the interaction process in the chose n space interval.)

K.A. Gorshkov, L.A. Ostrovsky and Y.A. Stepanyants

186

If now there is an infinite sequence of small-amplitude solitons in front of a large-amplitude soliton, then the latter will consecutively interact with each of them separately. As a result of that it will move non-uniformly, decelerating at the rear slope of each small soliton and accelerating at the frontal slope. The mean speed of the large-amplitude soliton can be determined by its own speed in a free space plus a correction to that speed caused by phase shifts arising each time when it passes through the next small-amplitude soliton . Thus, in the case of strong modulation, the motion can be understood as a propagation of a "dislocation" in the soliton lattice, as shown in Fig. 5.4. u 1.0

(a)

0.5

10

20

30

40

50 x

10

20

30

40

50 x

u 1.0

(b)

0.5

Fig. 5.4 Single soliton dislocation nonuniformly moving on the small-amplitude soliton chain . Number I - designates the soliton chain; number 2 - shows large-amplitude soliton .

If instead of one fast soliton there is a sequence of fast solitons , then a particular case of strong periodic modulation of a soliton lattice occurs (Fig. 5.5) . Such cases have been studied by Zakharov (1971) and Zaslavsky (1972). Actually, the dislocations represent to a certain extent a limiting kind of motion originated from the envelope waves considered above in the case of strong modulations of soliton sequences. It should be stressed, however, that the modulation waves in the form of envelope solitons are obtained by neglecting the inelastic effects associated with the radiation . In many cases the inelastic effects are either absent (e.g., when the governing equation is integrable) or are negligibly small. Description of the inelastic effects, when they are essential, requires a separate consideration. In the meantime, in some situations, the influence of inelastic effects is rather obvious. For instance, the radiation leaking from the region of interaction of the solitonic group of the initial wave field (this group forms the envelope solitary wave), leads

187

5 Dynamics of Soliton Chains u

u

x u 1.0

x u

x

x

Fig. 5.5 Interaction of two soliton chains.

188

K.A . Gorshkov, L.A. Ostrovsky and Y.A. Stepanyants

to i) either slow decay of the envelope solitary wave or ii) fast destruction of that wave, depending on the intensity of radiation.

5.3 Chains of solitons within the framework of the Gardner model In this section we consider a nontrivial generalization of stable soliton chains to the case when soliton interactions within the framework of particle-like approximation are described by equations which are different from the Newtonian equations of motion (5.4)-(5.6). This occurs when the basic model equation contains solitary solutions of a more complex structure than the simple one-parametric KdV solitons . One of the typical representatives of such a class of equations is the Gardner equation also known as the combined or extended KdV equation [see, e.g., (Ostrov sky and Stepanyants, 1989,2005; Apel et al., 2007)] . This equation in the dimensionless variables reads (Gorshkov and Soustova, 2001 ; Gorshkov et al., 2004) :

au au a3u at + 6u(1 - au) ax + ax3 = 0,

(5.13)

where a = ± I is the parameter characterising a sign of the cubic nonlinearity. The Gardner equation became popular in recent years as the model equation describing nonlinear wave processes in the case when wave amplitude is not small. Such situation occurs , for instance, for internal waves in oceans, see (Lee and Beardsley, 1974) as the pioneering paper in this field and reviews (Ostrovsky and Stepanyants, 1989, 2005 ; Ape! et al., 2007) for further references. Equation (5.13) with a = I possesses solitary solutions with exponential asymptoties [when a = -I , Equation (5.13) has solitary solutions of different types; among them there is a one-parametric family of solitons both with the exponential and algebraic asymptotics [see, e.g., (Grimshaw et al., 1997) and references therein] :

u(x,t)

= UG(x- Vt) = ~ {tanh [~(x- Vt+~)] -tanh [~(x- Vt-~)]},

(5.14a) where V = k 2,~ = (I 12k) In [(I +k) /(1 - k)] and k is a free parameter varying from o to I. Depending on the value of this parameter, soliton (5.14) may be very close to the KdV soliton when k --+ 0, or to a superposition of separated kink and antikink, which form the so called "fat" or "tabletop" soliton when k --+ I (see Fig. 5.6). For k = I, solution (5.14a) degenerates into a single kink or antikink :

u(x,t)

=Ukink(X-t) = ~ [I ±tanh (x~t)] .

(5.14b)

5 Dynamics of Soliton Chains

189

u

- 20

- 15

- 10

- 15

o

5

10

20 x

15

Fig.5.6 Shape of Gardner solitons (5.14a) for different parameter e = I - k. (from the smallest to the highest solitons e = 0.5 , 0.1, 10- 2 , 10- 4 , 10- 6 , 10- 8). For very small e, one of the soliton slopes reduces to the kink (5.14b), whereas another slope reduces to the antikink. Width of the widest fat soliton is shown by horizontal lines with the arrows.

The peculiarity of tabletop solitons is the logarithmic divergence of their widths j!": Va (x)dx when k --+ I . Because of that, when two tabletop solitons collide , their neighbouring fronts begin to interact , whereas their other slopes are not influenced by such interaction for awhile . This fact is not reflected in the point-particle model (5.4) or (5.6) : the left-hand side of Eq. (5.6), dP / dt, does not adequately describe strong variations of the solitons' widths and momentums for large values of k when solitons cannot be treated as point particles . Details of tabletop soliton interactions can nevertheless be described within the framework of the same asymptotic method (Gorshkov and Ostrovsky, 1981; Ostrovsky and Gorshkov, 2000) if the interaction of tabletop solitons is treated as the interaction of kinks and antikinks which constitute such solitons . Application of that method to the kink-antikink interaction results in the following equation for their coordinates (Gorshkov and Soustova, 200 I; Gorshkov et al., 2004): ~ rv In (I - k) and momentums P =

dSn dt

= -4 [e - (Sn+I -Sn) +e- (Sn-Sn-I )] +D

(5.15) '

where D is an arbitrary constant. Equations (5.15) are symmetric with respect to coordinates of kinks (even index numbers) and antikinks (odd index numbers) or, in other words, solitons ' fronts and rear slopes. These equations, known as the Kac-Moerbeke system [see, e.g., (Toda, 1989)], describe well both soliton's widths and distances between solitons . The Kac-Moerbeke system is completely integrable and represents the Backlund transformation of the Toda-Iattice equations. From a physical point of view one may say that the evolution of tabletop solitons is adequately described in terms of the positions of their front and rear slopes rather than the coord inate of soliton centres. This reflects the non-synchronous motion of soliton fronts and rear slopes in the external non-uniform field (e.g., in inhomogeneous medium) . In particular, when two solitons approach each other, the front of one of them begins interacting with the rear slope of another, and only later do their other slopes enter into the interaction. We focus here on an infinite series of solitons (soliton chains) leaving aside the details of two-soliton interactions [they are presented in terms of exact solu-

190

K.A . Gorshkov, L.A . Ostrovsky and Y.A. Stepanyants

tions of the Gardner equation in (Slyunaev and Pelinovski , 1999; Slyunyaev, 200 I; Grimshaw et al., 2002) for both signs of the parameter a in Eq. (5.13)]. Differenti ation of Eqs. (5.15) on t and elimination of derivatives dS,,/dt with the help of the very same Eqs. (5.15) leads to the two independent sets of Toda-chain equationsfor odd and even numbers n : (5.16) Each of these sets describes the evolution of soliton fronts and rears, whereas Eqs. (5.15), playing the role of links between them, are the Backlund transformation for each set. As a result of this, solutions ofEqs. (5.15) can be presented as the solutions of two Toda-chain sets of Eqs. (5.16) , provided they are linked with each other by the Backlund transformation (5.15). In the general case, the constitutive portions of such a composite solution have different structures. In the degenerative case of D =0, the Backlund transformation does not change the solutions structure of both subsets described by Eq. (5.16) and only imposes conditions which relate the parameters of these solutions . Such solutions for the finite chains describe N-soliton interactions of tabletop solitons. Although the character of interaction of tabletop solitons is qualitatively similar to the interaction of particles with the repulsive potential between them, there is no direct mechanical interpretation for the Eqs. (5.15), as was mentioned above. The specifics of the tabletop solitons ' interaction can be easily seen from the example of two-soliton collision when the solutions for the pairs of soliton fronts and rears are the same but shifted in time and space with respect to each other. When two solitons approach each other, the front of one of them interacts with (repeals from) the rear of another one, and then after the corresponding delays front-front, rear-front and rear-rear interactions occur. This picture is in the correspondence with the exact solution of the Gardner equation obtained in (Slyunaev and Pelinovsky, 1999; Grimshaw et aI., 2002) for the case of Eq. (5.13) with a = I. Returning to the description of infinite chains of Gardner solitons, note first that there exist trivial solutions to Eqs. (5.15):

S" = { nA + Vt + 8, nA+ Vt,

for even n,

(5.17)

for odd n.

Such solutions correspond to periodic nonlinear waves in the original PDE model (5.13) with the spatial period 2A and the given on-off time ratio 8 which ranges in -A < 8 < A (Fig. 5.7) Substitution of solution (5.17) into Eqs. (5.15) yields the dispersion equation v = -2e- A cosh 8, (5.18) which relates the speed of a periodic wave with the parameters A and 8 . Wide classes of solutions can be constructed for an infinite soliton chain described by Eqs. (5.15). Among these classes there are modulation waves on the background of a periodic sequence of tabletop solitons (5.17). Bearing in mind the

5 Dynamics of Soliton Chains

191 u

o

8

A

8 +4A

8 +2A

8 +6A

x

Fig. 5.7 Sketch of a typical chain of Gardner solitons . Positions of kink and antikink fronts are indicated underneath .

compound character of solutions of Eqs . (5.15) (see above), we conclude that the N soliton solution of one of the subsets of Eqs . (5.16) for soliton fronts or rear slopes corresponds to the (N + 1)-soliton solution of another subset of these equations:

Sn(t) =

S,~N)(t),

for even n (odd n),

{ SI~N+I )(t) ,

for odd n (even n).

(5.19)

Within each subset of Eq. (5.19) asymptotics of solutions S~N\t) and S,~N+l ) (t) when t ----+ 00 represent sequences of Toda-chain solitons ordered in amplitudes and speeds . As the solution S,~N+ I) (r) is obtained by a single application of Backlund transformation to solution S~N) (r), N solitons in each of the subsets are the same in pairs. This means that solution (5.19) describes a collision of N pairs of linked solitons from the different subsets plus one extra soliton in one of these subsets. Collisions between such solitons possess all attributes of soliton interactions in the integrable systems: there is no radiation or energy leakage from solitons, the number of solitons and their parameters are preserved after the interaction, and the solitons acquire only phase shifts in the course of interaction . However, in contrast with the single Toda chain, the set (5.15) allows the existence of two types of localised formations that correspond to the presence of Toda solitons either in only one of the subsets or simultaneously in both of them :

Sn(t) = { where V

nA + Vt + In {Cosh [A,(n - 2) - f3 t]} , cosh(A,n - f3t)

even n (odd n) ,

nA+Vt ,

oddn (evenn) ,

= -2 cosh (2A.) exp( -A) , 13 = sinh (2A.) exp( -A) , <5 = ±2A.; <5

nA+Vt+ Sn(t) = {

(5.20)

+

I {Cosh [A,(n-2-d)-f3 t]} n

cosh[A,(n-d)-f3t]

cosh [A,(n - d) - f3 t ]} I { nA+Vt+n cosh ('Iln - 13) t

,

even (odd n), (5.21)

,

odd n (even n),

192

K.A. Gorshkov, L.A. Ostrovsky and Y.A. Stepanyants

where V is as above, but f3 = sinh (lA)exp(A) and sinh2 A(I - d) = sinh [A2 (1 + d)] exp (28) (note that in this case d and 8 are independent parame ters). Solutions (5.20) describe mod ulation waves with respect to the original soliton sequence (5.17). When the modulatio n waves of the first kind propagate, the kinks (soliton fronts or rear slopes) of only one of the subsets shift at a distance of 4A. Thereby the specific exchange takes place: spatial widths of solitons and distances betwee n them in the final state are equal to distances betwee n solitons and their spatial widths in the initial state, i.e., (initial widths of solitons) --+ (distances between solitons in the final state), (initial distances between solitons) --+ (widths of solitons in the final stage). As the modulation waves of this kind reorder the initial field structure, they can be called envelope kinks (Fig. 5.8). u ----.,.

rrr>.

rr>.

rr-.

I.~

rr-;

r-..

"

05

rI

o

- 100

- 200

100

200

x

Fig. 5.8 Examp le of the envelope kink for thc sequence of Gardner solitons (the kink realizes a smooth transition from one periodic sequence of table-top solitons in the left to another periodic sequence of solitons in the right).

Modulation waves of the second kind realise shift of the kinks (soliton fronts or rear slopes) of both subsets at the same distance 4A. Therefore, wave profiles in the initial and final states are the same. Such modulation waves can be natura lly called envelope solitons (Fig. 5.9) which are somewhat similar to a freque ncy-modulated quasi- harmonic wave. u

""'

'""'

'""'

rr>;

Ito"" -..,

rr>.

'""'

'""'

r:

.5

\

f\

-200

- 100

o

100

200

x

Fig. 5.9 Example of the envelope soliton on the sequence of initial Gardner solitons (the soliton represents a smooth transition from the periodic sequence of table-top solitons in the left to wider soliton in the center and then, back to the same periodic sequence of solitons in the right).

5 Dynamics of Soliton Chains

193

The shift parameter A for an envelope kink is unambiguously related to the on-off time ratio 0 of the periodic chain (5.17) in the limit of 0 ----+ 21. . Envelope solitons exist on the background of any periodic chain (5.15); they are presented by the family (5.21) with the parameter A independent of O. This situation is similar to that occurring for solitons and kinks in the basic model (5.13) , where the unique kink (or antikink) solution (5.14b) exists in parallel with the entire family of solitons (5.14a) depending on the parameter k. The kink (antikink) solution (5.14b) corresponds to envelope kinks related to the excitation of Toda solitons in different subsets (5.20). Envelope solitons (5.21) can be treated as compound formations which are formed by a pair of envelope kink-antikink from different subsets (5.20). With the help of envelope solitons or envelope kinks one can construct an infinite periodic chain . Quasi-sinusoidal perturbations of such a chain can be described in terms of the NLS equation (see Fig. 5.2), whereas large-amplitude perturbations of the chain can be described in terms of second-order envelope solitons or kinks with all the aforementioned properties. Repeating this argument, one can construct a family of solutions in the form of a hierarchy of envelope waves of various orders described by multi-periodic functions of aliquant periods in general. It is important to note that the composite character of two-parametric Gardner solitons is repro duced on each hierarchical level of description; this property is inherited from the basic Gardner model (5.13). Similar solutions in the form of a hierarchy of envelope waves within the models having "simple" one-parametric solitons (such as KdV or sine-Gordon solitons) demonstrate only the same "simple" envelope solitons on either hierarchical level.

5.4 Unstable soliton lattices and stochastization Soliton chain dynamics becomes much richer and multifarious when the solitons are described by non-integrable equations. Here, rather complex phenomena up to stochastisation, may occur even in a progressive wave. In particular, an interesting dynamics is possible in chains consisting of solitons with non-monotonous asymptoties in the form of oscillatory "tails," as shown in Fig. 5.10. 4.5 u

- 15

Fig.5.10 A solitary wave with oscillatory tails.

15

x

194

K.A. Gorshkov , L.A . Ostrovsky and Y.A. Stepanyants

The first example of such solitons was apparently obtained by Kawahara (1972) , who numerically constructed a solitary solution with oscillatory tails for the fifthorder KdV equation: au au a 3u a 5u + u ax + ax3 + Yax5 = 0, (5 .22a)

°

at

where Y > is the dimensionless parameter (when y < 0, Eq. (5.22a) has only solitons with monotonic tails). Equation (5.22a) presented here in the dimensionless form was earlier derived by (Kakutani and Ono , 1969) for magnetosonic waves in plasma [a similar equation was later derived for many other types of waves, among them the gravity-capillary waves on thin liquid films (Hasimoto, 1970; Stepanyants, 2005), electromagnetic waves in nonlinear electric circuits (Gorshkov and Papko, 1977, Nagashima, 1979)]. Similar properties demonstrates an equation describing nonlinear waves in with two types of dispersion: small-scale (KdV-type) and large-scale (waveguide-type) dispersion : (5.22b) Similar to the previous case, here solitons with oscillating tails exist only when y > 0, whereas there are no soliton solutions at all when y < (Leonov, 1981; Galkin and Stepanyants, 1991; Liu and Varlamov, 2004). Equation (5.22b), also presented in a dimensionless form, was derived for the first time by Ostrovsky (1978) for surface and internal waves in a rotating ocean (for these types of waves y < 0). Later a similar equation with both signs of y was derived for other types of waves, among them elastic waves in curved thin rods (Rybak and Skrynnikov, 1990), waves in relaxing media (Vakhnenko, 1999), and oblique magnetosonic waves in rotating plasma (Obregon and Stepanyants, 1998), for further references see, e.g., (Stepanyants, 2006) . The stationary solutions of these equations satisfy the ordinary differential equations : d4 u d2 u u2 (5 .23a) Yd~4 + d~2 -Vu+ 2 =0,

°

d 4u

I d 2u2 d~4 -v d~2 +yu+"2 d~2 =0, d 2u

(5.23b)

where ~ = x - V t , V is the wave speed . (Note that the linear parts of these equations are similar). The asymptotic wave field, which corresponds to a soliton with oscillating tails, can be described by the formula U(S) = e-}'1 S cos A2S, see Eq. (5.6) . By substituting this expression into Eqs. (5.23a) or (5.23b) and assuming that Y > 0, one can find the constants AI and A2. For Eq . (5.23a) they are

A 2

=~J2V-YV+I 2 y '

(5 .24a)

5 Dynamics of Soliton Chains

195

where the wave speed V is negative and restricted from the top, V < - I / 4y (otherwise, the expo nent Al becomes imagi nary). For Eq. (5.23b) these constants are (5 .24b) and - 2yY < V < 2yY. The decay rate of oscillatio ns depends on the para meter V. This is illustrated in Fig . 5. 11.

16

u

u

12 6 4

°-15

4

4

2

2

15°135 t (a)

u

165° 10 t

14 (b)

90 t (c)

Fig. 5.11 Possible shapes of a stationary soliton depending on its velocity for Eq. (5.23b): (a) soliton with aperiodic tails, (b) soliton with oscillatory tails, (c) envelope soliton (Fraunie and Stepanyants, 2002).

Due to the non-monotonous character of the potential functio n U(5), see Eq. (5.6), for the oscillating solitons, interaction between them may be both rep ulsive and attractive depending on the distance between their maxi ma. Respectively, the chain becomes stable or unstable when dista nces between solito ns vary. Boundaries between stable and unstable zones in the soliton chai n are dete rmined by zeros of function U'(5). Considering only two interacting solitons with oscillating tails, one can construct statio nary solutions in the form of stable or unstable bisolitons. Exam ples of such solutions numerically constructed in (Obregon and Stepanyants, 1998) for Eqs . (5.23) with y = I are shown in Fig. 5. 12. Stable biso litons correspond to the cases when the maximum of one soliton is located in one of the local minima C'potential wells") of another soliton (suc h exa mples are shown in Figs. 5. 12b and d), whereas unstable bisolitons correspond to the cases when the maxim um of one soliton is located in one of the local maxima of another soliton (one of such examples is shown in Fig . 5.12c). Similar bisolitons were obtained for Eq . (5.23a) and even observed experimen tally in a specially constructed electromagnetic transmission line (Gors hkov et al., 1979). Analogous solutions in the form of bisolitons were also obtained for 2D models such as Kadomtsev-Petviashvili (KP) equation and its generalisations (Abramyan and Stepanyants, 1985a, b, 1987). In the case of KP equation bisoliton solution and even more complex stationary multisoliton solutio ns were derived analytically (Pelinovs ky and Stepanyants, 1993), but all these constructions were found to be unstable. Biso litons also have osci llatory tails which means that even more co mplex stationary structures - bound states of multisolitons - can be constructed. This state-

K.A. Gorshkov, L.A . Ostrovsky and Y.A. Stepanyants

196

4

2

(a)

0 -+--~-~-~---=::::=''--T--+-r+-+---,<;:=",-~-.----~-x - 0 - 20 - 10 10 20

-2

(b)

20

x

- 20

20

x

- 20

20

x

-30

-20

- 30

- 30

o

(c)

(d)

Fig. 5.12 Examp les of stable and unstab le bisolitons for Eq , (5.23) with Y = I (Obregon and Stepanyants 1998).

ment was proven more rigorously for Eq. (5.23a) in (Gorshkov et a\., 1979). In the four-dime nsional phase space of Eq. (5.23a) solitons are represe nted by homoclinic orbits, i.e. orbits which tend to the origin for ~ ----+ 00 . All orbits leaving the coordinate origin form a two-dimensional unstable manifold (surface) \¥", which corre -

5 Dynamics of Soliton Chains

197

sponds to the eigenvalues Al ± iA2, whereas the orbits tending to the origin lie on a stable manifold W, symmetric with respect to Ht;, and corresponding to the eigenvalues -AI ± iA2. It was shown numerically that Wu and W, intersect transversely, and their intersection line corresponds to a soliton . As was first noted by Poincare and then proven by Devaney (1976), the existence of a single transverse homoclinic orbit leads immediately to an infinite, countable number of intersections of the surfaces Ht;, and W,. Thus, a countable number of stationary multi solitons exists for Eq. (5.23a) that corresponds to these intersection lines. The existence of a homoclinic structure of such a kind also provides a proof of the non-integrability of Eq. (5.23a) and, con sequently, of (5.22a) from which it follows . Note that Eq. (5.23a) also describes a family of one-dimensional stationary solutions of a relativistically invariant system (5.25)

a at

a ax

where D == 2 / 2 - c2 2 / 2 is the D' Alambert operator. The system represents one of the popular models in the field theory. Hence, the above arguments regarding the existence of multiple solitons may testify to the potential existence of a countable set of field particles in the classical limit of the equations of quantum electrodynamics. Developing further the idea of the existence of soliton bound states, one can construct even a random infinite sequence of stationary solitons . When solitons comprising such stationary waves are slightly perturbed, i.e., deviated from their equilibrium positions, further dynamics of this chain of solitons may be different depending on whether the solitons are in stable or unstable zones. For simplicity, consider further the case when the soliton chain is periodic. For small but finite perturbations of solitons in stable zones , a solution in the form of a stationary envelope wave can be obtained. Exact solutions for such periodic waves may be unknown in general ; however, for small soliton deviations from the mean positions, function Vi (5) can be approximated by Eq. (5.12). Then , the approximate equation reduces again to the equation of quadratic (or cubic) FPU chain as described at the end of the previous section . What is important is that in the long-wave approximation, this equation reduces in turn to the KdV (or mKdV) equation for envelope waves. Both the KdV and mKdV equations give rise to the exponential solitons on the basis of which the Toda-NLS hierarchy can be constructed in accordance with the scheme shown in Fig. 5.2. If the equilibrium distances between solitons are such that they fall into one of the unstable zones , then small perturbations lead to large deviations of solitons from their equilibrium positions. As a result, they can reach a neighbouring stable zone. The aforementioned quadratic approximation represented by Eq. (5.12) for the forces being exerted on the solitons is no longer valid. The resulting system dynamics may be very complex and, apparently, even chaotic . The latter presumption looks quite realistic . Indeed, if soliton displacement from the equilibrium position is of the order of the chain period Ao, then the potential energy of interacting solitons with oscillating tails resembles a non-symmetric profile of the Lennard-Jones potential (Lennard-Jones, 1924),

198

K.A. Gorshkov, L.A. Ostrovsky and Y.A. Stepanyants

A

B

(5.26)

Uw(x) = 12 - 6' X

X

with relatively steep part corresponding to repulsion (at small distances between solitons) and flat part corresponding to attraction at large distances between solitons (see, e.g., soliton profile shown in Fig. 5.10 in the vicinity of its first minimum). In numerical experiments by Bocchieri et al. (1970) with the chain of particles controlled by the Lennard-Jones interaction potential , a stochastisation in the particle motion was discovered. Therefore, one may expect a stochastisation in the chain of oscillating solitons as well, due to the qualitative analogy in the structure of the potential functions . Experimental and numerical results obtained in different papers (Gorshkov and Papko, 1977; Kawahara and Toh, 1988; Kawahara and Takaoka, 1988, 1989) confirm this guess . To illustrate possible soliton dynamics in the chain of oscillating solitons , consider, following Kawahara and Toh (1988), Kawahara and Takaoka (1988 , 1989) and Obregon (1993), a periodic perturbation of a soliton chain described by Eq. (5.22b) with only three solitons within each period . Such periodic chain of soliton s can be treated in terms of three particles interacting on the ring as shown in Fig. 5.13b. Equations of motion for the three-particle system follow from Eq. (5.6) and in normalised variables they read

X = - [f(y) - 2f(x) + f(z)], {

.9 = - [f(z)-2f(y)+f(x)], Z = - [f(x) - 2f(z) + f(y)],

(5.27)

where X rv 52 - 5"y rv 53 - 52,Z rv 5, - 53 are normalised distances between the neighbour solitons (see Fig. 5.13a) and dots stand for time derivatives . It is assumed that x, y and z are greater than 8, where 8 is a characteristic width of a single soliton in the normalised variables . As a dynamical system, Eqs. (5.27) represent a conservative set of three degrees of freedom . It has two integrals of motion and one constraint due to periodicity : z

x

- 30

- 20

20 (a)

(b)

Fig. 5.13 Positioning of two neighboring solitons (a) and an equivalent three-particle system on the ring (b).

199

5 Dynamics of Soliton Chains

x+y+ z= L,

(5.28a)

x+ y+z=o,

(5.28b) (5.28c)

where L is the period of soliton-chain perturbation (or the circumference of the ring shown in Fig . 5.l3b), E is the total energy of three-particle system , and W(x,y,z) is the effective potential energy :

W(x,y, z) = -3 [U(x) + U(y) + U(z)],

(5.29a)

where

U(x) =

x

J -

00

f(x')ctx' = -.V I

A

+

,Ve- A1X [A.I COS(A.2 X - cp) - A.2 sin(A.2x- cp) ] 2

(5.29b) and A and cp are some constants which can be determined from the comparison of the potential function U(x) with the asymptotic of a soliton solution of Eq. (5.24b) for a given velocity V . In the particular case V = 0, the parameters are cp = - n I 4 - 0.05 and 1.1 = 1.2 = h 12, see Eq . (5.24b). Note that, despite of the existence of three integrals (5.28), the system (5.27) is not completely integrable, because only two of these integrals are independent. Namely, Eqs . (5.28b), and (5.28c) represent the momentum and energy conservation, respectively, whereas Eq. (28a) is just a constraint (differentiation of Eq. (5.28a) over time obviously gives Eq . (5.28b), whereas in general the momentum conservation integral is an arbitrary constant). One of three variables in Eqs. (5.27) can be excluded with the help of the first equation of the set (5.28) and, correspondingly, the total 3D potential W (x,y, z) can be presented as a function of two variables, e.g., x and y :

W(x,y) = -3 [U(x) + U(y) + U(L-x- y)] . The structure of the reduced potential function W (x,y), including number of extremes, depends on the period L. Two simplest examples of the reduced potential function W(x,y) are shown in Fig . 5.14. Depending on soliton locations at the potential extremes, three qualitatively different configurations of stationary soliton arrangement are possible: i) an equidistant arrangement when Xo = Yo = zo = L13, ii) an arrangement with two equal distances Xo = Yo -I- zo, and iii) an arrangement with all three different distances Xo -I- Yo -I- zo: these three arrangements are illustrated in Fig . 5.15 with the help of interacting particles in the ring . Stability of equilibrium states can be studied as usual by substitution into Eqs. (5.26) of slightly perturbed coordinates: x = Xo + ~ ,y = Yo + 11 ,z = zo + " where ~ « xo, 11 « YO, ' « zo and ~ + 11 + , = O. After substitution of x, y and z in Eqs . (5.26), subsequent linearization of these equations and elimination of zo and' one obtains:

200

K.A. Gorshkov, L.A. Ostrovsky and Y.A. Stepanyants

y 15.J.:--

-

-

-

-

-

-

--,

y 20 -1.--

white

14

18

13

16

12

14

II

12

blue

-

-

-

-

-

-

--,

blue (a)

(b)

Fig. 5.14 Line levels of the reduced potential function W (x,y) for two values of the parameter L: (a) - L = 35; (b) - L = 40. Blue color corresponds to minima, white color - to maxima of the potential. z

z

z

(a)

(b)

(c)

Fig. 5.15 Three qualitatively different configurations of soliton arrangement: (a) an equidistant arrangement when x = y = z = L!3 , (b) an arrangement with two equal distances x = y 1= z, and (c) an arrangement with all three different distances x 1= y 1= z.

~ = - [f' (Yo)11 - 2f' (xog - f'( L - xo- YO)( ~ + 11 )], { 11 = - [f '(xog - 2f' (Yo)11 - f'(L - xo- YO)(~ + 11 )].

(5.30)

Solution of these linear equations can be sought in the exponential form : ( ~ , 11 ) rv eP1• Substitution of that into Eqs . (5.30) yields the following characteristi c equation:

pi = f'(xo) + f'(yo) + f'( zo ) ±R,

(5.31 )

where R = J [f' (xo) - f'(Yo)]2 + [f '(xo) - f' (zo)]2 + [f ' (yo) - f'(zo)]2. The cond ition of solution stability reduces to the requ irement < 0, otherwi se a real part of one of the characteristic expon ents Pl ,2 is positive, which corresponds to the growing solution and, hence , to instability. The latter case may be subdivided on two subcases: i) when imaginary parts of all roots are zero; i.e., p~ > 0, p~ > 0 (pure exponenti al instability) and ii) when they are nonzero ; i.e., p~ > 0, p~ < o (oscillating instability). Stability of equilibrium states naturall y depend s on the length L. This is illustrated by Fig. 5.16 (Obregon, 1993).

pi

5 Dynamics of Soliton Ch ains

201

25

20

15

10

5L-_ _.L-_ _.L-_ _.L-_ _..I..-_ _..I..-_ _..L-_ _ 15

20

25

30

35

40

45

L

Fig. 5.16 Equilibrium state s and their stability as functions of the period L for Eq. (5.23) . Gra y lines - exponentially unstable equilibriums. green lines - oscillatory unstable equilibriums. red lines- stable equilibriums.

In the particular case of Eq. (5.24b) with V = 0 and Xo = Yo = zo = L/3, Eq. (5.31) simplifies : p2

= -3e-L/ 3V2 cos (~_ ~ +0.05) .

3V2

4

(5.32)

For non small perturbations, solutions of reduced Eqs. (5.31) (with excluded variable z) were obtained numerically subject to the following initial conditions:

x(O)

= xo(1 + 8), y(O) = Yo( I - 8), x(O) = y(O) = 0,

(5.33)

where 8 is the dimensionless deviation from equ ilibrium . Results of calculations show that for small deviations 8 ;S 0.02 in the system of the period L = 30, oscillations are practically sinusoidal and such that two particles oscillate in antiphase (in opposite directions), while the third particle remains in rest in the equilibrium. When deviat ion becomes greater, numerical solutions demonstrate quasi-periodic oscillations. An example of such oscillations is shown in Fig. 5.17. 14 x

12 10 8 '--~-'~~---'-~---'--~--'--~--'------'-_

8000 12000 16000 20000 T Fig. 5.17 Example of qua si-periodic soliton oscillations as described by Eq. (5.27) subject to the initial conditions (5.33) with 8 = 0 .1.

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K.A. Gorshkov, L.A. Ostrovsky and Y.A. Stepanyants

Eventually, when the deviation becomes too large, solutions demonstrate very complex behaviour, which are possibly chaotic (see Fig. 5.18). With further increasing of parameter 8, the chaotic oscillations are replaced by regular although not sinusoidal oscillations, then again quasi-chaotic oscillations appear and so on. Such behaviour is typical for Hamiltonian systems with the intermittency of regular and chaotic motions (Zaslavsky, 2005) . 14 x

12 10

8 8000 12000 16000 20000 r Fig. 5.18 Example of random soliton oscillations as described by Eq. (5.27) subject to initial conditions (5.33) with 0 = 0.325 .

Thus , one can conclude that in the case of a partial differential equation having soliton solutions with nonmonotonic profiles, complex stationary solutions in the form of multisolitons can also exist. If soliton profiles contain several maxima and minima, infinite soliton chains of such solitons can be constructed. Depending on the distance between solitons in the chain, its oscillations may be regular, simple (quasi-sinusoidal) or complex (quasi-periodic), or irregular - possibly, random and chaotic . Examples of such soliton chains may be constructed for the fifth-order KdV equation (5.22a) (Kawahara and Toh, 1988; Kawahara and Takaoka, 1988, 1989) and model equation (5.22b) with y > 0 in both cases . Both these equations are rather universal models applicable for description wave processes in many physical systems . Even in the simplest case of a periodic chain containing only three solitons at the period the chain dynamics may be fairly complex representing an example of Hamiltonian chaos . One may expect even more complex oscillations when the number of solitons is more than three in each period . In conclusion note that all physical processes described and numerically modelled in this section were observed experimentally for electromagnetic waves in nonlinear transmission lines (Gorshkov and Papko, 1977).

5.5 Soliton stochastization and strong wave turbulence in a resonator with external sinusoidal pumping As long as solitons are considered as interacting particles, it is natural to put forth the problem of existence of a "soliton gas," i.e. a stochastic ensemble of solitons. This problem has been discussed since the famous work by Fermi, Pasta and Ulam

5 Dynamics of Soliton Chains

203

(1955), where an expected equipartition of modes failed to be established in an equation of a nonlinear string. After Zabusky and Kruskal (1965) demonstrated the particle-like features of solitons in the KdV equation, this unusual behaviour was related to mode synchronisation with the formation of solitons . The question of possible stochastization of solitons was given a negative answer (Zakharov, 1971): as long as in the KdV equation, solitons are not changed upon collisions, and their energy distribution remains unchanged, which is analogous to the case of a onedimensional ideal gas with pair collisions. This feature was discovered to be generic for all integrable systems. However, even in such simple equation as the KdV, the stochastization is possible if the phase shifts between solitons become significant in the process of their interaction with each other and external fields. An example of such stochastization was considered by (Gorshkov et al., 1977). In that work, a limited-length electromagnetic line consisting of Nnonlinear LC oscillatory circuits was studied. Both ends of the line were completely reflecting (so that the line was an electromagnetic resonator), with a periodic forcing (pump) at one end. Such a system is not completely conservative: a soliton, propagating back and forth along the line, periodically interacts (collides) with the end pump and, depending on the phase of these collisions, increases or decreases its energy. If there are several (or many) solitons in the line, interaction of the trial soliton with all others affects its propagation time and, hence, the phase of interaction with the pump changes at each period which can result in stochastization of the solitons or, in other words, strong wave turbulence. In the same paper, an estimate for the condition of such stochastic "heating" of the "soliton gas" in the resonator was made . The solitons were considered as the analogs of particles moving between two oscillating walls: a model used before in relation to the so called Fermi acceleration [details and further references can be found in (Zaslavsky, 1984, 1985,2005)]. According to these models, the stochastization criterion is K » I, whereK is the "phase stretching coefficient," characterizing change of phase upon to consequent interact ions of a soliton with the end pump. This issue was discussed with George Zaslavsky whose remarks and advices were very useful in the preparation of the paper (Gorshkov et a\., 1977). The experiment with the aforementioned electromagnetic line of N nonlinear cells showed that only regular soliton dynamics occurs when N < 80 [such a dynamic regime was observed earlier (Gorshkov et a\., 1973) and can be called the parametric pulse generation] .

s

(a)

(b)

w

Fig.5.19 A fragment of random soliton sequence (a) in the resonator and its Fourier spectrum (b) when K :::Y I.

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K.A. Gorshkov, L.A. Ostrovsky and Y.A. Stepanyants

But for N > 100, a random soliton sequence, a "soliton gas," was observed in accordance with the theoretical prediction based on the Chirikov-Zaslavsky criterion (Gorshkov et al., 1977). An example of such soliton gas and corresponding Fourier spectrum are shown in Fig. 5.19 .

5.6 Chains of two-dimensional solitons in positive-dispersion media Description of wave processes in terms of ensembles of interacting solitons appears to be very useful not only in one-dimensional but in multidimensional cases as well. In this section we describe the peculiarities of the dynamics of two-dimensional solitons using the known Kadomtsev-Petviashvili equation as an example: (5.34) Here the equation is presented in the dimensionless form with only one parameter (J' = ± I which controls the dispersion. Namely, (J' = I corresponds to the case of positive dispersion which is considered here . In such a case Eq. (5.34) is dubbed the KP I-equation, whereas in the case of negative-dispersion media «(J' = -I), the corresponding equation is dubbed the KP2-equation. Figure 5.20 illustrates the qualitative difference in the character of dispersion. aJ(k) 1.0

0.8

0.6

0.4

0.2 0.0 / / / '/

0.0

0.2

0.4

0.6

0.8

1.0

k

Fig. 5.20 Qualitative sketch of the dispersion relation for plane sinusoidal waves of infinitesimal amplitude described by linearized KdV equation (5.1).

5 Dynamics of Soliton Chains

205

It is evident that in the one-dimensional case when propagation of a plane wave is considered, one can let a = 0 and reduce the KP equation (5.34) to the KdV equation (5.1) (note that the character of the dispersion in this case in insignificant - the KdV equation can be presented in the dimensionless form regardless of the dispersion character in the real system). Hence, all known solutions to the KdV equation, including the soliton (5.2), are also particular solutions to the KP equation. It is currently well known that in isotropic nonlinear media, the evolution of multidimensional perturbations essentially depends on the character of the dispersion . In particular, Kadomtsev and Petviashvili (1970) discovered that in the case of positive dispersion, plane solitons moving in the x-direction and described by Eq. (5.2) are unstable with respect to the self-focusing, whereas in the negative-dispersion case, plane solitons (5.2) are stable . This conclusion was later confirmed by Zakharov (I975) who derived an exact formula for the instability growth rate. To be more precise, the self-focusing instability of plane solitons is determined by the "decaying" spectrum of small perturbations, i.e., by dispersion relation which allows the three-wave resonances between quasi-monochromatic waves. The spectrum is indeed decaying in such a sense in isotropic homogeneous media with weak positive dispersion. In anisotropic media, however, the spectrum may be decaying even in the case of negative dispersion (Abramyan et al., 1992). The linear and early nonlinear stages of the self-focusing instability within the framework of the KPI-equation have been studied by many authors [see, e.g., (Pelinovsky and Stepanyants, 1993) and references therein] . As was ultimately clarified , the nonlinear development of small perturbations of a soliton front gives rise, at the intermediate stage of instability, to periodic chains of two-dimensional solitons (the so-called KP lumps), which have larger amplitude and smaller velocity than the initial plane soliton (note that in the positive-dispersion media, solitons of greater amplitude move slowly). The initial plane soliton emitting lumps decreases in amplitude and moves faster than the chain of lumps. A lump is described by the formula (see also Fig. 5.21)

Fig.5.21 Qualitative sketch of the KP lump described by Eq. (5.35).

206

K.A. Gorshkov, L.A. Ostrov sky and Y.A. Stepany ants

9+ 16V 2y2 -12V~2

u(~ , y) =48V (9+ 16V2 y2+ 12V~2)2 '

(5.35)

where ~ = x + Vr, and V > 0 is the dimensionless velocity of the lump. Note that Eqs . (5.4) in application to KP lumps (5.35) can be transferred to a completely integrable system, the Calogero-Moser system of particles on a complex plane (Gorshkov et a1. , 1993). The corresponding set of equations follows from the Hamiltonian (Calogero, 1976) H

.2 = '"' i..JZk

k

'"' 2 )2' i..J ( l
(5.36)

where the complex coordinate of a lump with a number k is zk = (sk)' x + i(sk)'y ; the prime denotes differentiation on x or y, respectively, and the dot denotes differentiation with respect to time . As any system of classical particles , the system described by the Hamiltonian (5.36) has a particular equilibrium configuration in the form of an equidistant 10 chain of particles positioned either along the x-axis (the direction of motion) or along the y-axis (perpendicular to the direction of motion .) In both cases the equilibrium configuration is unstable. In particular, in the latter case, when the particles (lump solitons) situated along the y-direction move in the x-direction, the frequency of infinitesimal perturbations of the chain with the main period So and perturbation wave number is purely imaginary (Potapov et a1., 2001) : (5.37) The same result was obtained earlier for stationary periodic chains of lumps by Zaitsev (1983) and independently by Zhdanov and Trubnikov (1984) . In the latter work a hypothesis was also expressed that such periodic solution chain is the result of the development of the instability in a plane soliton of the KP I equation. However, Burtsev (1985) later discovered that the chains of lumps are also unstable against periodic deformations of their fronts . Based on this fact, he argued that the unstable chain of lumps cannot be a result of instability of a plane soliton . Many researchers inclined to the opinion that instability of a plane soliton simply leads to the spreading of wave energy in space [see, e.g., (Zakharov, 1975)]. In the meantime, the fact of stability of individual lumps was very well known . Further study of the problem showed (Pelinovsky and Stepanyants, 1993) that when the original chain of lumps is initially sinusoidally modulated, it decays into two other chain s with greater separation between the lumps . The Burtsev's solution describes only the early stage of such decay. Each of the secondary chains are also unstable, although the rate of such instability becomes smaller at each step. Hence, one can expect that the development of unstable quasi-plane solitary structures under the action of random perturbations gives rise, through a series of secondary instabilities of two-dimensional lump chains , to the creation of a disordered ensemble of chaotically moving lumps.

5 Dynamics of Soliton Chains

207

The analysis of the plane soliton instability also shows that the instability is closely associated with the resonant interaction of plane solitons with the chains of lumps. The soliton resonance is well known for negat ive-dispersion media (for the KP2-equation in particular). A simple example of such resonance was discovered in (Newell and Redekopp, 1977; Miles , 1977). In those papers the exact solutions for the KP2-equation were constructed; the solution describes triads of plane solitons at certain amplitudes and the angles between them. The soliton resonance is also possible in positive-dispersion media described by the KPI-equation, and this resonance plays an important role in the decay of quasi-plane initial perturbations. In the simplest case when there is only one plane soliton propagating in the x-direction and having a front slightly sinusoidally modulated in phase in the ydirection, the result of the self-focusing instability is the formation of a 2D-soliton chain (the chain of lumps) located parallel to the front of the original plane soliton ; the distance between the lumps moving side-by-side is equal to the period of the initial soliton modulation. This process is shown in Fig. 5.22. Analysis of the exact solution describing such a process reveals that the selffocusing instability occurs within a limited range of transversal wave numbers [see details in (Pelinovsky and Stepanyants, 1993)]. At one edge of this range when the modulation wavelength is very large, the solution describes a chain of rarefied lumps moving side by side behind the residual plane soliton, parameters of which (amplitude, velocity, half-width) differ little from the parameters of the initial plane soliton . At the other edge, the distance between the lumps is so small that they merge to reconstruct the initial, almost plane soliton (5.2) with a small-amplitude modulation of its front. The amplitude of another residual plane soliton (5.2), emerging in the process of such decay is infinitesimal. If the amplitude of phase modulation is not small, more than one chain of lumps may be created behind the residual small-amplitude plane soliton . Such a process was computed by Infeld et al. (1995). As was shown in that paper, the process of the chains ' emergence occurs step by step: one chain emerges first with a period equal to the period of initial perturbation, and then another chain of lumps of smaller amplitudes but with the same period as the first one emerges from the residual quasi-plane soliton. Calculations show that when the initial modulation is sufficiently strong , the third and, perhaps, even more chains may emerge. Chains may interact with each other resulting in rather complex patterns of lumps formation. Similarly, one can consider a more complicated process of plane soliton decay under the action of a quasi-periodic transverse perturbation. If the perturbation consists of only two sinusoidal modes whose growth rates are y, and Yz, then at the final stage two chains of lumps will be generated with different periods corresponding to two periods of transverse modulation of the initial quasi-plane soliton . In addition to those two chains of lumps, a residual plane soliton of small amplitude will be also generated. Although the result of the plane soliton decay in this case is qualitatively the same in the asymptotic (representing two chains of lumps and the residual plane soliton) , the intermediate stage of such process may be different depending on the relationship between the growth rates y, and yz . If y, > yz, chains of lumps are formed due to the subsequent single-mode decay of the plane soliton first to the one

208

K.A. Gorshkov, L.A. Ostrovsky and Y.A. Stepanyants x 12 - --

-

-

-

-

-

-

-

-

-

-,

3.3

- 12 "_ - II

...........

.....1._

..

o

11

Y

x 12 - --

-

-

-

-

-

-

-

-

-

-,

6.4

(b)

0

- 12 '__ - II

....1.-_

....1...

o

x

(c)

o

1.1

o - 12 "_ - II

o

0

0

...........

.....1._

..

o

11

Y

Fig. 5.22 The birth of 2D soliton chain as a result of the plane soliton instability . (a) t

=

0; (b)

t = 10; (c) t = 20. The number s indicate maxima of the wave field.

chain of lumps and the modulated plane residual soliton . Then , this residual soliton decays to another chain of lumps and to the final non-modulated residual plane soliton . This process is illustrated in Figs. 5.23a-c.

5 Dynamics of Soliton Chai ns

209 .r

.r 12

12

a

z 4.9

(a)

o

o

.r

(b)

120 Y

x

(c)

0

2.0 - 18 - 120

(c)

0

120 Y

- 24 - 120

.r

.r

42

24

0

(I)

0

120 Y

21.8

0

1.1

2.0

o

Y

o

Y

120 Y

Fig. 5.23 Plane soliton decay for a two-period initial perturbation. Left column - YI > /'2 . (a) t = 0, (b) t = 15. (c) t = 40; right colum n - YI < /'2 . (d) t = 0, (e) t = 10, (f) t = 25. Numbers indicate maxima of the wave field.

Alternatively. if Yl < Y2, a perturbed chain of lumps separa tes first in the course of the primary decay of the initially modulated plane soliton with the formation of residual small amplitude plane soliton . Then , the chain of lumps decays into two separate chains of different perio ds. This process is illustrated in Figs. 5.23d-f. In the part icular case of Yl = Yz, both scenarios of the plane soliton decay take place simultaneously, and no metastable intermediate structure can be clearly distingu ished in the process of decay.

210

K.A . Gorshkov , L.A. Ostrovsky and Y.A. Stepanyants

The results presented above can be generalized for an initial perturbation with any number M of discrete sinusoidal modes . In this case, the development of the self-focusing instability gives rise to M parallel chains of lumps propagating one after another with transverse wave numbers and to a single plane soliton with the smallest possible amplitude and highest velocity. Thus, the process of plane soliton decay for an M-mode perturbation is the M-fold decay of one of the two structures (either the perturbed plane soliton or the perturbed chain of lumps) formed at each intermediate stage of plane soliton decay. The specifics of this process depend on the ratio of the growth rates of unstable modes . These results can be generalized to the case when there is a localised perturbation on the front of the initial plane soliton . In this case the analysis shows (Pelinovsky and Stepanyants, 1993) that asymptotically, when t ----+ 0, the corresponding initial perturbation gives rise not only to the structures described above (plane soliton and chains oflumps), but also to separate two-dimensional solitons following the leading residual plane soliton . The number of such separate solitons, their amplitudes and relative position depends on the intensity of the initial perturbation of the plane soliton. An example of such a process was presented in (Infeld et aI., 1995) on the basis of numerical calculations. A self-focusing mechanism in the case of an arbitrary perturbation may, in general, give rise to a disordered ensemble of individual lumps and periodic chains of lumps as well as a residual plane soliton whose amplitude may be infinitesimal. If the decay of a plane soliton occurs in a closed system, the self-focusing instability results in creation of numerous two-dimensional solitons - lumps. Reflecting from the boundaries, the lumps may undergo multiple elastic collisions similar to collisions of KdV solitons (see, e.g., Ablowitz and Segur, 1981). As a result, the stochastic ensemble of lumps - a "gas" of 2D solitons - may be formed performing a specific sort of strong wave turbulence. This hypothesis, however, has not been examined thus far. To underpin and supplement the aforementioned hypothesis, consider also the evolution of a periodic chain of lumps arranged along the axis y and moving in the x-direction. As mentioned, such a chain is unstable against small transversal perturbations, which is similar to the self-focusing instability of a plane soliton (Burtsev, 1985). As a result of instability development, the chain disintegrates into secondary chains of lumps (Pelinovsky and Stepanyants, 1993). The typical process of chain disintegration for the case of simple periodic perturbation is shown in Fig. 5.24 . The analysis of decaying processes in the simplest case of plane solitary wave instability allows us to understand the problem of plane-wave instability in positivedispersion media . As nonlinear quasi-plane structures decay, the energy is not lost in small oscillations of the medium . Instead , it is condensed in two-dimensional and plane solitons (although in non-integrable systems some portion of energy may be scattered in the non-solitonic form of quasi-linear ripples). Since the model considered is conservative, these processes are reversible ; i.e., the soliton merging is also possible. Note that although all the results presented here are based on the integrability of the KP I-equation, it is believed that for other similar models with positive dispersion where the instability of a plane soliton relative to self-focusing

211

5 Dy namics of Soliton Chains x 9

38. 1

(a)

0

- 9 '--

...J.....+.

o

- 37

37 y

x 9 47.1

(b)

0

- 9 '--

---L_+_

o

- 37

37 y

x

9

(c)

54.2

0 13.2

-9

-37

ciJ

00 0

37 y

Fig. 5.24 Decay of a chain of lumps sin usoida lly modulated in y-direction: (a) (c) I = 20.

I =

0, (b)

I =

10,

was discovered, similar plane soliton decay may be observed provided the inelasticity effects of soliton interaction are negligible. However, the question of the origin

212

K.A . Gorshkov, L.A . Ostrovsky and Y.A. Stepanyants

of small oscillations of the medium which can accompany soliton decay needs an independent investigation.

5.7 Conclusion As was shown in this review, in many cases solitons can be treated as particles interacting through the potentials which are determined by asymptotics of their own fields. A specific feature of such particles , in contrast to the Newtonian particles, is that their asymptotics can be rather different (exponential, algebraic , oscillating). Note that in 2D and 3D cases the effective masses of solitons can be tensors rather than scalars (see, e.g., Gorshkov et al., 1993), although such cases were not considered here. A periodic chain of solitons may be stable or unstable depending on the basic governing equation, and soliton interactions within the chain may be elastic or inelastic. In the latter case, their interaction may be accompanied by a weak radiation . It is important to emphasise that the perturbation method described in this paper is equally applicable both to integrable and non-integrable governing equations and even to dissipative equations. Moreover, it can be used not only in the one-dimensional case, but in multidimensional cases as well (Gorshkov et al., 1993; Gorshkov, 2007) . On the basis of this method, soliton dynamics in two- or even three-dimensional lattices may be studied . As was shown in this paper, unstable soliton chains may result in a rather complex soliton dynamics which may even become stochastic under certain conditions. Here we presented only relatively simple examples of quasi-stochastic behaviour of solitons occurring as a result of chain instability . The problem of soliton stochastization in closed or periodic systems seems topical and promising from the point of view of application of the concept described in this review.

Few words in memory of George M. Zaslavsky The authors had numerous scientific and personal contacts with George while in Russia and later in the USA. He always impressed us by combining a seriou s and almost meticulous approach to scientific work with friendly and unbiased personal communications. George regularly participated in the Gorky Scientific Schools on Nonlinear Oscillations and Waves. He was one of the most popular lecturers. His lectures at the Schools provoked great interest, and after the lectures George was usually "attacked" in the lobby by numerous questions (one such after-lecture discussion is shown in the photo Fig. 5.25). On several occasions George visited our laboratory at the Radio physical Research Institute and later at the Institute of Applied Physics of the Russian Academy of Sciences in Nizhny Novgorod . Discussions with him were always interesting and fruitful and we acknowledged his useful

5 Dynamics of Soliton Chains

213

advice [see, e.g., (Gorshkov et al. 1977)]. We were also pleased to meet George at many other conferences and discuss with him not only scientific problems but general issues in art, literature, history, politics, etc. One of these informal meetings is reflected in the photo Fig. 5.26.

Fig.5.25 G.M. Zaslavsky at the Gorky School on Nonlinear Oscillations and Waves. Village Zholnino (near Gorky), March, 1973.

Fig. 5.26 G.M. Zaslavsky and L.A. Ostrovsky at a conference taking place on board of a ship cruising on the Yenisey River (early 1980s).

214

K.A. Gorshkov , L.A . Ostrovsky and Y.A. Stepanyants

References Ablowitz MJ. and Segur H., 1981, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia. Abramyan L.A. and Stepanyants YA., 1985a, Two-dimensional multisolitons: stationary solutions of Kadomtsev-Petviashvili equation, lzvestiya VUZov, Radiofizika, 28, 27-35 (in Russian. Engl. transl.: Radiophysics and Quantum Electronics, 1985a, 28, 20-26). Abramyan L.A. and Stepanyants YA., 1985b, The structure of two-dimensional solitons in media with anomalously small dispersion. Zh. Eksp. Teor.Fi., 88, 1616-1621 (in Russian. Engl. transl. : Sov. Phys. JETP, 1985,61,963-966). Abramyan L.A. and Stepanyants YA., 1987, Structure of two-dimensional solitons in the context of a generalised Kadomtsev-Petviashvili equation, lzvestiya VUZov, Radiofizika, 30, 1175-1180 (in Russian. Eng\. trans\.: Radiophysics and Quantum Electronics, 1987, 30, 861-865). Abramyan L.A., Stepanyants Y.A. and Shrira V\., 1992, Multidimensional solitons in shear flows of boundary layer type, DAN 327, 460-466 (in Russian. Engl. trans\. : Sov. Phys. Doklady, 1992,37,575-578). Apel L, Ostrovsky L.A., Stepanyants YA. and Lynch LE, 2007, Internal solitons in the ocean and their effect on underwater sound, J. Acoust. Soc . Am., 121, 695722. Bocchieri P., Scotti A., Bearzi B. and Loinger A., 1970, An harmonic chain with Lennard-Jones interaction , Phys. Rev. A ., 2, 2013-2019. Burtsev S.P., 1985, Instability of a periodic chain of two-dimensional solitons, Zh. Eksp. Teor.Fir., 88,1609-1615 (in Russian . Engl. trans\.: Sov. Phys. JETP, 1985, 61, 959-962). Calogero E , 1976, Exactly solvable two-dimensional many-body problems , Lett. Nuovo Cimento, 16,35-38. Dauxois T., 2008, Fermi, Pasta, Ulam, and a mysterious lady, Physics Today, January,55-57 . Devaney R.L., 1976, Homoc1inic orbits in Hamiltonian systems, 1. Diff. Eq., 21, 431-438. Dubrovin B.A., Matveev Y.B. and Novikov S.P., 1976, Non-linear equations of the KdV type, finite zone linear operators and Abelian varieties, Uspekhi Mat. Nauk, 31,55-136 (in Russian. Engl. transl. : Russian Math. Surveys, 1976,31, 59-146). Fermi E., Pasta J. and Ulam S., 1955, Los Alamos Scientific Laboratory Report LA1940, Later published in Segre E, ed., 1965, Collected Papers of Enrico Fermi, Y. 2., Univ. of Chicago Press, Chicago . Fraunie P. and Stepanyants Y, 2002, Decay of cylindrical and spherical solitons in rotating media, Phys . Lett. A, 293, 166-172. Galkin Y.M. and Stepanyants YA., 1991, On the existence of stationary solitary waves in a rotating fluid, Prikl. Matamat. i Mekhanika , 55, 1051-1055 (in Russian. Eng\. trans\. : J. Appl. Maths. Mechs., 1991,55,939-943.)

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Gerdjikov V.S., Kaup OJ., Uzinov I.M. and Evstatiev E.G., 1996, Asymptotic behavior of N -soliton trains of the nonlinear Schrodinger equation, Phys. Rev. Lett., 77, 3943-3946. Gorshkov KA., 2007, Perturbation Theory in the Soliton Dynam ics, The Doctoral Degree Thesis, Institute of Apllied Physics, Russian Academy of Sciences, Nizhny Novgorod, 252 p (in Russian) . Gorshkov K.A., Ostrovskii L.A . and Papko V.v., 1973, Parametric amplification and generation of pulses in nonlinear distributed systems, lzvetiya VUZov, Radiofizika, 16, 1195-1204 (in Russian. Eng\. trans\. : Radiophys. and Quantum Electronics, 1973, 16, 919-926). Gorshkov K.A., Ostrovsky L.A . and Papko V.v., 1976, Interactions and bound states of solitons as classical particles, Zh. Eksp. Teo r.Fiz; 71, 585-593 (in Russian. Eng\. trans\. : Sov. Phys. JETP , 1976,44,306-311). Gorshkov KA ., Ostrovsky L.A . and Papko V.v., 1977 , Soliton turbulence in the system with weak dispersion, DAN SSSR , 235, 70-73 (in Russian). Gorshkov K.A., Ostrovsky L.A ., Papko V.V. and Pikovsky A.S ., 1979, On the existence of stationary multisolitons, Phys. Lett. A , 74, 177-179. Gorshkov KA. and Papko V. V. , 1977, Dynamic and stochastic oscillations of soliton lattices, Zh. Eksp. TeorFiz., 73, 178-187 (in Russian. Eng\. trans\. : Sov. Phys. JETP, 1977,46,92-97). Gorshkov K.A. and Ostrovsky L.A ., 1981, Interaction of solitons in nonintegrable systems: direct perturbation method and applications, Physica D, 3, 428-438. Gorshkov KA. , Ostrovsky L.A. , Soustova l.A. and Irisov V.G., 2004, Perturbation theory for kinks and its application for multi soliton interactions in hydrodynamics, Phys. Rev. E, 69,016614. Gorshkov K.A., Pelinovsky D.E. and Stepanyants Y.A., 1993, Normal and anomalous scattering, formation and decay of bound-states of two-dimensional solitons described by the Kadomtsev-Petviashvili equation, Zh. Eksp. Teor.Fiz. , 104, 2704-2720 (in Russian. Engl. transl. : Sov. Phys. JETP, 1993,77, 237-245). Gorshkov KA. and Soustova l.A. , 2001 , Interaction of solitons as compound structures in the Gardner mode\. lzvetiya VUZov, Radiofizika , 44, 502-514 (in Russian. Eng\. transl. : Radiophys. and Quantum Electronics, 2001 , 44, 465-476). Grimshaw R., 1979, Slowly varying solitary waves. I Korteweg-de Vries equation, Proc. Roy. Soc., 368A, 359-375; Slowly varying solitary waves, II Nonlinear Schrodinger equation, Proc. Roy. Soc ., 368A, 377-388. Grimshaw R., Pelinovsky D., Pelinovsky E. and Slyunyaev A., 2002, Generation of large-amplitude solitons in the extended Korteweg-de Vries equation, Chaos, 12, 1070-1076. Grimshaw R., Pelinovsky D. and Talipova T., 1997, The modified Korteweg-de Vries equation in the theory of large-amplitude internal waves, Nonlin. Proc. in Geophy s, 4,237-250. Hasimoto H., 1970, Kagaku, 40, 401-403 (in Japanese). Infeld E., Senatorski A. and Skorupski A.A., 1995, Numerical simulations of Kadomtsev-Petviashvili soliton interactions, Phys. Rev. E, 51,3183-3191 .

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Kadomtsev B.B., Petviashvili VI., 1970, On the stability of solitary waves in weakly dispersive media. DAN SSSR 192, 753-756 (in Russian. Engl. transl. : Sov. Phys. Doklady, 1970, 15, 539-54\). Kakutani T. and Ono H., 1969, Weak non-linear electromagnetic waves in a cold, collision-free plasma, J. Phys. Soc. Japan , 26, 1305-1318. Karpman VI. and Maslov E.M., 1977, Perturbation theory for solitons, Zh. Eksp. Teor.Fiz. 73,537-559 (in Russian. Engl. transl. : Sov. Phys. JETP, 1977,46, 281295). Kaup OJ . and Newell AC., 1978, Soliton as particles, oscillator and in slowly changing media : a singular perturbation theory, Proc. Roy. Soc. London A ., 301, 413-446. Kawahara T., 1972, Oscillatory solitary waves in dispersive media, J. Phys. Soc. Japan, 33, 260-264. Kawahara T. and Takaoka M., 1988, Chaotic motions in oscillatory soliton lattice, ~ Phys.Soc.Japan,57,3714-3732. Kawahara T. and Takaoka M., 1989, Chaotic behavior of soliton lattice in an unstable dissipative-dispersive nonlinear system, Phys ica D, 39, 43-58. Kawahara T. and Toh S., 1988, Pulse interactions in an unstable dissipative dispersive nonlinear system, Phy s. Fluids, 31, 2103-2111. Keener J.P. and McLaughlin D.W. I977a, Soliton under perturbation, Phys. Rev. A ., 16,777-790. Keener J.P. and McLaughlin D.W. 1977b, Green function for a linear equation associated with solitons, J. Math. Phys., 18, 2008-2013. Kivshar Yu.S. and Malomed B.A, 1989, Dynamics of solitons in nearly integrable systems, Rev. Modern Phys., 61, 763-915. Kuznetsov E.A. and Mikhailov A.V , 1974, Stability of stationary waves in nonlinear weakly dispersive media, Zh. Eksp. TeorFiz., 67, 1019-1027 (in Russian . Engl. transl. : Sov. Phys. JETP, 1974,40,855-859). Lee Ch.- Y. and Beardsley RC; 1974, The generation of long nonlinear intemal waves in a weakly stratified shear flows, J. Geophys. Res., 79,453-457. Lennard-Jones J.E., 1924, On the determination of molecular fields. II. From the equation of state of a gas, Proc. Roy. Soc. A, 106,463-477 (see also Wikipedia: http ://en.wikipedia.org/wi kilLen nard-Jones-potential). Leonov AI., 1981, The effect of Earth rotation on the propagation of weak nonlinear surface and intemal long oceanic waves, Ann . New YorkAcad. Sci., 373, 150-159. Liu Y. and Varlamov V, 2004, Stability of solitary waves and weak rotation limit for the Ostrovsky equation , J. Diff. Eq. , 203, 159-183. Manakov S.V, 1974, Complete integrability and stochastization of discrete dynamical systems, Zh. Eksp. Teor.Fiz., 67,543-555 (in Russian. Engl. transl. : Sov. Phys. JETP, 1974,40,269-274). Miles J.w. , 1977, Obliquely interacting solitary waves, J. Fluid Mech., 79, 157-169. Resonantly interacting solitary waves. ibid. 171-179. Nagashima H., 1979, Experiment on solitary waves in the non-linear transmissionline described by the equation du /dt+udu /d z-d 5u /dz5 = 0, J. Phys. Soc. Japan, 47, 1387-1388.

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Newell A.C. and Redekopp L., 1977, Breakdown of Zakharov-Shabat theory and soliton creation, Phys. Rev. Lett., 38, 377-380. Obregon M.A., 1993, Chaotic dynamics of soliton chains in rotating media with positive dispersion, Master Degree Thesis, NNGU, Nizhny Novgorod . Obregon M.A and Stepanyants Yu.A, 1998, Oblique magneto-acoustic solitons in a rotating plasma, Phys Lett A 249, 315-323. Ostrovsky L.A., 1978, Nonlinear internal waves in a rotating ocean. Okeanologia, 18, 181-191 (in Russian. Engl. transl.: Oceanology, 18, 119-125). Ostrovsky L.A and Gorshkov K.A, 2000, Perturbation theories for nonlinear waves, In: Christiansen P, Soerensen M (eds.) Nonlinear science at the down at the XXI century, 47-65. Elsevier, Amsterdam . Ostrovskii L.A and Pelinovskii E.N., 1972, Method of averaging and the generalized variational principle for nonsinusoidal waves, Prikladnaya Matematika i Mekhanka (PMM), 36, 71-78 (in Russian. Engl. transl.: Appl. Math. Mech., 1972, 36,63-70). Ostrovsky L.A. and Stepanyants YA ., 1989, Do internal solitons exist in the ocean, Rev. Geophy s., 27,293-310. Ostrovsky L.A and Stepanyants Yu.A, 2005, Internal solitons in laboratory experiments: Comparison with theoretical models, Chaos, 15,037111. Pelinovsky D.E. and Stepanyants Yu.A., 1993, New multisoliton solutions of the Kadomtsev-Petviashvili equation, Pis'ma v ZhETF, 57, 25-29 (in Russian. Engl. trans!': JETP Lett ., 1993,57,24-28). Potapov Al. , Pavlov l.S ., Gorshkov K.A and Maugin G.A, 2001, Nonlinear interactions of solitary waves in 2D lattice, Wave Motion, 34, 83-95. Rybak S.A. and Skrynnikov Yu.I., 1990, A solitary wave in a uniformly curved thin rod, Akust Zhurn, 36, 730-732 (in Russian. Engl. transl: A single wave in a thin rod of constant curvature . Sov. Phys. Acoustics, 1990, 36, 410-411). Scott A.C., Chu EYE and McLaughlin D.W., 1973, The soliton: a new concept in applied science, Proc. IEEE, 61, 1443-1483. Slyunyaev A V, 200 I, Dynamics of localized waves with large amplitude in a weakly dispersive medium with a quadratic and positive cubic nonlinearity, Zh. Eksp. Teor.Fiz. , 119,606-612 (in Russian. Eng\. trans\.: JETP, 2001, 92, 529534). Slyunyaev A.V and Pelinovski E.N., 1999, Dynamics of large-amplitude solitons, Zh. Eksp. Teor.Fiz., 116,318-335 (in Russian. Engl. transl.: JETP, 1999,89, 173181.) Stepanyants YA , 2005, Dispersion of long gravity-capillary surface waves and asymptotic equations for solitons, Proc. Russ. Acad. Eng. Sci. Ser. Appl. Math. And Mech, 14, 33-40 (in Russian). Stepanyants YA ., 2006, On stationary solutions of the reduced Ostrovsky equation: periodic waves, compactons and compound solitons, Chaos, Soliton s and Fractals, 28, 193-204. Toda M., 1989, Theory ofNonlinear Lattices 2nd ed., Springer-Verlag, Berlin. Vakhnenko VA, 1999, High-frequency soliton-like waves in a relaxing medium, J. Math. Phys. , 40, 2011-2020 (Early version: Vakhnenko VA, 1991, Peri-

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odic short-wave perturb ations in a relaxing medium , Prep rint, Inst. of Geophy s., Ukrainian Acad, Sci., Kiev). Weisstein E.W., 2003 , CRC Conc ise Encyclop edia of Mathematics 2nd ed., Chapman & Hall/CRC , Boca Raton et al. Whitham G.B., 1965, Nonlinear dispersive waves, Proc. Roy. Soc. A., 283,238-261 . Whitham G.B., 1974, Linear and Nonlinear Waves, Wiley-Interscience, New York. Zabu sky N.J. and Kruskal M.D., 1965, Interaction of "solitons" in a collis ionless plasma and the recurrence of initial states, Phys. Rev. Lett., 15,240-243. Zaitsev A.A., 1983, Formation of stationary nonline ar waves by superposition of soliton s, DAN SSSR , 272,583-587 (in Russian. Engl . transl .: Sov. Phys. Doklady, 1983, 28, 720-722). Zakharo v, 1971, Kinetic equation for solitons, Zh. Eksp. Teor.Fiz., 60,993-1000 (in Russian. Engl. transl.: Sov. Phys. JETP, 1971,33,538-541). Zakharov, 1975, Instability and nonlinear oscillations of solitons, Pis 'ma v ZhETF, 22, 364-367 (in Russian . Eng!. transl .: JETP Lett., 1975,22, 172-173). Zakharov Y.E., Manako v S.Y. , Noviko v S.P. and Pitaevsky L.P., 1980, Theory of Solitons: The Inverse Scatt ering Method, Nauka, Moscow (in Russian. Engl. trans!.: Zakharov Y.B., Manakov S.Y. , Novikov S.P. and Pitaevsky L.P., 1984, Theory ofSolitons, Consultant Bureau, New York). Zaslavsky G.M., 1972, Scattering and transformat ion of nonlinear period ic waves in an inhomo geneou s medium , Zh. Eksp. Teor Fiz., 62,2129-2140 (in Russian. Engl. trans!.: Sov. Phys. JETP, 1972,34, 622-625). Zaslavsky G.M., 1984, Stochastisity of Dynamical Sys tems, Nauka , Moscow (in Russian). Zaslavsky G.M., 1985, Chaos in Dynamic Sys tems, Harwood Academic Publi shers, NY. Zaslavsky G.M., 2005, Hamiltonian Chao s and Fractional Dynamics, Oxford University Press, New York. Zhdanov S.K., Trubnikov B.A., 1984, Pis 'ma Zh. Eksp. Teor. n«, 39, 110-113 (in Russian. Engl. transl.: JETP Lett., 1984,39, 129-132).

Chapter 6

What is Control of Thrbulence in Crossed Fields? - Don't Even Think of Eliminating All Vortexes! Dimitri Volchenkov

Abstract Convective instabil ity in the cross-field system of thermonuclear reacto rs can be overridden by poloidal drifts. While in crossed fields, a long-tim e, large-scale turbulent regime, in which the eddie s of some particular size are destined to persist longer than usual, would come into being. Perhap s, we may keep such vortexes using them as tools for mainta ining the stability of still an illusory con struct of plasma fusion .

Councilor Hamann : Down here, sometimes I think about all those people still plugged into the Matrix and when I look at these machines I can't help thinking that in a way we are plugged into them . Nco: But we control these machines; they don 't control us. Councilor Hamann: Of course not. How could they? The idea is pure nonsense. But ... it does make one wonder... j ust... what is control? Nco: If we wanted, we could shut these machines down. Councilor Hamann: Of course. That 's it. You hit it. That's control , isn't it? If we wanted we could smash them to bits! ... Although, if we did, we'd have to consider what would happen to our lights, our heat, our air... Nco: So we need machines and they need us. Is that your point, Councilor? Councilor Hamann: No. No point. Old men like me don't bother with making points. There's no point. Nco: Is that why there are no young men on the counc il? Councilor Hamann: ... Good point.

"The Matrix Reloaded" , the second film in The Matrix franchise, Writt en and directed by Andy & Larry Wachowski.

Dimitri Volchenkov I he Center 01 Excellence Cogmtlve Interaction Iechnology (Cl lEe), Umverslty 01 BIelefeld, Postfach 100131, D-33501 , Bielefeld, German y, e-mail: [email protected]

A. C. J. Luo et al. (eds.), Long-range Interactions, Stochasticity and Fractional Dynamics © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010

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6.1 Introduction The international project on magnetic confinement fusion is designed to make the transition from today's studies of plasma physics to future electricity-producing fusion power plants . A successful fusion device has to contain the particles in a small enough volume for a long enough time for much of the plasma to fuse. Once fusion has begun, neutrons having a vast kinetic energy radiate from the reactive regions of the plasma, crossing magnetic field lines easily due to charge neutrality, barraging, together with charged particles, the wall blanket of the containment chamber, and degrading its structure. The reliable confinement (or control) of very energetic particles is one of the crucial problems arisen in course of the fusion project. Despite lasting efforts , the strategy of effective plasma flow control of a turbulent boundary layer is still mostly unclear that threatens our hopes for the successful implementation of the project in the near future. Here, we show that control of turbulence being understood in the framework of traditional paradigm as elimination of all long-living turbulent fluctuations in plasma flows is by no means compatible with symmetry of the crossed-field system and inevitably breaks down its stability. While trying to gain control over turbulent patterns in crossed fields, we are perhaps plugged into vortexes keeping some of them as tools for maintaining the stability of still an illusory construct of plasma fusion. In the forthcoming section (Sect. 6.2), we demonstrate that while in crossed fields, an alternative long-time, large-scale sate would exist in which the eddies of some particular size are destined to persist during essentially long time. In Sect. 6.3, we investigate the stochastic problem of the long-range turbulent transport in the Scrape-Off Layer of thermonuclear reactors and calculate (in the one-loop approximation) the magnitude of poloidal drift required to override convective instability in the cross-field system. We conclude in the last section. In our study of the stochastic counterparts of models in nonlinear dynamics, deterministic trajectories are replaced by random trial trajectories of some well defined stochastic processes. The proposed approach is closely related to the Nelson stochastic mechanics, the probabilistic interpretation of dynamical equations, and the critical phenomena theory. We thoroughly use the renormalization group (RG) method - one of the most important non-perturbative techniques developed in the framework of the quantum-field theory. Asymptotic solutions for the models in stochastic dynamics are obtained in the form of a perturbation theory which can be studied by means of Feynman functional integrals . Diagram series of the perturbation theory can sometimes be studied by means of renormalization group techniques . In statistical mechanics, the RG (which is, in fact, a semi-group since the transformations are not invertible) forms an ensemble of transformations that map a Hamiltonian into another Hamiltonian by the elimination of degrees of freedom with respect to which the partition function of the system remains invariant. The RG allows calculating the critical exponents related to phase transitions in renormalizable models .

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221

6.2 Stochastic theory of turbulence in crossed fields: vortexes of all sizes die out, but one A model of the fully developed turbulence based on the stochastic Navier-Stokes equation with an external random force and a model of magneto hydrodynamic equation s supplemented with stochastic force terms can be formulated as the quan tum field theorie s. We use the powerful method s developed in the Quantum Field Theory to investigate the critical regimes in turbulence and their stability. The existence of dissipation minimum in the sub-leading dissipation regime predicts essentially long lifetime for eddie s of some prefer able size.

6.2.1 The method ofrenormalization group Ultraviolet renormalization has been developed in the framework of quantum field theory in 1953. An article by E.e.G. Stueckelberg and A. Peterm an in 1953 and another one by M. Gell-Mann and EE. Low in 1954 opened the field by a study of the fact of invariance of the renorm alized quantum field action under the variation of bare parameters at the subtraction point. In the framework of quantum field theory, the renorm alization group (RG) was developed to its contemporary form in the wellknown book of Bogoliubov and Shirkov, in 1959. The technique was developed further by R. Feynman, J. Schw inger and S.-I. Tomonaga, who received the Nobel prize for their contributions to quantum electrodynamics. However, these techniques have not been implemented in critical phenomena theory until the works of Leo Kadanoff who had proposed a simple blocking procedure in 1966. In 1974-1975, Kenneth Wilson had used it in order to solve the famou s Kondo problem . In 1982, he was awarded by the Nobel Prize for this work . It is important to mention , in concern with the Kondo effect, the work of P. W. Anderson , D.R. Hamman , and A. Yuval (1970) , in which the techniques similar to that of RG had been used in critical phenom ena theory, independently of Wilson 's approach. The old-style RG in particle phy sics was reformulated in 1970 in more physical terms by e.G. Callan and K. Symanzik. Later (1974) , M. Fowler and A. Zawadowski developed the method of multiplicative renorm alization in the framework of quantum-field theory. It is remark able that the mathematical background beyond the RG is quite simple and has been known long time before Peterman and Kadano ff; it is called the compactification procedure. Logarithmic divergence s arise since the integration domain is not comp act. If we find a way how to project the model onto a compact manifold (in d + I dimensional space, the new dimensional parameter f.1 is called a renorm alization mass), we gain finite amplitudes for integral s (see Fig. 6.1). In general , such a "projection" is irrelevant since it breaks the natural physic al scales, however, it may have a sense if the model possesses a property of scaling invariance.

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Fig. 6.1 The compactification procedure .

Let us suppose that we have found a way how to redefine the model on a compact manifold, and therefore the diagram series converges . Physically relevant results should not depend upon fl. The interaction charges have to be independent of u, and the sum should be invariant with respect to uniform dilatations of all its arguments. The Green function has to be an eigenfunction of the dilatation operator belonging to some eigenvalue y. If now we make a simultaneous rescaling of momentum and mass by 1\., then the Green function G would rescale with a power factor. If G has a constant infrared asymptotic (in turbulence, it is called Kolmogorov constant), we obtain the infrared scaling for the long time large scale asymptotic behavior. Iterating the RG transformations R for the particular values of the initial bare parameters, it may be possible to attain a fixed point such that H*

= R(H*) ,

(6 .1)

where H is a Hamiltonian. In critical phenomena theory, the RG transformations R rely upon the rescaling of the system variables described by the Hamiltonian H at the fixed point H * that has the same appearance whatever the scale at which it is considered. This means that the correlation function of q>(x) (e.g. q>(x) may represent the magnetization density in a magnetic system or spins in the Ising model) must be of the form

(q>(r)q> (O))

rv

r- (d-2+1) ),

(6 .2)

i.e. that the system is at the critical point, in which a correlation length ~ = 00 . If we make a change to parameters of the Hamiltonian H, in the vicinity of the fixed point, H

= H * + LgiOi , i

(6.3)

6 What is Control of Turbulence

223

where O, are called operators and gi are called fields, then we can study how the Hamiltonian evolved under the action of the RG transformations. In order to clarify the idea of the method, let us imagine that transformations we like to study forms a continuous group . Then, the fields gi have to obey the equations of motion, (6.4) If we are interested in the stability analysis of the dynamical system described by (6.4), we linearize the function f3 in its r.h.s. It is clear that linearized equations have the solutions (6 .5)

for some parameter yt > 0, so that the field gi increases due to the renormalization transformation; it is said that gi is an essential field (or a "relevant field"). Otherwise, for Yi < 0, the field gi decreases under the action of renormalization transformations, and called an inessential field (or an "irrelevant field"). Finally, if yt = 0, the field gi does not vary in the linear order and is called a marginal field. In the later case, to investigate the stability of the fixed point we need to go beyond the linear order. In critical phenomena theory, temperature and the magnetic field are those fields pertinent near the critical point. Solving the renormalization group equations, we obtain that in the vicinity of critical point the free energy is of the form F

=

(T - T,:-)(2-a ) f

(

H

(T - Tc")y+f3

)

(6.6)

and therefore satisfies the Widom 's hypothes is of homogeneity. More generally, the RG allows to predict all critical exponents pertinent to the system at the fixed point by studying how its Hamiltonian is transformed by the RG at the fixed point. There are many ways to implement the RG techniques for real-world models . In large scales (small moments), the asymptotic behavior predicted by the RG can be modified . The corrections are calculated by means of the Short Distance Expansion method . They are related to scaling behavior of composite operators, the local averages with respect to a point. Namely these quantities can be measured experimentally . If their scaling dimensions are negative, they can alternate the asymptotic behavior. Scaling dimensions are inherent not to composite operators themselves, but to their certain linear combinations which have a physical meaning. If the coupling constant in quantum field theory is not small, we have deal with the essentially non-perturbative regime, and such a theory is said to be asymptotically free for low energies. The non-perturbative regime is difficult to study, because of in addition to the problem of divergences of Feynman diagrams in perturbative series we have to deal with the essentially non-perturbative contributions coming from the instantons which cannot be neglected . In quantum mechanics and quantum field theory, an instanton is a classic solution of equations of motion, i.e. one of local minimums of the action functional , but not the global one. Mathematical methods developed in quantum fields theory are beyond any doubt applicable also in Eu-

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Dimitri Volchenkov

clidean space to classical problems involving random fields. Models of stochastic nonlinear dynamics can be reformulated as models of quantum fields theory, and then the powerful techniques developed in that can be used.

6.2.2 Phenomenology offully developed isotropic turbulence Despite more than a century of work and a number of important insights, a complete understanding of turbulence remains elusive, as witnessed by the lack of fully satisfactory theories of such basic aspects as transition and the Kolmogorov "5 /3-spectrum". In phenomenological theory of turbulence formulated by A.N. Kolmogorov in 1941, it was conjectured that the correlation functions of velocity in some intermediate scales (called the inertial interval) depend upon the only dimensional parameter, the power of energy pump W . It was supposed that energy comes from large scale eddies which bifurcate due to nonlinear interactions until the small scale vortexes are dissipated in fluid at the minimal scale. The only physically relevant combination of energy pumping rate Wand momentum k gives the Kolmogorov asymptotic for the fully developed turbulence. It follows then that the velocity of fluid has a formal dimension -1 /3 , and the famous I-dimensional energy spectrum is -5 /3 . This result has been justified in the framework of RG techniques by many authors. The recent theoretical, compu tational and experimental results dealing with homogeneous turbulence dynamics have been summarized in (Sagaut and Cambon, 2009) . In the present section, we follow the seminal work (Adzhemyan et al., 1996). To describe the spectral properties of incompressible fluids in the inertial range of developed turbulence, one considers the stochastic Navier-Stokes equation with an external random force (Monin and Yaglom, 1971, 1975; Wyld, 1961). (6.7) here ~i is the vector velocity field, which is transverse due to the incompressibility condition, p and F; are the scalar pressure field and transverse external random force per unit mass (all these quantities depend on x == (t,x)), Vo is the kinematical coefficient of viscosity, and VI is the Galilean-invariant covariant derivative . Equation (6.7) is studied on the entire t axis and is supplemented by the retarda tion condition and by the condition that ~ vanish asymptotically for t ----+ - 0 0 . We take F to be a Gaussian distribution with zero average and correlator

(t - t') (F;() x Fj (x ' )) = 8(2;rr)d where

J () () .( ') dk Fij k dF k exp lk x - x ,

k.k,

PiIJ-(k) = 8--_ _kI_J IJ 2

(6.8)

(6.9)

225

6 What is Control of Turbulence

is the matrix of transverse projector in the momentum (Fourier) representation, dF(k) is some function of the momentum k == [k] and the model parameters, and d is the dimension of the physical space. The introduction of a random force phenomenologically models the stochastic drive (which in a real situation must arise spontaneously as a consequence of the instability of laminar flow) and, at the same time, the injection of energy into the system owing to the interaction with large-scale eddies. The average power W of the energy injection is related to the function d F in (6.8) by the equation

W

=

d-I 2(2n)"

j'dkdF(k) .

(6.10)

In the stochastic problem we can also do away with specific initial and boundary conditions and directly study homogeneous, fully developed turbulence (Monin and Yaglom, 1971, 1975 ; Wyld, 1961) . The field

(
)

and also the response functions, the variational derivatives of the correlation functions with respect to a pseudo-random external force introduced additively into the right-hand side of (6.7). Borrowing quantum field terminology, we shall refer to all these objects as Green functions . Due to the transversal invariance of the problem, all they depend only on the time variable and coordinate differences, so that the equal time correlation functions (
j(m/k) for mfk

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Dimitri Volchenkov

that in this region of applicability the pair correlation function of the Fourier components of the velocity of the d-dimensional problem can be written as

( cp (OJ ,k )cp(OJ' ,k' ))

= (2n)d+18(OJ+OJ')8(k' +k)~j(k)D(OJ ,k) ,

(6.11)

in which (6.12) where f is some as yet unknown function of two independent dimensionless arguments . Representations like the above one can also be written down for more complicated correlation functions involving any number of fields cp. They follow only the hypothesis 2, and altogether imply the existence of an infrared (because the conditions k « l'~i~' OJ « voll~?n of hypothesis 2 do not suggest a lower limit) scale invariance (scaling) with fully defined Kolmogorov dimensions of all infrared - relevant quantities F = {cp == cp(x) ,m,t rv OJ-I ,r rv k- I} for irrelevant W, vo: ,1cp

=

-1 /3 ,

,1{

=

-,1 w

=

-2/3;

,1k

=

-,1 r

=

,1111

=

l.

(6.13)

The scaling is generalized uniformly under a combined consistent dilatation F ----+ ALlFF with arbitrary parameter A > of all relevant quantities F for fixed irrelevant quantities (zero dimensions can be formally assigned to the latter) . The infrared (IR) asymptote corresponds to A ----+ 0, and the statement about the existence of scaling , strictly speaking, refers not to the exact Green functions, but only to the leading term of their infrared asymptote A ----+ 0, because in representations like (6.12) the corrections vanishing for klmin ----+ and l~inOJ /vO ----+ have already been discarded. The aim of the theory is to justify scaling for a given microscopic model and to calculate the corresponding critical dimen sions of all IR-relevant quantities. In the theory of critical behavior the analogous problem has been solved successfully using the renormalization group method .

°

°

°

6.2.3 Quantum field theory formulation ofstochastic Navier-Stokes turbulence It is well known (Janssen, 1976; Phythian, 1977) that any model of stochastic dynamics with the t-Iocal interaction (i.e ., in which the interaction term does not contain time derivatives) driven by the Gaussian random force is equivalent to the quantum field theory model of the doubled set of fields, ep = cp, cp' . This means that statistical averages (... ) of random quantities can be ident ified with functional average s with weight expS( ep), so that the generating functionals of the full [G(A)] and connected [W(A)] Green functions of the stochastic dynamical problem are represented by the functional integral

G(A) =expW(A)

=

J

Depexp [S(ep)+Aep]

(6.14)

227

6 What is Control of Turbulence

with arbitrary sources A == Arp ,Arpl in linear form :

A

J

dx [Arp (x)
(6.15)

The generating functional of I-irreducible Green funct ions is defined by the Legendre transform ofW (A) with respect toA:

r(

x

M(x ) .

(6.16)

Here

S(
= ~JJ dr dr'
J

dx
(6.17)

in which DF is the random-forcecorrelator (6.8). The auxiliary field 0 describes the deviation from logarithm ic behav ior. In this model it is independent and completely unrelated to the space dimension d (in contrast to the theory of critical behav ior, where usually e = 4 - d) . The logarithmic theory corre spond s to the value e = 0, and the injection (6.18) becomes infrared only for e > 2. In the region 0 < e < 2 the injection (6.18) is ultraviolet; the integral (6.10) for it diverges at large k. Then there is understood to be a cutoff at k :::; ll~in ' and the integral is dominated by contr ibutions at scales k rv ll~i~ ' Most studies on the RG theory of turbulenc e use a simpler, purely power-law inject ion: (6.19)

228

Dim itri Volchenkov

corresponding to m = 0 in (6.18). This is possible if one is interested only in establishing IR scaling and the critical dimensions (which must be independent of m for any injection) , and simple objects like the scaling functions are calculated using the graphs of perturbation theory only in the form of a-expansions . Then the passage to the theory with m = 0 is con sistent, because the coefficient of the s-expansions of the graphs always have finite limits for m --+ O. For E = 2 the parameter Do in (6.19) acquires the dimension of W.ln addition, idealized injection by infinitely large eddies corresponds to DF(k) rv o(k) , and for suitable choice of the amplitude function it can be assumed to be a power-law model of the d-dimensional o-function. The integral (6.14) is a standard construction of quantum field theory, and so all the Green functions have the standard Feynman diagram representations (Vasil'ev, 1998). The lines in the graphs correspond to the elements of 2 x 2-matrix of bare propag ators (<1><1» 0, related as (<1> <1» 0 = K - 1 to the matrix K in the free (quadratic in <1» part of the action So

=

I --<1>K<1>

2

(6.20)

'

in which we imply the summation over indices and integration over continuous variables . In momentum frequency representation, the bare propagators of the model (6.17) have the form

(q)(P')o =( Cp' CP) o =

-iOJ~vok2 ' dF(k)

(cpCP) o =

OJ2

( cp' cp') o = O, (6.21)

+ vJk'"

with the function dF(k) from (6.8). All lines of the diagram technique are proportional in the vector indexess to the transverse projector ~j , which was omitted in writing (6.21) but is always understood to be present. The interaction in (6.17) is the three-leg vertex with vertex factor

Vijs= i(kAs+ksoij ),

(6.22)

where k is the momentum flowing into the vertex via the field cp'. As an illustration, in Fig. 6.2 we give the graphs of the exact Green functions (cp cp) and (cp cp') in the one-loop approximation. The lines in the graphs correspond to the bare propagators (6.21), the vertices correspond to the factors (6.22), the external legs with a slash correspond to the auxiliary field cp', and those without a slash correspond to the field cp. This diagram technique is known as the Wyld diagram technique (Wyld, 1961). The expansion parameter in perturbation theory (the coupling constant or the charge in the terminology of quantum field theory) is go == Do /vg with Do . The functional formulation (6.17) significantly simplifies the derivation of exact functional relations like the Schwinger equations and, of particular importance, allows the standard quantum-field RG technique to be applied to the stochastic NavierStokes equation. Recently, the field theoretic renormalization group has been ap-

6 What is Control of Turbulence

229

+ ('{J'{J') =

+--+C>--

--------+

+ ...

+ ~ +

Fig. 6.2 The Wyld diagram technique in the theory of turbulence.

plied to the stochastic Navier Stokes equation in connection with the problem of construction of the 1/ d expansion for the fully developed fluid turbulence beyond the scope of the standard s-expansion in (Adzhemyan et al., 2008) .

6.2.4 Analytical properties of Feynman diagrams All the graphs of the pair velocity correlator converge at large momenta and frequencies for finite e > 0, so that they can be calculated without a ultraviolet (UV) cutoff. In the calculations with dimensional regularization, the UV divergences arising for E --+ 0 are manifested as poles in e, and the perturbation series for the correlator has the form

(cp cp) = (cp cp)o

[1 + I~l (gok- £Y'A (w/vok 2

n

2

,

mf k, e)]

,

(6.23)

where go = Do/va with Do from (6.18) . The poles in e are contained in the coefficients An. We see from (6.23) that to define the k --+ 0 asymptote for fixed charge go and coefficients All it is necessary to sum the entire series. This is the infrared problem to be solved by the RG method. It is nontrivial for any e > 0, including the region 0 < e < 2, when the injection (6.19) is ultraviolet. Even for that the perturbation series contains IR singularities, which will also be summed by the RG method . The solution of this problem justifies the second Kolmogorov hypothesis. However, the coefficients All in (6.23) can also contain singularities for m fk --+ O. These singularities present in the theory turbulence for finite e > O. This problem cannot be solved just by the re-summation of the perturbation series using the ordinary RG technique. Analogous problems also arise in models of critical behavior and are solves by the theory of renormalization of composite operators using the Wilson operator expansion (Collins , 1992; Zinn-lustin, 1990).

6.2.5 Ultraviolet renormalization and RG-equations Details on the RG procedure in the quantum field theory can be found in (Collins, 1992; Zinn-lustin, 1990) and in the comprehensive book of N.N. Bogolubov and

230

Dimitri Volchenkov

D. V. Shirkov (Bogolubov and Shirkov, 1980). The RG analysis of stochastic hydrodynamics had been discussed in (De Dominicis and Martin, 1979; Adzhemyan et al., 1983; Adzhemyan et al., 1998), it contains anomalously few UV divergences compared to the usual models of the critical phenomena theory like the famous cp4_ model. The standard analysis of UV divergences with regards to the canonical dimensions of fields shows that for d > 2 superficial divergences exist only in the 1irreducible functions (cp cp') and (cp cp'cp ), the corresponding counterterms must contain the symbol a. Therefore, the first function generates only the counterterm cp' .1cp without the addition cp'a, cp of the same dimension, and the second generates only the counterterm cp' (cpa) cp when the the fact that all fields are transversal is taken into account. Although this counterterm is allowed by the conventional dimensional analysis, it is actually forbidden by Galilean invariance, which requires that the differential operators at and cpa enter into the counterterm only as the covariant derivative V t = at + cpa, (De Dominicis and Martin , 1979). The absence of the counterterm cp'at cp implies the absence of cp' (cpa)cp . In the special case d = 2 a new superficial divergence appears in the function (cp' cp' ) generating the local counterterm cp'.1 cp'. In this section, we suppose that d > 2. All UV divergences can be removed from the perturbation theory of the renormalized action (6.24) in which Z, is the renormalization constant (the necessary summation over indexes and integrations with respect to time and space are implied). It is completely dimensionless and can be therefore depend only on the completely dimensionless renormalized parameter g. In renormalization using the minimal subtraction scheme, only the poles in care subtracted from the divergent expressions without changing the finite contributions, and the renormalization constants Z always have the following form : 00

Z = 1+ I>k(g)e -ke - k = 1+ k=1

00

11

11=1

k=l

L g" L allke- k.

(6.25)

The coefficients allk in our models can depend only on the space dimension d, and the absence of e in the residues ak(g) is a feature specific to the minimal subtraction scheme. One-loop graphs generate contributions of order g, two-loops ones generate contributions of order s'. and so on. The order of the pole in E never exceeds the number of loops in the graph . The renormalized action (6.25) is obtained from its non-renormalized analog by the following dilatations of the parameters: Vo

= V Z v , go = gJ1 2cZg ,

(6.26)

6 What is Control of Turbulence

231

Fields do not require renormalization, Z<j> = I. If the power-low injection is replaced by the realistic model, the bare mass parameter m also does not require renormalization, mo = Zmm, and Zm = I. The only independent renormalization constant Z, is calculated directly from the graphs of the perturbation theory. It determines the corresponding RG functions YF(g), the anomalous dimensions of a quantity F , and any charge g is used to introduce the corresponding /3-function : (6.27) in which eo

= {vo,go} are the bare parameters of the theory. In particular, one has (6.28)

and the RG operator takes the form (6.29) Calculation of the constant Z, for the model of stochastic hydrodynamics in the one-loop approximation gives, (Adzhemyan et al., 1983), 2n d / 2(d - I)g

Z,

= 1- 4(d + 2)r(d /2)(2n)d + O(g

2

(6.30)

).

The solutions of RG equation (6.31) are stable with respect to the infrared asymptotes if there exists a fixed point g; such that /3(g*) = 0, /3'(g*) > 0 in the physical region g > 0 for small e > O. When the IR-stable fixed point is present , the leading terms of the IR asymptote of the Green functions W;~ of any single charge model satisfy the RG equation with the replacement g ---+ g*. In particular, we obtain (6.32) Canonical scale invariance is expressed by the equations (6.33) in which F

= {t ,x ,,u,v,g ,m} is the set of all arguments of

W;~,

n = {nqJ ,n~} and

d}W are the canonical dimensions of F in the action functional (6.17): d~ d~1

=d+

I, d~',f1

=

I, dt

=

-2, d;

= 2£

=

(0 in the logarithmic theory) , d~

-I ,

=

I,

232

Dimitri Volchenkov

d:, = -I , d,C::,/l = 0, dc:l = I, d: = O. The canonical dimensions of w,~ are the sums of canonical dimensions of their arguments. We are interested in the scaling with dilatations of t , x, and m for fixed J1 , v, and g. Substituting the canonical dimensions into (6.32) and (6.33), after eliminating J1a/l and va v we obta in the equation of critical scaling (6.34) with the coefficients (6.35) which are the corresponding critical dimensions. Substituting the known values 'tv = -nqJ + (d + I )nqJI , and d$n = nqJ - nqJ' (the dimension of the connected function w,~ is equal to the sum of the dimensions of its fields) into (6.35), we obtain the following expressions for the critical dimensions:

2£ /3 , d~, =

,1qJ = I -

2£ /3, ,1 r

=

,1qJl

-,1 w

=d-

,1qJ,

= -2+2£/3 .

,1111

=

I,

(6.36)

They do not have terms of order £2 , £ 3 , and so on, and coincide with the Kolmogorov dimensions (6.13) for the real value e = 2. This main result has been reproduced in (De Dominicis and Martin, 1979) for the first time. Influence of weak uniaxial small-scale anisotropy on the stability of inertial range scaling regimes in a model of a passive transverse vector field advected by an incompressible turbulent flow was investigated in (Jurcisinova et al., 2006) by means of the field theoretic renormalization group . Weak anisotropy means that parameters which describe anisotropy are chosen to be close to zero, therefore in all expressions it is enough to leave only linear terms in anisotropy parameters. Turbulent fluctuations of the velocity field are taken to have the Gaussian statistics with zero mean and defined noise with finite correlations in time. In (Jurcisinova et al., 2006) , it was shown that stability of the inertial-range scaling regimes in the three-dimensional case is not destroyed by anisotropy but the corresponding stability of the two-dimensional system can be destroyed even by the presence of weak anisotropy. Critical behavior of a fluid, subjected to strongly anisotropic turbulent mixing, is studied by means of the field theoretic renormalization group in (Antonov and Ignatieva, 2006) in a simplified model where relaxation stochastic dynamics of a non-conserved scalar order parameter was coupled to a random velocity field with prescribed statistics . Existence of a new, non-equilibrium and strongly anisotropic, type of critical behavior (universality class) was established, and the corresponding critical dimensions were calculated. The scaling behavior appears anisotropic in the sense that the critical dimensions related to the directions parallel and perpendicular to the flow are essentially different.

6 What is Control of Turbulenc e

233

6.2.6 What do the RG representations sum? In the renormalized theory all quantities are calculated as series in the charge g of the type R(g , ... )

=

['gIlR Il(··· ).

(6.37)

11 =1

For any initial value of charge gO, the renormalized value g E [O,g. rv e], and so it can be assumed that g. c::: e. Owing to the smallness of g rv e and the absence of poles in e in the coefficient s of the series (which are elimin ated by the UV renorm alization) , it may seem that in the e scheme there is absolutely no need for any infinite summation of contributions of the series (6.37). This is true, but not for the critical region : direct calculations show that the coefficients RIl involve factors of the type (s-2 e - I)/ e, which are UV-finite (in the limit e ----+ 0, S = canst ) and of order unity for s = k/)1 c::: I, but become of order 1/ e and higher for [s In s] ?: I. The maximum number of such "large" factors of order 1/ e in the terms of the perturbation series never exceeds the number of "small " factors g rv e. These two numbers have to be approximately equal that means at small s we have the new parameter

~ == ~(s-2e -I) ,

e

(6.38)

all powers of which must be summed for each order in e. This is the statement of the first infrared problem in the language of the renorm alized theory with small e > 0.

6.2.7 Stochastic magnetic hydrodynamics In this section , we consider the field-theoretic analysis of several problems in magneto hydrodynamic statistics . These problems include the inertial-range scaling laws in incompressible fully developed turbulence of conductive fluid corre spond to various regime s of physical scaling behavior in a model of magneto hydrod ynamic (MHD) equations supplemented with stochastic force terms and the scaling exponents of some composite operators . The first attempt to study the MHD model has been performed in (Fournier et al., 1982), but it was incomplete (see the details below) . Then, the correct model had been propo sed in (Adzhemyan et al., 1985), but the renorm alization had been made erroneous. Here, we present the correct version of the renormali zed magnetohydrodynamics. The stochastic MHD equations for two transverse vector fields ( v is the velocity field and is the magnetic field) are written as

e

(6.39) (6.40)

234

Dimitri Volchenkov

r

where A = c2 / 4n O" y is the inverse magnetic Prandtl-type constant, and I" and are the Gaussian random force and curl of the random current with mean zero values and covariance

U? (r,t)f! (r' ,t' )) = o(t -

t')D~{3 (r - r'},

a, f3 = v, e.

(6.41)

where e = B/ .j4np, B is the magnetic induction, 0" is the conductivity, and p is the density of medium, p is the scalar pressure field, c is the speed of light. The transversal condition for the velocity field v follows from the incompressibility constraint:

J ·v=O. The purely longitudinal contribution of the pressure in (6.39) can be eliminated by inserting a transverse projection operator I1s = Ois - kikslk2 onto solenoidal vector fields in front of the longitudinal contributing factors : (vJ)v to V'tV and (eJ)e . The correction to (6.39) is given by the Lorenz force which is proportional to [curlB x B]

= (BJ)B - J(B 2 / 2),

where the second term is included into the pressure p, and by (6.40) which follow s from Ohm 's law for a moving medium in the simplest form, j

= O"(E+ [v x B]/ c),

and from Maxwell's equations without allowance for the displacement current. The Fourier transform of D~{3 or force spectrum,D~{3 (k), is necessarily non-negative. The canonical dimensions of these fields are the same, and the inverse magnetic Prandtl constant is dimensionless. In the massless models, the covariances for the random forces (6.41) are chosen in the power-law forms : 1'1' Dis

= gl Y 3 Pis d d.;

(6.42)

VI' ,

= k 4 - d - 2e ,

dee

= k4 - d -2ae ,

d I,e -- k 3 - d - ( I+a)e .

The amplitude factor gi (i = 1,2,3) in the correlation functions play the role of coupling constants (y 3 is separated from the amplitude factors for the future convenience) . The coupling constant g3, in the mixed correlation function can be defined as g3 = ~ .jglg2 where ~ is not in essence a charge but an arbitrary parameter of the theory. This situation is analogous to the gauge parameter found in quantum electrodynamics. ~ is subject to the inequality I~ I:::; I, which follows from the requirement that the matrix of correlation functions be positive. The value e = 2 corresponds in the momentum representation to o(k), this expressing the idea of "pumping of energy from the large-scale motion". The theory is renormalizable and logarithmic for E = O. The positive constant a in the exponent of the magnetic correlation function is an arbitrary parameter of the theory and models the difference between the spectra of magnetic and hydrodynamic energy pumping. Cism is the completely antisymmetric pseudo-tensor. The index structure of the correlators (6.42) is determined by the

6 What is Control of Turbulence

235

requirements that the fields be transverse and spatial parity be conserved: the field Vi is a vector, while i is a pseudovector, and so the mixed correlator is a pseudo-tensor. It is automatically transverse in the indexes i and s. In calculations performed in the spirit of dimensional regularization, the symbols Oik and Cism can formally be used for arbitrary dimension d , but in the final expres sions the symbol Cism, in contrast to Ois , is mean ingful only for the real dimen sion d = 3. In other dimensions, say, in d = 2, there is no pseudo-tensor which is transverse in both indexes, and the mixed correlator must be taken to be zero. The mixed correlator was not introduced in (Fournier et al., 1982) and had been introduced in (Adzhemyan et al., 1985).

e

6.2.8 Renormalization group in magnetic hydrodynamics Stochastic dynamic s allows for the path-integral repre sentation of the probabilit y generating functionals, using the so-called MSR action (Martin et al., 1973), (de Dominicis, 1976), (de Dominicis and Peliti, 1978). The generating function al of renormalized correlation functions takes the form G(A
=

J~

exp (SR (

= {v,v' , e,e'}

(6.43)

with twice the number of fields and with the action functional given by

SR = ~gIV3,u2cv'D""v' + ~g2v3,u2ace'Deee' + g3 v3,u(l +a)c v'D"ee' 2

2

+v' [- alv + ZI vL1 V - (va)v + Z3( ea )e] +e' [- al e + Z2V' L1 e - (va)e + (ea )v] ,

(6.44)

(the necessary integrations over x and summations over the indexes are understood). The operator det.4t ex: expaH(O), where H(x) is the Heaviside funct ion, it turns into a con stant if the convention is taken that H(O ) = O. The renormalization mass parameter u fixes the renorm alization procedure. Notice that action (6.44) possess the Galilean invariance, i.e., it is invariant under the following field transformations:

Vb(X ,t) = v(x +u(t ),t) - b(t ), v~(x ,t ) = v'(x +u(t ),t ), eb(x,t) = e (x +u(t) ,t ), e~(x ,t) = e'(x +u(t),t) ,

(6.45)

where b(t) is a parameter of transformation, and u(t) = i. : dt'b(t') . The structure of RG-tran sformation in the model of stochastic hydrodynamics is given by the following relation s between ZI ,2,3 and the renorm alization con stants of fields and charges: Z" =Z"I

=

I,

Z,,=ZI ,

Dimitri Volchenkov

236

(6.46) The renormalization consta nts ZI ,2,3 can be calculated from the corresponding 1irreducible functions (Adzhemyan et al., 1996) (all graphs needed for the one-loop calcu lation are shown in Fig. 6.3) In the diagrams shown on Fig. 6.3, the lines (prop agators) in the (k , t )-representation are

(vi(k ,t )vj(-k,O))o = ~jH(t)exp( - vk2t ), 2 2e (vi(k ,t )Vj(- k ,O))o = ~/IV2J1 k2-d -2eH(t)exp(-vk2 Itl),

(6.47)

(ei(k ,t )e;(-k, 0) )0 = ~jH(t) exp( -A vk 2t), g2v2kl- d- (I +a)eJ12ae 2 (eJ k ,t )ej(-k,O))o = ~j 2A H(t)exp( - YAk Itl), g3 v 2kl- d- (I +a)eJ1 ( I +a)e I +A [H(t )exp(- Avk2t ) +H( -t)exp(Avk2t)] .

(eJ k ,t )Vj(- k ,O))o = eisjk,\.

All vertexes contain the coordinate derivatives: Vivav and Vi eae are symmetric with respect to the basic fields V and the third vertex, aee-antisymmetric . From Fig. 6.3 and Eg. (6.47), one obtai ns (for d > 2):

e,

e'

= l _ g1d(d - l)_g2(d 2+d -4)

Z

4Be

1

4Bad2

'

Z = 1_ g1(d + 2)(d - I ) _ g2(d + 2)(d - 3)

2Bd(A + 1)

2

( VV') , = VK2he

2BaeA2(A + 1) ,

(6.48)

v' v v' v v' 0 0' v ~ - ~

~

~

o

0' V V' (00') ,= A.VK2 _ ~ - ~

~

he

(v'OO)I_ie=

~0 V

+

~

V'

v

+ V

V' V'

V'

0 0

0' 0 V V'

0

0

Fig. 6.3 The Feynman diagrams needed for the one-loop calculation of renormalization constan ts 21.2.3

237

6 What is Control of Turbulence

where B = 2d(d + 2)(2n)" / S" = d(d + 2)(4n)" /2r(d /2). it is convenient to eliminate the ratios g 1/ A and gd A2 by changing from g I and g2 to new charges (6.49)

In terms of new charges, the RG f3-functions are the following : f3;.. = A(YI - Y2) ,

f3g = g( -2£ + 2YI + Y2 ),

(6.50)

bg l =g'(-2a£ +YI + 2Y2- Y3 ). Calcul ating the Yi

= IlJJllnZi in the new variable s (6.49), for d = 3, we find Y3 = 2(g' - g),

(6.51)

from which using (6.50) we find the f3-function s in the one-loop approximation :

f3g= g[-2£ +6 gA +8 g' + ,1,1;1] ' (l

fJg 1

g] = g, [ -2a£ +3gf\,, +2g +2g, + 120 +,1, ' f3;.. =,1, [3gA +4l

(6.52)

- /~gA] ·

In contrast to a single charge theori es cons idered in the previous chapters, in a multi-charge one there may be several fixed point s gi* '" e. The type of each point is determined by the eigenvalues (Oa of the corresponding stability matrix (6.53) A point g* = {gi*} will be IR-attractive (stable with respect to long time large scale asymptotes) only if the real parts of all the eigenv alues (Oa are strictly positive. For such a point, any phase trajectory representing a solution of the RG differential equ ation passing through the neighborhood of g* will be subsequently attracted to it. The system of algebraic equations (6.52) has six non-trivial solutions f3i(gi*) = 0.

I. The line g* = g: =

°and ,1,* arb itrary;

2.g:=A*=0, g*=£ /5 ; 3. g* = ,1,* = 0, g: = ae: 4.g:=0, g*=£(1 +A*/15) , ,1,*( J 43/ 3 - I)/2 ~ 1.393; 5.,1,*=0 , g* = £(4a - I)/ 39, g:£(11 - 5a)/39 ;

238

Dimitri Volchenkov

6. g* = e( 1+,1,*)/15 , g: = e (10 - 3,1, * (,1,*+ I)) / 60, where ,1,* is the positive root of the equation 3,1, 2 + 7A + 54 = 60a, which exists for g ~ 0.9. It is easy to check that the points 3 and 4 are stable. These points have been found in (Fournier et al., 1982) : the "kinetic" point is stable for any a ~ 0.25 and the "magnetic" fixed point is stable for any a :::; 1.16. in the intermediate region 0 .25 :::; a :::; 1.16 both points are stable, and the critical regime depends on which of their basins of attraction the initial data of the RG-equations foe the invariant charges are located in. In this sense the critical behavior in the stochastic magnetic hydrodynamics is not universal. The basins of attraction of these points for the real case a = I are studied in (Camargo and Tasso, 1992). However, the results quoted there cannot be considered reliable, as the authors (Camargo and Tasso, 1992) did erroneously assuming not take into account the renormalization of the vertex Vi that it is an effect of higher order in the charges. Fully developed magneto-hydrodynamic turbulence near d = 2 up to d = 3 has been investigated in (Jurcisin and Stehlik, 2006) by means of renormalization group approach and double expansion regularization . A modification of standard minimal subtraction scheme has been used to analyze the stability of the Kolmogorov scaling regime which is governed by the renormalization group fixed point. In particular, it has been demonstrated that the magnetic stable fixed point has been calculated and its stability region has been also examined. This point losses stability: (I) below critical value of dimension de = 2.36 (independently of the a-parameter of the magnetic forcing) and , (2) below the value of a e = 0.146 (independently of the dimension).

ee,

6.2.9 Critical dimensions in magnetic hydrodynamics The problem of critical scaling in the magnetic hydrodynamic has not been discussed in (Fournier et al., 1982), but studied later in (Adzhemyan et al., 1996). The RG equation for the model in question for any renormalizable Green function FR has the form : (6.54) in which ne and ne' are the numbers of arguments satisfied by any function of the following type :

e and e' in FR . Eq . (6.54) is (6.55)

where R is a scaling function of dimensionless arguments with a finite IR asymptote, and g(s,g) is the solution of the Cauchy problem (6.56)

6 What is Control of Turbulence

239

with s = k] J1 . As the IR asymptote s ----+ 0, in (6.55) we have g ----+ g* and v and since R is assumed to be finite, it acquires a definite critical dimension:

----+

v*,

(6.57) However, this is not the case for the magnetic hydrodynamics. In the kinetic point , = and therefore the scaling function R (flfI ) vanishes as being proportional to s'. Introducing the correction exponent Wgl characterizing the falloff rate as g' ----+ 0,

g:

°

one obtains the asymptote for the scaling function R

rv

R (flfI )

as s ----+ 0:

s(J)l R(g*,O,A*).

(6.58)

Therefore, the scaling function R (flfI ) has a nonzero critical dimension LiR (88) = W gl that corresponds to the addition of wg l/2 to the critical dimension of the magnetic field

e,

Lifl

2e

= 1-:3 +e(l-a) .

(6.59)

The critical dimensions for the velocity field v and time t coincide (in d = 3, e = 2) with their values in the Kolmogorov theory of turbulence, Li v = - I/ 3 and Lit = -2/3 . The value (6.59) is exact (has no corrections of the order O(e2 ) , etc.) The result is true for all the Green functions in the kinetic regime . In (Adzhemyan et al., 1996), the correct critical dimension of magnetic field has been derived using the usual rules, but the magnetic fields have been subjected to a suitable dilatation: e = e/ Rand e' = e' R. In the magnetic point, A* = g* = 0, and g: = O(ae), and the variables Xand g fall off for s ----+ according to a power law:

°

(6.60) where w;. = 4ae + ... and wg = - 2e + 8ae + ..'. In contrast to the charges g and g' which appear in the nominators of scaling functions, the charge A would appear in the denominators giving rise to the singularities in the scaling asymptote in the magnetic critical regime, while A ----+ 0. In (Adzhemyan et al., 1996), the appropriate dilatations, v = Av, v' = v', e = vIe, e' = e'/ vI have been used. In the magnetic critical regime it leads to the following critical dimensions of fields (in d = 3): (6.61) The critical dimensions (6.61) determine the IR asymptotes of the dynamical Green functions and also of the corresponding static objects if they exist as g = and A = O. This result remains true for all Green functions excepting for the static functions (vv) and (vvv). When calculating the scaling functions R (vv), one can see that the

°

240

Dimitri Volchenkov

contribution of the bare graph vanishes , while, in the internal I-irreducible block , the integral over the frequency , J w - I / 2d w oc WI /2, diverges for W --+ 00 . This means that it is impossible to go directly from dynamical objects to static one, so that one has to analyze the asymptotes before taking this limit. The divergences of the frequency integral gives rise to a crossover phenomenon for the static correlation functions of the velocity field in the magnetic regime which had been discovered first in (Fournier et al., 1982). The divergence factor oc A. - I / 2 rv s -2ae arising from the I-irreducible block should be compared with the contribution of the bare propagator cc g/ A. rv s -2e+4ae . For a > 1/3 with s = k] J1 --+ 0, the loop contribution is more important, and for a < 1/3 the bare one is. This leads to a final result (Fournier et al., 1982; Adzhemyan et al., 1996): for

a > 1/3 ,

{ k - d+2-2e+4ae for

a < 1/3 .

k - d+2-2ae

(vv) lstat rv

(6.62)

6.2.10 Critical dimensions ofcomposite operators in magnetic hydrodynamics In phenomenological theory of turbulence, it is supposed that energy is pumped into the inertial range from a large scale of A . We take this scale into account in the infrared mass parameter m == A - I to consider corrections to fully developed turbulent spectra due to design of energy pump with the use of short distance expansion method. According to this, one can establish that

l/J(Xt,t)l/J(X2 ,t) c::: L Ci(r)Ff(x ,t) ,

l/J == {v, e},

(6.63)

i

where X == (Xl +x2 )/2 , r == Xl -x2 , and F; are the various compo site operators of the theory. Averaging (6.63) over fluctuations, one obtains asymptotes, mr --+ 0, for the I-partical correlation function

< l/J(XI ,t)l/J(X2 ,t) > c::: L Ci(r)mLlF;, l/J == {v,e} ,

(6.64)

i

In this expression .1 F; are the scaling exponents of the compo site operators. Hence, if there are any operators of negative scaling exponents, the contribution of the mass parameter m to the asymptotes (m --+ 0) becomes essential. In this section , we study the renormalization and compute the critical dimensions of the simplest local composite operators of the theory : l/Jil/Jk(X), l/J! l/Jk(X), as well as vector operators (l/Jl/J)l/Ji(X), and (Jl/Jl/JMx) . We note that these operators are multiplicatively renormalizable, i.e., there exists a matrix of renormalization constants Zik such that for the renormalized operator, Ff = ZikFt( eJJR ) , all the Greens functions with one F R and any number of the fields eJJ are finite (don't have divergences) in the logarithmic theory. If one adds the term L aiFjR with arbitrary finite coefficient s

6 What is Control of Turbulence

241

a, to the action functional of the renormalized theory (6.44), then the corresponding functional (6.44) is finite up to the terms linear in G inclusively. We now consider the renormalization of composite operators with mixing . In accordance with the general rule, only the composite operators of the same total canonical dimension in the logarithmic theory are mixed on renormalization. The critical dimensions of Fj are found from the usual power counting (Zinn Justin, 1990; Collins, 1992): d~ = Yik + d}Oik + d(j)d7! Oib where d} and d7! are the momentum canonical dimensions of operator considered, Yik = IlJJlln2ib and d(j) is a critical exponent of frequency , d(j) = 2 - 2e / 3 in the models with the power-law correlators. The entity dF is the matrix of critical dimensions of the system of the mixed operators FjR , and the critical exponents are the eigenvalues of this matrix . Definite critical dimensions belong to the linear combinations of operators make the matrix d F diagonal.

6.2.11 Operators of the canonical dimension d = 2 The tensor operator I/>il/>k is a sum of scalar and zero trace operator

they do not get mixed on renormalization. The operator GI = ~ViVb which is not Galilean invariant, is finite (211 = I) because of the Galilean invariance property of the theory (Lifschitz, 1989), and it does not get admixed to the operator G2 = ~ eiek , hence 221 = O. Notice , that the operator G3 = Viek is a pseudo-tensor, so that it does not be admixed to GJ and G2 on renormalization : 223 = 232 = 213 = 23J = O. Furthermore, the I-irreducible diagrams for G3 have no divergences: 233 = 2 l , where the renormalization constant of magnetic field can be expressed in term of the known constant 23: 2e = 2 31/2 , (Adzhemyan et al., 1985; Adzhemyan et al., 1996). For fixed space dimension, d > 2, in the lowest order of perturbation theory, the calculation yields

e

(6.65) where

_~)

c= I +Oik(d+2)(d-l) ( 2(A+I)Be

gJ

GA

.

The operators gJ GI and g2G2, in both "kinetic" and "magnetic" critical regimes, have definite critical dimensions themselves (and not a linear combination of them) The results about operator scaling exponents are shown in Table 6.1. Notice, that the exponent for vv is exact.

242

Dimi tri Volchenkov

Tab le 6.1 The critical dimensions of the compo site operators of the cfJicfJj-type Composite Operator v2

ViVk,

eiek ve

Viek, e2

Kinetic regime

Magnetic regime

2 -4/3 e

2

2 -2(a -3 /10 )e

2 +3ae

2 - (a + 1/3 )e

2 +ae

2 -2ae

2 + 12ae

6.2.12 Vector operators ofth e cano nical dim ension d

=

3

The operators 1/>'1/> do not get admixed to any other one and are not renormalized, as soon as all the appropriate I-irreducible functions are in reality equal to zero due to the presence of closed loops of the advanced functions . Formally, these diagrams are logarithmic, but in reality, the structure of interactions of action (6.44) provides the removing of one derivative from the loop to the external 1/>' -line and red uces the diagram index of divergence. Thus, the scaling dimension of these operators are equal to the space dimension, L1 (1/>'1/» = 3. The operators, with given derivative, (dl/>I/»i , ca n be reduced to a differential, d( 1/>1/», due to the fact that the fields I/> == {v, e} are transversal. The scaling dimensions of such operators are L1 (d(I/> I/> ))

= L1 ( I/> I/> ) + I ,

I/>

== {v, e} .

The rest of vector operators, FI = Viv2 and F2 = Vie2, are the true tensors, yet F3 = e ie 2 and F4 = ev 2 are pseudotensors, so that these pairs do not be admixed on renormalization . Due to Galilean invariance property of the action, one ca n prove that FI is finite and it does not be admixed to F2, as well as F3 does not be admixed to F4 , (ZII = I, Z21 = 0, Z34 = 0). At the "k inetic " fixed point these operators have different powers of the coupling constant g2. Yet at the "magnetic" point all the non-diagonal elements of a renormalization matrix are equal to zero as being proportional to 1.* = O. Thus, the scaling exponents are determined simply from the diagonal elements of renormalization matrix just as in case of a simple multiplicative renormalization, and we shall say that these dimensions are "associated" with the corresponding composite operators. A deviation of the scaling dimensions from the canonical dimensions are the following :

YII

= 0, 'Y22 = 2(CI - I )y, Y33 = - 6CIY, Y44 = - 2y,

where Y = gJ/ BA - g2/ BA2, CI = (d + 2)(d - 1)(1. + 1) -1 . The scaling exponents are listed in Table 6.2.

6 What is Control of Turbulence

243

Table 6.2 The critic al dimension s of the comp osite operators (1jJ 1jJ )1jJ; Composite Operator V;V

2

e2 e;e 2 V;

e;v2

Kinetic regime

Magnetic regime

3 - 2e 3 - 2(a + 0.507 )e 3 - 3(a + I)e 3 -(a + 0.6S)e

3 3 + ISae 3 +60ae 3 - 2ae

6.2.13 Instability in magnetic hydrodynamics In accordance with the SDE method, the inertial-range asymptote of a I-partical correlation function can be expressed as follows :

< 1/>I(k,t)/fJ2(-k,t)

>=Ak-d- L1

q> I -

L1 \l:2

(I + ~bi (~tF;),

{1/>I ,/fJ2} == {v,e} ,

(6.66) where A is a Kolmogorov-type constant, and .11jl are the scaling dimen sions of the fields. One can see that RG-predicted spectrum is secure as long as .1F; > O. If .1F; is negative, the appropriate contribution changes the scaling asymptote of correlation function in the inertial range . If we are interested in asymptote of static correlation functions.z.e., they do not depend on time , we do not con sider the contributions of tho se operator s that are not Galile an invariant. It is quite clear that such a contribution would depend on the parameter of Galilean tran sformation b(t), but these operators contribute to asymptote of dynamic correlation funct ions . For the real value of E = 2 some Galilean invariant operators have negative scaling exponents at the "kinetic" and "magnetic" fixed point s for some values of the pump parameter a:

Notice, that the last one doe s not contribute to the asymptotes of I-partical static correlation functions at the "k inetic" point. This operator is a pseudotensor, it cannot contribute to real tensor correlation functions, yet its contribution to the mixed I-partical correlation function , < ve >, is O(g2* ) and can be neglected. For the same reason, the operators e2 and eiek do not contribute to asymptotic of I-partical hydrodynamic correlation function , < vv >. Hence, in the static case, for < vv > and < ve > the RG-predicted scaling asymptotes is secure for the both "kinetic" and "magnetic" points. If a > ~ , the exponent for < ee > should be corrected at the "kinetic" point as follows .1( < ee

» = -d -

2.1e - .1 (e 2 ) .

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Dimitri Volchenkov

This result has a simple physical meaning . At the "kinetic" point the magnetic field is passively advected to hydrodynamics. When the value of the parameter a is comparable to I, the spectrum of magnetic pump is infrared localized . It means that the inertial-range motion is exposed to ambient magnetic field, which depends on the hydrodynamics. In this case, some instabilities arise in MHD system, (Lifschitz, 1989), which are driven by the magnetic pressure gradient J(B 2 /2) . Yet the velocity field correlation function is virtually unaffected by these instabilities, but they contribute to the magnetic field correlation function. Likewise the usual hydrodynamics, at the "kinetic" critical regime the inertialrange asymptotes of dynamic correlation functions have a lot of essential contributions from Galilean non-invariant operators. In this sense, one can say, (Eyink, 1994), that there are infinitely many fixed points in the fully developed turbulence. However, at the "magnetic" critical regime, the scaling asymptotes for the < vv > - and < ee > -functions still have the same value as predicted by RG. The mixed function exponent has a correction associated with the operator i V2 , which becomes essential, while a > ~ . It is important to notice that in this case the contributions from each of the eivll-type operators are also possible. We do not know their scaling dimensions, so that this scaling exponent would be corrected as follows

e

.1 « ve »

= -d -.1e -

.1" - max [.1 (evil)].

6.2.14 Long life to eddies of a preferable size In the present section, we compute the scaling asymptote of the spectral density tensor of energy dissipated in a unit time per unit mass by the magnetic hydrodynamical system being in the "kinetic" critical regime, (Landau and Lifshitz, 1995; Landau et al., 1995):

e = V'O'V+AV [V' x ef,

(6.67)

averaged with respect to the statistic of Gaussian distributed random force f . Here & is a tension tensor of the incompressible fluid. Doing some basic calculations, we arrive at

where the angular brackets denote the average with respect to configurations f(x ,t). The result of RG-transformation acting on a renormalized composite operator (the local average of fields and their derivatives with respect to one point) is always a linear combination of the renormalized composite operators having the same symmetry, structure, and canonical dimension . This fact is known as a miming of composite operators (Collins , 1992). Denoting the renormalized composite operators of a mixing set as FjR, we write the RG equation for them in the form :

6 What is Control of Turbulence

245

(6 .69) where ,1ij is the matrix of critical exponents. The linear combinations 2' {FjR}, for which ,1ij has the diagonal form, have the definite physical meaning and correspond to the certain physical processes. The unique property of the energy dissipation composite operator e is that two different eigenvectors of the RG-operator have the same eigenvalue corresponding to the zero anomalous dimension y = O. Consequently, the relevant critical dimension matrix ,1F can be transformed merely to the Jordan form

DRC2'{FjR} = ,1L2'{FjR} , { DRC2" {FjR} = ,1L2" {FjR} + 2'{FjR} ,

(6 .70)

in which 2' {FjR} is the eigenvector and 2" {FjR} is the adjacent vector. ,1L is the shared critical exponent of 2' and 2". Such a phenomenon has not been discussed neither in the quantum fields theory literature nor in the statistical physics before. In the asymptotic region k] J1 ----+ 0, one can solve (6.70) to obtain (6.71) where C] and C2 are the normalized amplitude factors . Now we calculate the critical exponents ,1L and the relevant linear combination s 2' of renormalized composite operators explicitly. Note that the energy dissipation operator e is a sum of local composite operators of the canonical dimension dF = 4. The result of the cat ion of the DRc-operator on e can be written in the form (6 .72)

where K, and K2 are some linear combinations of the renormalized composite operators V'/V'kG~ , in which dc = 2, and FjR with dF = 4, and there fore K] and K2 are renormalized separately with no mixing. The set of composite operators Gik with dc = 2 reduces to the tensor operator q,iq,k which is a sum of scalar and zero -trace operators,

q,

= {v, tl} .

We have studied the renormalization of this family in the previous subsection. All of them have their own critical dimensions independently of others and contribute to the following part of the energy dissipation function :

The family of composite operators dF

= 4 consists of 7 items :

Dimitri Volchenkov

246

FI= (V4 ),

F2= (v282),

F5 = (8· .18) ,

r"3 = (v-ziv), F4= (V·(8 ·V8 )), F7 = (8 4 ) .

F6 = (8· (8 · Vv) ),

Linear combinations of their renormalized analogs contribute to another part of the energy dissipation function :

The renormalized operators F{ are related to the not renormalized ones , Fk, by the linear equations Fj =ZikF{, in which Zik are the renormalization constants found from the requirements that all correlation functions with one FjR and any number of fields v, 8, v', and 8' are finite as the UV-cutoff A ----+ 00. The problem of computation of the entries Zik can be substantially simplified by the symmetry arguments. For instance, since the model of magnetic hydrodynamics is invariant under the Galilean transformation of fields, the composite operators which break this symmetry is not renormalized being finite. Moreover, they do not mix to any other Galilean invariant operator. Therefore, ZII

=

I,

Zj2

Zk3

Zil

= 0,

= 0,

"Ii> I ,

"Ij > 2,

(6.73)

= Zk4 = 0, 'ik > 4.

Then, one can use the Schwinger functional equations and the Ward identities expressing the Galilean invariance of the MHD model: (6.74) where cp == {v , 8} and cp' == {v', 8 '} , AIJ! and A IJ!I are the relevant source functions . The r.h.s. of (6.74) is UV-finite and has the definite critical exponents independently. Therefore, the operators in the I.h.s. should also be UV-finite having the definite critical exponents:

(VZ, V.L1 V+ Z3V. (8 . V8 )) { ()., vZ28· L1 8 + Z38· (8 · Vv))

< 00, < 00,

(6.75)

where Z, ,2,3 are the renormalization constants of MHD-action. UV-finiteness means that the divergent parts of the renormalization constants subtract each other in the combinations (6.75) , therefore,

Zi3+ a Z i4=0 ,

U:t5 ,6), Z55=Z2 1 ,

Z66=Z3

1

,

247

6 What is Control of Turbulence

where a = g(d - I )/2d£(1 + A )(4n )d/2r(d /2 ). Other nontrivial entries of Z - ikmatrix require evaluation of the diverging parts of the relevant I-irreducible diagram s and remain unknown . Neverthele ss, we show that (6.73) and (6.75) provide us enough information to define the critical exponents of the operator £, . The matrix Z ik appears to be triangular and its diagonal elements give us the complete set of anomalou s exponents, 11k = -2£ · dJg logZik + 0 (£2 ), (6.76) The zero eigenvalue is twice degenerated, y" = Y22 = 0, the entries "tn , Y22, Y43 and Y33 so not equal to zero and still unknown , however, they do not contribute to £], £,

= EWijFj ,

Wij

= diag [O,O, 1,0,1.,0,0] .

i

Denote the matrix which transform s Wij into the Jordan form as U , then, the linear combin ations of renorm alized compo site operators which corre spond to £, in the process of UV-renormalization are

There are two such combinations:

in which a] ,2 are some analytical coefficients expre ssed via the unknown entrie s (which are obviously finite even as E --+ 0). We have to emphasize that is a vector defined in the two-dimen sional eigensubspace of the RG-operator having the single eigenvalue Y= where

Zik]

Lf

°

L'/ = (v·L1 v + Z3Z1]v , ( e · ve) ),

L'{R

= (Abf e· L1e + z:; ' e · (e · vv) + zl ' ( 4 )

(6 .77)

are the linearly independent vectors spanning this eigensubspace. The critical expon ent relevant to is L1 L 1 = 4 - 4£ /3 = 4/3 (for E = 2). The anomalou s exponent corre spond ing to the second combin ation, L~ , is - Y2 that gives L1L2 = 4 - 2£ = (s = 2). The critical dimensions of the compo site operators G ik are L1c = 2 + L1t/J;I!Jk where L1Mk are the critical dimensions of composite operators with d F = 2 studied in the previous subsection. Certa inly, we have:

Lf

°

(6.78) All value s are computed for

e = 2.

248

Dimitri Volchenkov

We conclude this section collecting the results on the critical exponents of the energy dissipation function f. In Fig. 6.4, we have presented the different asymptotic contributions into the energy dissipation function E == f(k) /£o via the dimensionless scaling parameter s == k] j1 where £0 is a constant energy dissipation rate. The sum of all contributions is drawn with the black bold line. The uniform rate £0 is given by the thin gray horizontal line, two power-law asymptotic contributions are represented by the dotted lines. At a decided disadvantage for the small scale eddied (s » I) (in the far-dissipation range) the total dissipation rate increases considerably, therefore, they dissipate very fast. Fig. 6.4 displays that there are two opportunities for the long-time , large-scale asymptotic behavior of f . In the inertial range, indeed, the constant dissipation rate £0 dominates the dissipation process. Nevertheless, in the MHD model, the alternative asymptotic "steady state" exist, and it would come into play when the regime characterized by the constant dissipation rate looses the stability. One can see on Fig. 6.4 that both dissipating regimes meet precisely at the dissipation wave number k" = j1 ~ A and at the pumping scale kj1-1 ~ O. The alternative dissipating regime has a minimum somewhere in between these two points . The existence of dissipation minimum in the sub-leading dissipation regime predicts essentially long lifetime for the eddies of some preferable size €. The linear combination is responsible for the long -time breaking of the size equivalence of eddies in the inertial range . From (6.77) one can see that it describes a kind of feedback controlling loop, accentuating the eddies of particular size and suppressing the turbulence in other scales by shadowing one of the infinitely many unstable periodic orbits embedded in the chaotic turbulent attractor. When the trajectory converges to the optimal orbit, the feedback term vanishes identically,

Lf

The energy dissipation rate 5

4

o

0.5

1.5

S

2

2.5

3

Fig. 6.4 The critical exponents of composite operators of the canonical dimension 4 in the model of magnetic hydrodynamics in the kinetic critical regime .

6 What is Control of Turbulence

249

The latter equation defines the configurations of fields fJ and v relevant to the alternative dissipation regime .

6.3 In search of lost stability The functional formulation of long-range turbulent transport problem in the ScrapeOff Layer (SOL) of thermonuclear reactors reveals convective instability in the cross-field system which can be override by a finely tuned poloidal drift. We also consider a simple qualitative discrete time model of anomalous transport in the SOL which exhibits a surprising qualitative similarity to the actual flux driven anomalous transport events reported in experiments.

6.3.1 Phenomenology of long-range turbulent transport in the scrape-offlayer (SOL) ofthermonuclear reactors Turbulence stabilization in plasma close to the wall blanket of the ITER divertor is the important technical problem determining the performance of the next step device. Long range transport in the scrape-off layer (SOL) provokes the plasma wall interactions in areas that are not designed for this purpose . Evidence of the strong outward bursts of particle density propagating ballistically with rather high velocities far beyond the e-folding length in the SOL has been observed recently in several experiments (Rudakov et aI., 2002 ; Antar et al., 2001) and in the numerical simulations (Ghendrih et al., 2003) . These events do not appear to fit into the standard view of diffusive transport: the probability distribution function (pdf) of the particle flux departs from the Gaussian distribution forming a long tail which dominates at high positive flux of particles (Ghendrih et al., 2003) . Theoretical investigations of the reported phenomena remain an important task. In the forthcoming sections, we consider a variety of two dimensional fluid models based on the interchange instability in plasma studied in (Nedospasov, 1989; Garbet, 1991) and discussed recently in (Ghendrih et al., 2003) exerted to the Gaussian distributed external random forces to get an insight into the properties of turbulent transport in the cross-field system . The E x B drift motion of charged test particle dynamics in the SOL was analyzed to investigate a transport control strategy based on Hamiltonian dynamics in (Ciraolo et al., 2007) . A method of control which is able to create barriers to magnetic field line diffusion by a small modification of the magnetic perturbation has been proposed in (Chandre et al., 2006). This method of control is based on a localized control of chaos in Hamiltonian systems .

Dimitri Volchenkov

250

Neglecting for the dissipation processes in plasma under the constant temperatures T; » Ti, this problem is reduced to the interactions between the normalized particle density field n(x,y,t) and the normalized vorticity field w(x,y,t) related to the electric potential field ep(x,y,t), (6.79) defined in the 20 plane transversal to ez , the direction of axial magnetic field Bo. In (6.79), x and yare the normalized radial and poloidal coordinates respectively. The Poisson's brackets are defined by

When g = 0, Equations (6.79) describe the 2D-rotations of the density and vorticity gradients around the cross-field drift v

= -elBo V'ep x ez,

in which V' == (~r,()y) . Their laminar solutions (with w = 0) are given by any spatially homogeneous electric potential ep = 1/1, (z ) and any stationary particle density distribution n = 1/12 (x,y). Other configurations satisfying (6.79) at g = 0 are characterized by the radially symmetric stationary vorticity fields

with the electric potentials invariant with respect to the Galilean transformation

ep(x) ---., ep(x) +xq>, (r) + C!'2(t) , where the parameters of transformations q>' ,2 (z ) are the arbitrary integrable functions of time decaying at t ---., - 0 0 . The relevant density configurations

n = q>3 (x, y-

[ 00 vy(x,t/) dt')

have the form of profile-preserving waves convected in the poloidal direction by the poloidal cross field drift vy(x,t) . The poloidal component of cross field drift itself remains invariant with respect to the Galilean transformation v y ---., v y + q>, (z ), while its radial component Vx = O. Configurations that satisfy (6.79) for g > 0 have the Boltzmann density distribution of particles in the poloidal direction. In particular, those solutions compatible with the Galilean symmetry discussed above (with v y "* 0) are the solitons (solitary waves) of density convected by the poloidal electric drift,

nocexp-

/

gT x,y

) Iy-jt Vy(x,y,t')dt'l , -

00

6 What is Control of Turbulence

251

where T (x, y) is an arbitrary function twice integrable over its domain. In addition to them, for g > 0, Equation. (6.79) allows forthe radially homogeneous configurations a.rn = 0, IV = U(y) mod2n with v y = 0 which do not fit into the Galilean symmetry, these are the steady waves,

11

Y

n ex: exp -U(y, )dy, . g 0

The latter solution does not possess a reference angle and can be considered as an infinitely degenerated state of the system since the relevant configurations {n, w} can be made equal at any number of points by the appropriate choice of U : U (Yl) = U(Y2) = ... = U(YIl), and Jg 1 U(y')dy' = Jg2 U(y')dy' = ... = Jg" U(y')dy' . For instance, it can be represented by the periodic lattice potential controlled by the spokes of high particle density radiating from the center. With two concurrent symmetries there can occur either the frustration of one of them or the vanishing of both with the consequent appearance of a complicated dynamic picture that is most likely stochastic . The latter case corresponds to a maximally symmetric motion resulting from the destruction of unperturbed symmetries (Sagdeev and Zaslavsky, 1986). In particular, instability in the system (6.79) occurs with respect to any small perturbation either of density or vorticity. Accounting for the dissipation processes in plasma smears the picture, so that the small scale fluctuations would acquire stability. We demonstrate that the small scales fluctuations can be stable provided there exist the reciprocal correlations between the stochastic sources of density and vorticity in the dynamical equations. The large scale stability of a fluctuation can be characterized by the order parameter

in the momentum space where kx and ky are the radial and poloidal components of momenta respectively. For the uncorrelated random forces (under the white noise assumption), a fluctuation with ~ > 0 is unstable with respect to the large scale asymptote in the stochastic problem. The accounting for the convection of particles by the random vortexes introduces a finite reciprocal correlation time 'rc ( [r - r'[) between the density and vorticity random forces. Then there exists the critical scale ~c , in the stochastic model, such that a fluctuation with ~ < ~c vanishes with time, but its amplitude grows up unboundedly with time, for ~ > ~c. Biasing of wall components can locally modify turbulent transport and is considered to be beneficial if one aims to insulate the Tokamak main chambers from the bursts of density (Ghendrih et aI., 2003). Indeed, the generation of a uniform electric drift in the poloidal direction, v y ---+ v y - V, would frustrate one of the symmetries in (6.79) reestablishing the Galilean invariance in the system . For instance, those configurations characterized by the trivial poloidal component of electric drift v y = 0 would be eradicated. We investigate the problem of turbulence stabilization close to the divertor wall in the first order of perturbation theory and shown that there exists a critical value IVc I < 00 of the poloidal electric drift which would suppress the

Dimitri Volchcnkov

252

large scale instability in the stochastic system with the correlated statistics of random forces, ~c > O. However, for the uncorrelated random sources in the stochastic problem, ~c = 0 and I Vc I ----+ 00 as k ----+ O. Correlations between the unstable fluctuations of density at different points are described by the advanced Green 's functions which are trivial for t > O. In particular, these functions determine the concentration profile of the unstable fluctuation s of density which increases steeply toward the wall. The size of such fluctuations grows linearly with time. In this case, the statistics of the transport events responsible for the long tail of the flux pdf is featured by the distribution of the characteristic wandering times of growing blobs convected by the highly irregular turbulent flow in the close proximity of the divertor wall. In our model, we have replaced this complicated dynamics with the one dimensional (the radial symmetry is implied in the problem) discrete time random walks. Such a discrete time model would have another interpretation: the advanced Green 's function is a kernel of an integral equation which relates the amplitudes of the growing fluctuations apart from the wall with those on the wall, in the stochastic dynamical problem. Indeed , this equation is rather complicated and hardly allows for a rigorous solution. Therefore, being interested in the qualitative understanding of statistics of the turbulent transport in the SOL , we develop a Monte Carlo discrete time simulation procedure which would help us to evaluate the asymptotic solutions of the given integral equation. General approach to the probability distributions of arrival times in such a discrete time model has been developed recently in (Flori ani et aI., 2003) . In general, its statistics can exhibit the multi-variant asymptotic behavior. Referring the reader to (Floriani et aI., 2003) for the details , we have shown that the statistics of arrival times for the unstable fluctuations is either exponential or bounded by the exponentials (in particular, the latter would be true in the case of the randomly roaming wall) that is in a qualitative agreement with the data of numerical simulations and experiments (Ghendrih et aI., 2003).

6.3.2 Stochastic models ofturbulent transport in cross-field systems The stochastic models of cross field turbulent transport used in the forthcoming sections refers to the effectively two-dimensional fluid model of plasma based on the interchange instability in the SOL (Nedospasov, 1989; Garbet, 199 I) recently discussed in (Ghendrih et aI., 2003) . In this model, one assumes the temperatures of ions and electrons to be constant, 'L « Te . Then the problem is reduced to that of two coupled fields, the fluctuations of normalized particle density n(x,y, t) and that of vorticity field w(x,y, t), governed by the following equations t7

_

vIW -

A

UOV Ll l- W -

( _ I )k8k ::l k '-k~ 1 - k - uyn

,,

+ I,IV ,

(6.80)

6 What is Control of Turbulence

253

written in the polar frame of reference with the normalized radial x = (r - a)/ p, and y = Ps poloidal coordinates. Time and space are normalized respectively to Qi- 1, the inverse ion cyclotron frequency, and to Ps, the hybrid Larmor radius. The covariant derivative is

a8/

in which V' == ({Jr , Oy), and il l- is the Laplace operator defined on the plane transversal to the axial magnetic field. The effective drive 0<: Oy log ( I + n/ ii) risen in the cross-field system due to the curvature of magnetic lines is represented by the series in oyn k with the coefficients gk rv Ps/ Rii k where R is the major radius of torus and ii is the mean normalized particle density. The curvature coefficients gk averaged along the lines of magnetic field are considered to be constant and small parameters in the problem . The diffusion coefficients v and Uo v both are normalized to the Bohm 's value T; / eB and govern the damping in the small scales, herewith Uo is the dimensionless Prandtl number where the knot distinguishes its value in the free theory from its effective value u in the renormalized theory. The Gaussian distributed random forces f" and fw in (6.80) play the role of stochastic sources in the dynamical problem maintaining the system out of equilibrium and simultaneously modeling the Bohm's boundary conditions at the sheath which have not been explicitly included in (6.80), in contrast to the original models (Ghendrih et al., 2003 ; Garbet , 1991). Herewith, the physically important effect of particle escape at the sheath is replaced with a quenched loss of particles in the SOL at the points for which f,,(x,y, t) < O. Simultaneously, the particles are supposed to arrive in the SOL in the areas where f,,(x,y, r) > 0 modeling the injection of particles from the divertor core along with the perturbations risen in the system due to the Langmuir probes (Gunn, 200 I; Labombard, 2002). For a simplicity, we assume that the processes of gain and loss of particles are balanced in average therefore (f,,) = O. The stochastic source of particles is used instead of the continuously acting radial Gaussian shaped source (localized at x = 0) studied in the numerical simulations (Ghendrih et al., 2003). Similarly, we impose the random helicity source fw exerting onto the vorticity dynamics in (6.80). Furthermore, the random sources f" and fw account for the internal noise risen due to the microscopic degrees of freedom eliminated from the phenomenological equations (6.80). From the technical point of view, the random forces help to construct a forthright statistical approach to the turbulent transport in the SOL. In particular, it allows for the quantum field theory formulation of the stochastic dynamical problem (6.80) (based on the Martin-Siggia-Rose (MSR) formalism (Martin et al., 1973» that gives a key for the use of advanced analytical methods of modern critical phenomena theory (Ma, 1976). The Gaussian statistics of random forces in (6.80) is determined by their covariances,

DIlIl(r- r', t - t ') == (f,,(r, t )f" (r',t') ) ,

Dww(r - r', t - t')

== \fw(r, t )fw(r', t ') ) ,

Dimitri Volchcnkov

254

where r == (x,y). describing the detailed microscopic properties of the stochastic dynamical system . We discuss the large scale asymptotic behavior of the response functions ( 8n(r ,t) / 81,,(0 ,0) ) and ( 8n(r ,t) / 8fw(0,0) ) quantifying the reaction of system onto the external perturbation and corresponding to the r-distributions of particle density fluctuations expected at time t > 0 in a response to the external disturbances of density and vorticity occurring at the origin at time t = O. The high order response functions are related to the analogous multipoint distribution F.Il ( rl ,tl, "' , rll,tll,. r ' I ,t(, " "' , r Il,tll ) as . f unctions

8" [n(rl,tl )'" n(rll,tll)] ) ( 81" (r'j , t] ) ... 1" (r'll ' til ) -

"( " ') z: F" rl,tl , "' , rll,tll; r '(,tl' "' , r ",t"

permut ation s

with summation over all n! permutations of their arguments rl ,tl , ... , r" , t". We consider a variety of microscopic models for the random forces 1" and tin the stochastic problem (6.80) . Under the statistically simplest "white noise" assumption, these random forces are uncorrelated in space and time, DIlIl (r - r', t - t')

= 1" 8(r' - r) 8(t - t'),

Dww(r - r' , t - t') = T;v 8(r' - r) 8(t - t') ,

(6.81)

in which 1" and T;v are the related Onsager coefficients. Recent studies reported on the statistics of transport events in the cross- field systems (Ghendrih et aI., 2003 ; Carreras, 1996) pointed out the virtual importance of correlations existing between density and vorticity fluctuations in the dynamical problem . In particular, this effect is referred to the formation of large density blobs of particles close to the divertor walls by attracting particles via the cross field flow, the latter being the larger for strong blobs with strong potential gradients (Ghen drih et aI., 2003) . Indeed, in the physically realistic models of turbulent transport in the SOL, it seems natural to assume that the random perturbations enter into the system in a correlated way. To be specific, let us suppose that there exists a finite reciprocal correlation time 'rc (I r' - r I) > 0 between the random sources fw(r, 'rc ) and f,,(r',O) in the stochastic problem (6.80) . For a simplicity, we suppose that the relevant relaxation dynamics is given by the Langevin equation, (6.82) in which f3 ~ ( f~) > O. In the momentum representation, the non-local covariance operator 'r(-:-I can be specified by the pseudo-differential operator with the kernel (6.83) which specifies the characteristic viscoelastic interactions between the "fast" modes of density and vorticity fluctuations. The coupling constant A, > 0 naturally establishes the time scale separation between "fast" and "slow" modes. In the case

255

6 What is Control of Turbulence

of 2y « I , the Langevin equation (6.82) with the kernel (6.83) reproduces the asymptotic dispersion relation typical for the Langmuir waves traveling in plasma, W rv k2 - Th as k ----+ 0 with n, c::: 0.0804 (in three dimensional space) (Pelletier, 1980) . Alternatively, for the exponents 2y ----+ I, it corresponds to the ion-acoustic waves traveling in the collisionless plasma with the velocity /\. v rv JTe / M where M is the ion mass . Intermediate values of y correspond to the various types of interactions between these two types of plasma waves described by the Zakharov 's equations (Zakharov, 1972) . The relaxation dynamics (6.82 - 6.83) establishes the relation between the covariances of random sources in (6.80), I

DIlIl(r,t) = 4;rrf3

J ' , t"

drdt Jo dp

Jo(pr')exp(-/\. v p 2- 2 yt,) , , /\' l - 2y Dww(r -r,t -t) Vp

(6 .84) where r == Ir I, and Jo is the Bessel function of the first kind . In the present section, we choose the covariance of random vorticity source,

\!w(r,t)!w(r',t') ) =

J~:

J(::)2

Dww(w,k) exp [- iW(t - t' )+ik (r - r' )],

k == Ik I, in the form of white noise (6.81), in which the relevant On sager coefficient l,v is found from the following physical reasons. Namely, the instantaneous spectral balance of particle flux, I W(k) = 2

JdW 2;rr ( f,,(k, W)f,,(- k ,W)) ,

(6.85)

derived from (6.84) should be independent from the reciprocal correlation time t"c( k) at any k that is true provided Dww(w , k) ex: /\. k- 2y . Furthermore, the On sager coefficient r w has to fit into the appropriate physical dimension which is assembled from the relevant dimensional parameters, Uo v and k. Collecting these factors, one obtains the Ansatz D ww ex: /\. u6v3 k6- d - 2y , in which d = 2 is the dimension of space . The power law model for the covariance of random helicity force ex: k6 - d - 2y does not meet the white noise assumption since 8(r - r' ) rv kO and therefore calls for another control parameter 2£ > O. Eventually, we use the model (6 .86) with the actual value of regularization parameter 2£ = 4 , for d = 2. Let us note that the Ansatz (6.86) is enough flexible to include the various particular models of particle pump into the SOL. For instance, the alternative to the white noise assumption spatially uniform particle pump for which the covariance D ww c:::

. u5:() k = lim

~ ---;o

J (xPs)-~ dx

-

e ikx

r(d /2) . (J: ) = k- d ~ /2 lim ':> k ps ,

2;rr'

~ ---; o

256

Dimitri Volchenkov

in the large scales, can be represented by the Ansatz (6.86) with the actual value 2£ = 3. In the rapid-change limit of the stochastic model, A ----+ 00 (i.e., 't"c ----+ 0), the covariance (6.84) turns into I f, (k

\

Il\

V , m)f,I lI\ - k ,m) ) - Af3 r--;»

k2 -d-2e+2y

'

(6.87)

and recovers the white noise statistics (6.81) along the line e = r, in d = 2. Alternatively, in the case of A ----+ 0 (that corresponds to 't"c ----+ 00), the time integration is effectively withdrawn from (6.84), so that the resulting configuration relevant to (6.86) appears to be static oc k 4- d - 2e and uncorrelated in space (at d = 2) for 2£ = 2. The power-law models for the covariances of random forces has been used in the statistical theory of turbulence (Adzhemyan et al., 1998, see also the references therein). The models of random walks in random environment with long-range correlations based on the Langevin equation (6.82) have been discussed in concern with the problem of anomalous scaling of a passive scalar advected by the synthetic compressible turbulent flow (Antonov, 1999), then in (Volchenkov et al., 2002) , for the purpose of establishing the time scale separation, in the models of self organized criticality (Bak et al., 1987; Bak, 1996). Recently, the renormalization-group methodology have been applied in order to prove the breakdown of magnetic flux conservation for ideal plasmas, by nonlinear effects (Eyink and Hussein, 2006) . The analysis of (Eyink and Hussein, 2006) is based upon an effective equation for magneto-hydrodynamic (MHD) modes ; it is proven that flux-conservation can be violated for an arbitrarily small length-scale that is similar to the decay of magnetic flux through a narrow superconductive ring, by phase-slip of quantized flux lines . Being analogous to Onsagers result on energy dissipation anomaly in hydrodynamic turbulence, this result gives analytical support to and rigorous constraints on theories of fast turbulent reconnection .

6.3.3 Iterative solutions in crossed fields The linearized homogeneous problem, for the fluctuations of density n and vorticity w vanishing at t ----+ 00,

[at - V.1-1-]'X 1l = 8(r)8(t), [at -

Uo

v .1-1- ]'Xw (r , r) + g Iay 'XIl (r , t) = 8( r ) 8 (t ),

is satisfied by the retarded Green 's functions, +-

.1 1l (r,t )

(r4vt 2

e(t) = - exp - - ) , 4nvt

(6.88)

257

6 What is Control of Turbulence <----

,1

r t

w( , )

e-(t ) ex =4nuvt p

(r - -2

)

4uvt

x

2g I e(r ) + v(u + I) r2 '

fitting into the retarding conditions, :111(r, t) = :1w(r, t) = 0, for t < 0, that expre ss the casualty principle in the dynamical problem. Nonl inearities in (6.88) can then be taken into account by the perturbation theory,

n(r ,t) w(r ,t)

J =J =

dr 'dr'

:1 (r 11

r' , t -t') [f,,(r' ,t' ) - v(r' ,t') x V n(r' ,t') ] '

dr'dt':1 w(r-r' ,t-t') [fw(r' ,t')

+ L (- ltgk oynk(r' ,t' )k2:1

v(r' ,t') x v w(r',t')] + g l

J

dr'dt ':1 w(r-r' ,t-t')

J

dr" dt":1 11(r' - r" , t' - t") x Oy [f" (r" , t") - v(r" , t" ) x Vn(r" ,t") ] . (6.89)

The solutions (6.89) allow for the diagram representation (see Fig. 6.5), where the external line (a tail) stands for the field n, the double external line denotes the field w, and the bold line repre sents the magnetic flux v. The triangle s stay for the random force f" , and the filled triangles represent fw . The retarded Green functions :111 are marked by the lines with an arrow which corresponds to the arguments (r' ,t' ) and (r" ,t" ) (the direction of arrows marks the time ordering of arguments in the lines). Similarly, the double lines with an arrow corre spond to the retarded Green function s :1 w . Slashes corre spond to the differenti al operator V. Circles surrounding vertexes representing the antisymmetric interaction v x V, squares present the vertexes proportional to the poloidal grad ient Oy. All correlation function s of fluctuating fields and funct ions expre ssing the system response for the extern al perturbations could be found by the multiplication of trees (6.89) displayed on Fig. 6.5 followed by the averaging over all possibl e configurations of random forces f,,( r , t) and fw( r ,t ). In diagrams , this procedure corresponds to all possible contractions of lines ended

~ - o-<-I>-~

• •..

=<---< -g,=< .g, -.{. ...

-- =

--

Fig. 6.5 The diagrammatic representation for Eq. (6.89).

258

Dimitri Volchenkov

with the identical triangles . Thereat, the diagrams having an odd number of external triangles (correspondent to the random forces) give zero contributions in average. As a result of these contractions, the following new elements (lines) appear in the diagrams of perturbation theory: Llllll(r - r' ,t' - t)

=

J J dr, dr,

dr2dt2 "lll(r - r[ ,t]- t)DIlI1(r] - r2,t[ - t2)

X +Ll n ( r2 - r 1, t: - t ') ,

Llww(r-r',t'-t) =

J J drdr,

dr2dt2 "lw(r-r[,t] -t)Dww(r] -r2,t] -t2)

x Lf w(r2 - r',t2 - t') + gTdiLlllll(r - r',t' - t),

Llwll(r - r',t ' - t)

= g]

Lfwn(r-r' ,t'-t) = g[

(6.90)

J J

dr, dr, "lll(r - r[ ,t[ - t)dyLlww(r[ - r',t[ - r'), drj dr, Lfn(r-r[ ,t, -t)dyLfw(r[ -r' ,t[ -t') ,

which are the free propagators of particle density and vorticity fluctuations, the mixed correlator, and the retarded mixed Green's function. In diagrams, we present the free propagators (6.90) by the correspondent lines without an arrow, and the retarded mixed Green's functions by the composite directed lines (see Fig. 6.6). The cross-field drift function v( r, t) is not involved into the linear homogeneous problem (6.88) and, therefore, it does not appear in the free propagators (6.90), however, it is presented in the nonlinear part of dynamical equations and therefore appears in the diagrams of perturbation theory. Due to the simple relation w = V xv, the prop+agators containing the field v are the same as those with w: Llvv , Ll nv, and Ll nv. The bold lines representing v in Fig. 6.5 can be replaced in the diagrams of perturbation theory with the double lines (which correspond to the field w) with the additional

o , II ,

+-D- +-D- +'"

Fig. 6.6 First diagrams for the simplest response functions (5n/ 5j ;,) and (5w/5j ;,,) .

259

6 What is Control of Turbulence

factor (in the momentum representation) -i Ezms km / k2 , where m, S == x,y and Ez.xy is the antisymmetric pseudo-tensor, for each v. In this framework, the exact correlation functions of fields and the response functions can be found from the Dyson equations,

/ iSn \ iSf"

)-1 = f--l .1/1

- 1:/1,

/ iSw

)-1

\ iSfw

(6.91)

where 1:/1 and 1:w are the infinite diagram series, in which the first diagrams are shown in Fig. 6.6. The diagram technique introduced in the present section is suitable for the system preserving the continuous symmetry of (6.79), apart from the sheath .

6.3.4 Functional integral formulation of cross-field turbulent transport In the present section, we study the properties of diagram series resulting from the iterations of the stochastic dynamical equations with the consequent averaging with respect to all possible configurations of random forces. The diagrams for some correlation functions diverge in small scales. The use of conventional arguments borrowed from the quantum field renormalization group (Zinn-Justin, 1990) helps to prove the consequent subtraction of the logarithmic divergent terms in all orders of perturbation theory out from the diagrams. The set of diagrams arisen in the perturbation theory by the iterations of (6.80) is equivalent to the standard Feynman diagrams of some quantum field theory with the doubled set of stochastic fields: the fluctuations nand w, the flux function v, the auxiliary fields n', w' funct ionally conjugated to the Gaussian distributed random forces f" and fw in (6.80), and the Lagrange multiplier v' for the binding relation w = V x v. The coincidence of diagrams is a particular consequence of the general equivalence between the t-Iocal stochastic dynamical problems (in which the interactions contain no time derivatives) and the relevant quantum field theories (De Dominicis and Martin, 1979) with the action functional :/' found in accordance to the MSR formalism (Martin et aI., 1973). Statistical averages with respect to all admissible configurations of random forces in a stochastic dynamical problem can be identified with the functional averages with the weight exp .v", In particular, for the stochastic problem (6.80), the generating functional of the Green functions , 0' (Aep) , with the arbitrary source fields Aep(r,t) where ep == {n,n', w,w', V, v'} can be represented by the functional integral (6.92) in which

Dimitri Volchenkov

260

J -J ~

.9'( ep) =

drdr' dt dt' [n' (r , t )D/1/1 (r - r' , t - t')n' (r ' ,t' )

+w' (r , t )Dww(r- r' , t - tl)W' (r' , t')] drdtn'(r ,t) [dln(r, t ) + v(r,t) x V'n(r,t) - v,1l-n(r,t)]

- j 'drdt Wi (r ,t) [dtW(r ,t) + v(r,t)

X

V'w(r,t)

+L k21

-UO V,1l-w(r,t )]

+

J

dr drv'(r.r) [w(r, t ) - V'

X

gk

(-I )k

dynk(r,t)

k

v(r,t)] ,

(6.93)

The source functions A/1 '( r , r ) and A w' (r, t) in (6.92) are interpreted as the not random external forces, so that the Green functions

<w(r , r) Wi (r' , t' )) ,
'.1'lIl,(k, OJ) =

(-iOJ + vk 2 ) -

,1wv,(k) = -

fz;~ks ,

----+

'.1ww' (k, OJ) = (-iOJ + uovk2 ) -

1,

'.1/1/1'(k ,OJ) =

if~/1,(k,OJ),

f--

,1/1/1 (k, OJ) = ,1/1/1'D/1/1 (k, OJ) ,1/1'," ,1ww( k, OJ) = gTk;,1/1/1 (k, OJ) + if ww'Dww( k, OJ) '.1 w'w , ,1/Hv(k, OJ) = f--

,

19l

f--

,1/1/1' (k, OJ )ky,1 ww(k, OJ),

f--

f--

,1/Hv,(k,OJ) = ig, ,1/1/1 ,(k,OJ)ky,1 ww,(k,OJ) ,

1,

6 What is Control of Turbulence

261 +--

----+ *

.1 llwl(k, ro ) =.1 Wlll(k,ro).

Propagators including the field v co incide with those of the field w up to the multiplicative factor - i e zms km / k2 for each field v. Propagators of auxiliary fields .1<1>1<1>' (k, ro) = O. The action functional (6.93) is invariant with respect to the generalized Galile an tran sform ation s of fields in the polo idal direct ion,

vy(r,t ) ----+ vy(r ,t) - a(t) , n(x,y,t)

----+

n(x,y - b(t) ,t)

(6.94)

with the parameter of transformations a( r ) (the integrable function decaying at ----+ -00) and b(t ) = i. : a(t' )dt'. Furthermore, any quant ity Q in (6.93) can be characterized with respect to the independent scale tran sform ation s, in space and time, by its momentum dimen sion d~ and the frequency dimension d Q. In the "logarithmic" theory (which is free of interactions that is analogous to the linearized problem (6.88», these scale transformations are coupled due to the relation rv V2 in the dynamical equ ation s. This allow s for the introduction of the relevant total canonical dimen sion

t

a,

and the analysis of UV divergences arisen in the diagrams of perturbation theory based on the conventional dimension counting arguments (Zinn-Justin, 1990 ; Coll ins, 1992) . In dyn amic al models, dQ plays the same role as the ordinary (momentum) dimension in the cr itical static problems. Let us note that the poloidal gradient term oc y in (6.88) is respon sible for the stationary contribution s into the Green's function as t > 0, so that the above definition of dQ rem ain s unambiguous. Stipulating the natural normalization conventions,

a

one can find all relevant canonical dimensions from the simultaneous momentum and frequency scaling invariance of all terms in (6.93) (see Table . 6.3).

Table 6.3 Canonical dimensions of fields and parameters in the action functional (6.93)

v -2 I

o

Uo

o o o

n

W

v

V'

o o o

o

- I I 1

d

I

2

o

d

n' d

o

d

W'

d

2 d +4

-2 I

o

- I

2 3

Integrals correspondent to the diagrams of perturbation theory repre senting the I-irreducible Green ' s function s G diverge at the large momenta (small scales ) if the correspondent UV-divergence index oGis a nonnegative integer in the logarithmic theory, (6.95)

262

Dimitri Volchenkov

where d is the dimens ion of space, d

(n'nv) ,

(n'v),

(n'vv),

(n'n ... n)

(with an arbitrary number of fields n ), the only Galilean invariant Green 's function admissible in the theory (6.93) which actually diverges at large momenta is (n'n). The inclusion of the relevant counterterm subtracting their superficially divergent contribution is reproduced by the multiplicative renormalization of the Prandtl number UD = uZ u , where UD and U are the bare and renormalized values of Prandtl's number. In principle, the relevant renormalization constant Z; can be calculated implicitly from the graphs of perturbation theory up to a finite part of the relevant counterterm. However, the standard approach of the critical phenomena theory is useless for the determining of the large scale asymptotes in the problem in question , since the severe instability frustrates the critical behavior in the system preventing its approaching to the formal asymptote predicted by the conventional renormalization group method . As a consequence, the critical dimensions of fields and parameters which can be computed from the renormalization procedure would have just a formal meaning .

6.3.5 Large-scale instability of iterative solutions The iterative solutions for the stochastic problem (6.80) constructed in the previous subsections would be asymptotically stable in the large scales provided all small perturbations of both density and vorticity damp out with time. In particul ar, the exact response functions found from the Dyson equations (6.91) should have poles located in the lower half-plane of the frequency space. The stability of free response function

which effectively corre sponds to the linearized problem (6.88) is ensured in the large scales by the correct sign of the dissipation term vk 2 > O. In a "proper" perturbation theory, apart from a crossover region, the stability of exact response functions is also

6 What is Control of Turbulence

263

secured by the dissipation term which dominates over the dispersion relation in the large scales, (6.96) w(k ) = - ivk 2 + iLIlI1(k,w ), where the self-energy operator L nn (k , zo ) is the infinite series of all relevant 1- irreducible diagrams of perturbation theory. However, in the perturbation theory, the leading contribution into the self-energy operator is L nn :=: O( ky ) that could lift up the pole of the respon se function into the upper half-plane of the complex w-plane rising the instability in the system as k --+ O. Such an anomalously strong contribu tion come s from the diagrams which simultaneously include both the antisymmetric interaction vertex c- i ez · Cijz vk, together with the poloidal grad ient c- is, ky and the free propagator of particle density L1'lIl (k, w). Such diagrams appear in all orders of perturbation theory for the response functions ( on and ( on fw ) indicating that the instability could arise due to the random fluctuations of both density and vorticity. The first order I-irreducible diagrams for them are displayed in Fig. 6.7. The small parameters gn with n > 2 appear in the forthcom ing orders of perturb ation theory. In particular, the last diagrams in both series shown in Fig. 6.7 generate the "anomalous" contributions in the large scales . To be certain , let us consider the second diagram in the series for ( on which corresponds to the following analytical expre ssion:

/ 0/" )

/0

/ 0/,, )

Lnnll _looP:=: g2ky

J J J27r dp (2n)2

dw

2n

dw' kxPx + kypy - p; - P~

XL1 1l1l (p, w') Lfw(k - p, w - w' ).

(k-p)2

(6.97)

Fig. 6.7 The simplest I-irreducible diagrams contributing into the self-energy corre ctions L nn and L m v for the response function s (onloj ;,) and (onloj ;,,). The field indexes denot e the type of propagator s and simultaneously the type of vertexes. The slashes mark the position s of deri vatives V', and the skewed trian gles denotes the inverse operator curr ' .

264

Dimitri Volchcnkov

Being interested in the O( ky )-contribution into (6.97), one can neglect the kdependence in the integrand . The analytical properties of this contribution depends very much upon the certain assumption on the covariances of random forces since it changes the free propagator Li,1I1• For instance, under the white noise assumption (6.81) Li/l/l c::: (co 2 + y 2 p 4 ) - I , the integral in (6.97) diverges at the small scales (large momenta) for e < I and diverges at the large scales (small momenta) for e > I. Introducing the relevant cut-off parameters, one obtains the anomalous contribution Y s: (A 2-2 £ - m2 - 2 £) (6.98) y L/l/lII _loop c::: -k 8n(-I+£)(u+l) , in which u is the renormalized value of Prandtl's number. In the preceding section, we have shown that the logarithmic divergencies risen in diagrams for the response function (on/ Of,, ) in the small scales (large momenta) can be eliminated from the perturbation theory by the appropriate renormalization. The singularity in (6.98) arisen at the small momenta m --+ 0 for e > I would compensate the smallness of g2, so that any density fluctuation with ky > 0 appears to be unstable . Accounting for the finite reciprocal correlation time 'rc > 0 between the vorticity and density random sources in (6.80) introduces the new dimensional parameter A > ointo the particle density propagator, Li/l/l c::: (co 2 + y2 p4 (I + A p -2 y) ) - I . Then, the integral (6.97) can be computed by its analytic continuation for any momenta excepting for the isolated points , -I + E = ymod I, L/l/l

= YA(£ ,y) ky ,

U ) (- I+e)!Y g2

A(£ ,y)II -loop c::: ( I

8uycsc

(n(-I+£)) Y .

(6.99) The dispersion relation (6.96) determines the region of asymptotic stability in the phase space of cross-field transport system . Namely, a density fluctuation arisen in the SOL with some random momenta (kx , ky ) would be asymptotically stable with respect to the large scales k --+ 0 if (6.100) and be unstable otherwise. In the first order of perturbation theory, the amplitude factor A(£, y) is given by (6.99). For different values A (s, y), the stability condi tion (6.100) determines the set of circles (see Fig. 6.8) osculating at the origin which bound the unstable segments of phase space. One can see that the density fluctuations with ky --+ 0 (i.e. extended in the poloidal direction) are asymptotically stable for any I k, I > O. Density fluctuations characterized by I ky I > 0 would be asymptotically stable in a certain stochastic model provided sign (ky ) = - sign (A ( e, y)) , for the given values of E and y. The signature of the I-loop order contribution (6.99) into A( £, y) is displayed on the diagram in Fig. 6.9 (black is for + I, white is for -I) at different values of yand d = 6 - 2£ - 2y, the space dimension related to the actual value of regularization parameter e under the statistical assumption (6.86). It is important to note that the stability condition (6.100) can be formulated as an up-

6 What is Control of Turbulence

265

- 0.5

-I

L...--

-I

- - - - , - - - - - - -.........

- 0.5

0.5

Fig. 6.8 The admissibility condition (6.100) defines the set of elliptic curves which bound the unstable segments in the phase space. Density fluctuations with k; ---> 0, (i.e. extended in the poloidal direction) appear to be stable in the large scales for any Ik, I > O. Those fluctuations charact erized by Ik; I > 0 would also be stable in the large scales provided sign ( ky ) = - sign (A (s, y) ) for the given values e and y. 3

d

y Fig . 6.9 The signature of I-loop order contribution (6.99) into A (s, y) (black is for + I , white is for - I ) at different values of y and d = 6 - 2£ - 2y, the space dimension related to the actual value of regulari zation parameter e under the statistical assumption (6.86).

266

Dimitri Volchenkov

I

per bound for the order parameter ~ = ky k 2 : ~ < ~c where ~c = A ( e, y) - 1. For the uncorrelated statistics of random forces, in the stochastic dynamical problem , A( e, y) (6.98) diverges as k ----+ 0 and therefore ~c ----+ O.

6.3.6 Turbulence stabilization by the Poloidal electric drift To promote the stochastic cross-field turbulent transport system (6.80) from the instability to a stable regime , it seems natural to frustrate the symmetry which break s the Galilean invariance in (6.80). This can be achieved by generating a constant uniform drift in the poloidal direction v y ----+ v y - V (by biasing the limiter surface, c{>(x) ----+ c{>(x) + xV) that would eradicate those configurations with the trivial poloidal component of electric drift v y = O. In general, the relevant dispersion equation w (k , V) = 0 could have many solutions Vk for k « I . Herewith, the turbulence stabilization is achieved for the drifts V from the intervals Vk-I < V < Vk for which

lm(w) < O. To be certain , let us consider the dispersion equation correspondent to the simplest response function 01,, ), in the I-loop order. The leading contribution into the dispersion equation is given in the large scale region by the diagram (6.97). Under the white noise assumption (6.81), the free propagator accounting for the uniform electric drift V is .11l1l c::: (w2 + v 2k4 + V 2ky 2 ) - I. Then , for Iky I < I, the dispersion relation reads as following,

(onl

w(k) V

. k2 + 1'k sss: Yu+l

~ -I V

,-

(

2 2e) v(m 8re(-I+f)

-

V

2(u+2)r(I rz: V /2+f)IOgm) re ----'--------,----'------'------'--------,----'----=--8v(l+u)r(2+f) '

(6.101) The latter relation shows that for any finite m > 0 there exists finite Vc < 00 such that for any V > Vc one obtains lm( w) < 0 in (6.10 I), however, Vc ----+ 00 as m ----+ O. In contrast to it, in the case of correlated statistics (6.82 - 6.83,6.86), the dispersion relation is not singular for k ----+ 0 excepting some particular values of e and y, and the correspondent stabilizing electric drift, in the I-loop order, equals to (uA)I /2Y(1 +u)I + (2e+1 )/2Y (( I +u)I H !Y

-I)

sin re(2f+ 1) /2y sin ref jy (6.102) In the range 0 < y < 1/2, u > 0, this expression is singular at the points e / y E Z. (I + u)2+ (2e+1)/2y+ u (2f + I) - 2 Y (I + 2u)

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6 What is Control of Turbulence

6.3.7 Qualitative discrete time model of anomalous transport in the SOL Large scale instability developed in the cross-field model (6.80) is related to the appearance and unbounded growth of fluctuations of part icle density close to the wall. In accordance to the fluctuation-dissipation theorem, the fluctuations arisen in the stochastic dynam ical system are related to its dissipative properties. In particular, the matrix of the exact response functions JR.( k, OJ ) expressing the perturbations of fields nand w risen due to the random sources f" and f w determines the matrix of exact dynamical Green 's function s G ( k, OJ) ,

JR. ( k, OJ) - JR.i· ( -k,

-OJ)

= i OJ G ( k, OJ )

(6.103)

where R f is the transposed R In the large scale limit k ---+ 0, we take into account for the leading contributions into the self-energy operators in the elements of JR., R nn

.

~ ( -/OJ -

vAl ky + v k 2) -1 ,

.

Rll\v~ ( -/OJ-VUA2ky

+vuk2) -1 , (6.104)

R wn

.

~ ( -/OJ

+ vk

2) -1

,

R ww

=

. ( -/OJ

+ vuk

2) -1

,

in which A 1,2 are the amplitudes of the anomalou s contributions competing with the dissipation 0<: O(k2 ) in the large scales . Fluctuations of particle density arisen in the model (6.80) grow up unboundedly provided either AI ky > k 2 or A2ky > k 2 for the given values of e and y. The corre spondent advanced Green 's funct ions appear to be analytic in the lower half-pl ane of the frequency space, a

vu ·Gnw(k,t)= - (

e(-t)

e(-t)

2) ---2- ' (6.105) A2ky-k k

being trivial for t > O. For instance , let us consider the advanced Green 's function G~n which relates the density of particles 8n( r , t) in the fluctuations characteri zed with AI ky > k 2 and arisen at the point r E .Q inside the divertor at time t with the particle density 8n( r' , t ' ) of those achieved the divertor wall at some subsequent moment of time t' > t at the point r ' E d.Q:

/ 8n(r ,t' )lc)Q

=1

1 <1'

dt

r dr8n (r,t)G~n(r/-r)

JQ

(6.106)

where

(6.107)

Dimitri Volchenkov

268

J

in which Ir - r'l == (x - x')2 + (y - y')2, Jo and Ho are the Bessel and Struve functions respectively. The integral in the r.h.s . of (6.106) is finite for any .Q provided IA I I > I, but for the compact .Q as IA I I < I. To be specific , let us consider the circle CR of radius R as the relevant domain boundary and suppose for a simplicity that the density of particles incorporated into the growing fluctuations inside the domain is independent of time and maintained at the stationary rate 8no( r) . Then the r-integral in the r.h.s of (6.106) can be calculated at least numerically and gives the growth rate B(R) for those density fluctuations,

8n(R, r )

= -r ·B(R) ,

(6.108)

where -r is the traveling time of the density blob to achieve the divertor wall that can be effectively considered as a random quantity. It is the distribution of such wandering times that determines the anomalous transport statistics described by the flux pdf in our simplified model. The discrete time model we discuss below is similar to the toy model of systems close to a threshold of instability studied in (Flori ani et aI., 2003) recently. Despite its obvious simplicity (the convection of a high density blob of particles by the turbulent flow of the cross field system is substituted by the discrete time I-dimensional (in the radial direction) random walks characterized with some given distribution function) , its exhibits a surprising qualitative similarity to the actual flux driven anomalous transport events reported in (Ghendrih et aI., 2003). We specify the random radial coordinate of a growing fluctuation by the real number x E [0 , I]. Another real number R E [0, I] is for the coordinate of wall. The fluctuation is supposed to be convected by the turbulent flow and grown as long as x < R and is destroyed otherwise (x 2 R). We consider x as a random variable distributed with respect to some given probability distribution function lP' {x < u} = F (u). It is natural to consider the coordinate of wall R as a fixed number, nevertheless, we discuss here a more general case when R is also considered as a random variable distributed over the unit interval with respect to another probability distribution function (pdf) lP' {R < u} = Q(u). In general, F and Q are two arbitrary left-continuous increasing functions satisfying the normalization conditions

F(O)

= Q(O) = O,F(CXl) = Q(CXl) =

1.

Given a fixed real number 11 E [0 , I], we define a discrete time random process in the following way. At time t = 0, the variable x is chosen with respect to pdf F, and R is chosen with respect to pdf Q. If x < R, the process continues and goes to time t = I. Otherwise, provided x 2 R, the process is eliminated. At time t 2 I , the following events happen: i) with probability 11, the random variable x is chosen with pdf F, but the threshold R keeps the value it had at time t - I. Otherwise, ii) with probability I - 11 , the random variable x is chosen with pdf F , and R is chosen with pdf Q.

269

6 What is Control of Turbulence

If x :2: R , the process ends; if x

< R, the process continues and goes to time t + I.

Eventually, at some time step r , when the coordinate of the blob, x, drops "beyond" R, the process stops, and the integer value -r resulted from such a random process limits the duration of convectional phase. The new blob then arises within the domain, and the simulation process starts again. While studying the above model, we are interested in the distribution of durations of convection phases PI) (r: F, Q) (denoted as P( r) in the what following) provided the probability distributions F and Q are known, and the control parameter 1] is fixed. The motionless wall corresponds to 1] = O. Alternatively, the position of wall is randomly changed at each time step as 1] = I . The proposed model resembles to the coherent-noise models (Newman and Sneppen, 1996; Sneppen and Newman, 1997) discussed in connection with a standard sandpile model (Bak et al., 1987) in self-organized criticality, where the statistics of avalanche sizes and durations take power law forms . We introduce the generating function of P( r ) such that 00

L s' P( r ),

P(s) =

P(-r)

, =0

= ~ d' P(s) r!

ds'

I

s=o'

(6.109)

and define the following auxiliary functions

1 00

K(n) =

F(u)"dQ(u) , OK(n) = K(n) - K(n+ I),

p(l) = 1]' K(l + I) , q(l) = (1-1])' K(l)'-I , r(l) = 1]' [1] OK(l+ I) + (1-1] )K(l+ I) OK(O)] , P = 1] oK(I) + (I - 1]) K(I )OK(0) .

for I :2: I , for! :2: I , for I :2: I ,

p(O) = 0 , q(O) = 0 , r(O) = 0 , (6 .110)

Then we find ,

P(s)=OK(O)+ps+

s

'( r() [r(s)+pp(s)q(s) +P K(l)q(s)+K(l) q(s)r(s)] , I-psqs

(6.111) where p(s),q(s),r(s) are the generating functions corresponding to p(l) ,q(l) ,r(l) , respectively. In the marginal cases 1] = 0 and 1] = I, the probability P(r) can be readily calculated, (6.112) The above equation shows that in the case of 1] = 0 , for any choice of the pdf F and Q, the probability P(r) decays exponentially. In the opposite case 1] = I, many different types of behavior are possible, depending upon the particular choice of F and Q. To estimate the upper and lower bounds for P(r) for any 1], one can use the fact that

270

Dimitri Volchenkov

K(IY' < K(n) < K(I)

and 0

< 8K(n) < K(I),

n E N.

Then the upper bound for K(n) is trivial, since 0 ~ F(u) ~ I for any u E [0 , I]. The upper bound for K (n) exists if the interval of the random variable u is bounded and therefore can be mapped onto [0 , I] (as a consequence of Jensen 's inequality, and of the fact that the function u :----+ u" is convex on the interval ]0, I [ for any integer n). The calculation given in (Floriani et al., 2003) allows for the following estimation for the upper bound,

P1)(r) ~ T/T 8K(r) + (I - T/)K(I) 8K(0) [T/ + (1 - T/) K(l)]T-I +T/K(l) { [T/+(l-T/)K(l)]T-l_T/T-l} ,

(6 .113)

and, for the lower bound,

P1)(r) ~ T/T 8K(r) + (1- T/) K(1)T 8K(0)

= T/TP1) =I(r)+(l-T/)P1)=o(r).

(6.114)

We thus see that, for any 0 ~ T/ < I, the decay of distribution P( r) is bounded by exponentials. Furthermore, the bounds (6.113) and (6.114) turns into exact equalities, in the marginal cases T/ = 0 and T/ = 1. The simpler and explicit expressions can be given for P(r) provided the densities are uniform dF(u) = dQ(u) = du for all u E [0 , I]. Then Eqs . (6.112) give, I

P1) =1 (r ) = (r + 1)(r + 2)

(6.115)

For the intermediate values of T/, the upper and lower bounds are

T/ T

(r+ 1)(r+2)

I (I__T/ + )T

+ (1- T/)2- (HI ) < P( r ) < _ -

- 2

2

(6.116)

The above results are displayed in Fig. 6.10. The accounting for the dissipation processes introduces the order parameter ~ = Ikyl / (k; + k;) and its critical value ~c such that the particle density fluctuation 8n(~) grows unboundedly with time as ~ > ~c and damps out otherwise. We compute the value of ~c, in the first order of perturbation theory developed with respect to the small parameter Ps/ Rii where Ps is the Larmor radius, R is the major radius of torus, and ii is the mean normalized density of particles. Our results demonstrate convincingly that the possible correlations between density and vorticity fluctuations would drastically change the value ~c modifying the stability of model. Characterizing the possible reciprocal correlations between the density and vorticity fluctuations by the specific correlation time rc , we demon strate that any fluctuation of particle density grows up with time in the large scale limit (k ----+ 0) as rc ----+ 00 (the density and vorticity fluctuations are uncorrelated) and therefore ~c = O. Alternatively, ~c > 0 provided rc < 00.

6 What is Control of Turbulence

271

-2 1

P I (1) = (t+I) (/+2)

-4

q=!

-6

q=O. 7

-8 -1

Po

q=O .5

(1) = T(t+!)

-12 -14 q =O

-16 -18 5

10

15

20

25

Fig. 6.10 The distributions of wandering times near the wall in the discrete time model, in the case of the uniform densities dF (u) = dG(u ) = du for all u E [0,00) at different values of control parameter 1].

The reciprocal correlations between the fluctuations in the divertor is of vital importance for a possibility to stabilize the turbulent cross field system, in the large scales, by biasing the limiter surface discussed in the literature before (Ghendrih et al., 2003) . Namely, if ~c > 0, there would be a number of intervals [Vk-l ,Vk] for the uniform electric poloidal drifts V such that all fluctuations arisen in the system are damped out fast. In particular, in the first order of perturbation theory, there exists one threshold value Vc such that the instability in the system is bent down as V > Vc . However, Vc --+ 00 as ~c --+ O. To get an insight into the statistics of growing fluctuations of particle density that appear as high-density blobs of particles close to the reactor wall, we note that their growth rates are determined by the advanced Green's functions analytical in the lower half plain of the frequency space. We replace the rather complicated dynamical process of creation and convection of growing density fluctuations by the turbulent flow with the problem of discrete time random walks concluding at a boundary. Such a substitution can be naturally interpreted as a Monte Carlo simulation procedure for the particle flux. Herewith, the wandering time spectra which determine the pdf of the particle flux in such a toy model are either exponential or bounded by the exponential from above . This observation is in a qualitative agreement with the numerical data reported in (Ghendrih et al., 2003) .

272

Dimitri Volchenkov

6.4 Conclusion Applications of methods developed in quantum field theory to the problems of statistical physics and critical phenomena have a long history . These powerful methods became an important tool in studies of nonlinear dynamical systems . In this report, we have developed a strategy of use the RG method in purpose of study the longtime large-scale asymptotic behaviors in stochastic magneto-hydrodynamics. The main conclusion of the study in magneto-hydrodynamics is that the RG transformations are characterized by two different fixed points stable with respect to long-time large-scale asymptotic behavior that can be naturally interpreted as "kinetic" and "magnetic" critical regimes , in which fields and parameters of the MHD theory acquires different critical dimensions. We have investigated long-time large-scale asymptotic behavior of correlation functions and composite operators (the local averages of fields and their derivatives , which can be observed in real experiments) in both critical regimes . The immediate observation of our study is that the MHD system is thoroughly unstable. Perhaps, the most fascinating result of our approach to MHD is the prediction of "optimal size" eddies that could survive in cross-fields much longer than others . In fact, we claim that if the cross-field system losses stability, it becomes transparent for certain plasma vortexes .We have cons idered two -dimensional models of the cross-field turbulent transport close to the "scrape-off layer" (SOL) in thermonuclear reactors. Stochastic perturbations of electron density and vorticity are responsible for the aggregation of electrons into bulbs which then propagate ballistically towards the wall blanket. The operation stability of the "next step" device crucially depends upon correlations between the fluctuations of electron density and vorticity . We have studied possible mechanisms which break the operation stability and proposed a simple discrete-time "toy model" resembling a Markov chain that correctly reproduces the statistics of the burst-like events observed experimentally in the ITER facility, in Cadarache (France).

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Lifschitz A.E., 1989, Magnetohydrodynamics and Spectral Theory, Kluwer Academic Publishers, Dordrecht. Ma S.K., 1976, Modern Theory of Critical Phenomena , Benjam in, Reading. Martin P.c., Siggia E.D. and H. A. Rose, 1973, Statistical Dynam ics of Classical Systems, Phys. Rev. A, 8, 423. Monin A.S. and Yaglom A.M., 1971, Statistical Fluid Mechani cs, 1, MIT Press, Cambridge, Mass. Monin A.S. and Yaglom A.M., 1975, Statistical Fluid Mechanics, 2, MIT Press, Cambridge, Mass. Nedospasov A.V. et aI., 1989, Turbulence near wall in tokamaks, Sov. J. Plasma Phys., 15, 659. Newman M.EJ . and Sneppen K., 1996, Avalanches, scaling, and coherent noise, Phys. Rev. E, 54, 6226. Obukhov A.M., 1941, On the distribution of energy in the spectrum of turbulent flow, Dokladi Akademii Nauk USSR, 32, I, 22 (in Russian). Pelletier G., 1980, Langmuir turbulence as a critical phenomenon. II Application of the dynam ical renormalization group method, 1. Plasma Phys., 24 , 421. Phythian R., 1977, The functional formalism of classical statistical dynamics , J. Phys. A , 10, 777. Rudakov D.L., l .A . Boedo, R.A. Moyer et aI., 2002, Fluctuation-driven transport in the DIII-D boundar y, Plasma Phys. Control Fusion, 44, 717. Sagaut P. and Cambon c., 2009, Homo geneous Turbulence Dynam ics, Cambridge University Press, Cambrige . Sagdeev R.Z. and Zaslavsky G.M., 1986, in Nonlin ear Phenomena in Plasma Physics and hydrodynami cs, Ed. R.Z. Sagdeev, Mir Publishers, Moscow. Sneppen K. and Newman M.EJ., 1997, Coherent noise, scale invariance and intermittency in large systems, Physica D, 110,209. Stangeby P.c. and McCracken G.M., 1990, Plasma boundary phenomen a in tokamaks, Nucl. Fusion, 30, 1225. Vasil'ev A.N., 1998, Functional Methods in Quantum Field theory and Statistics, Gordom and Breach, New York. Volchenkov D., 1997, Composite operators of the canonical dimension d=3 in magneto-h ydrodynamic turbulence, Acts ofSt.-Peterburg University: Physics and Chemistry, 2, 9-16 (in Russian). Volchenkov D., 2000, Field-theoretic approach to a stochastic magnetoh ydrodynam ics: the dimensions of composite operators, Phys. Lett. A , 265, 122-117. Volchenkov D., 200 I, The Bending Instability in the Vorticity Transport Through a Turbulent Flow, Intern. Jour. ofMod. Phys. B, 15, 1147-1164. Volchenkov D., Cessac B. and Blanchard Ph., 2002, Quantum field theory renormalization group approach to self-organized criticality : the case of random boundaries, Int. 1. Mod. Phys. B, 16, 1171. Volchenkov D., 2005, Stochastic models of edge turbulent transport in the thermonuclear reactors, 1. Phys.: Con! Ser., 7, 214-226. Wyld H.W., 1961, Formulation of the theory of turbulence in an incompressible fluid, Ann. Phys., 14, 143.

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Chapter 7

Entropy and Transport in Billiards M. Courbage and S.M. Saberi Fathi

Abstract Recent progress of the theory of dynamical systems and billiards sheds new light on the nonequilibrium statistical mechanics. Mixing, weak mixing and continuous spectrum are associated to relaxation to equilibrium via entropy increase. The properties of the relaxation time are reflected in the transport properties, which could be anomalous both in Sinai billiard with infinite horizon and in the barrier billiard. Numerical simulations are presented to corroborate these properties.

7.1 Introduction Recent progress in the ergodic theory of dynamical systems allowed to reconsider some old long debated problems as entropy increase and transport in conservative motions . The motion in billiards is, in this respect, the most studied model. Divergence of trajectories, mixing and weak mixing are among the main properties that are leading to an increase of the Gibbs coarse-graining entropy . These properties studied in the billiard by Krylov and Sinai were the main motivations for a series of simulations of the time evolution of the Gibbs entropy . We shall present some of them in the first part of this chapter. On the other hand, anomalous transport was discovered as one of the most important properties of chaotic and pseudo-chaotic motion in billiards, a field to which M. Courbage Laboratoire Matiere et Systemcs Complexes (MSC), UMR 7057 CNRS et Universite Paris 7- Denis Diderot , Case 7056, Batirnent Condorcct, 10, rue Alice Domon et Lonie Duquet , 75205 Paris Cedex 13, France, e-mail : [email protected] .fr S. M. Saberi Fathi Department of Physics , University of Wisconsin-Milwaukee, 1900 E. Kenwood Blvd ., Milwaukee, WI 53211 , USA, e-mail: [email protected]

A. C. J. Luo et al. (eds.), Long-range Interactions, Stochasticity and Fractional Dynamics © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010

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M. Courbage and S.M. Saberi Fathi

the prominent physicist George Zaslavsky greatly contributed. In fact, two types of irregular motions can be found in billiards, the first corresponding to exponentially diverging trajectories, and the second corresponding to linearly diverging trajectories. We shall present a series of simulations of the transport in the second part of the chapter.

7.2 Entropy There are two concepts of entropy in the theory of dynamical systems : the first one is the famous Kolmogorov-Sinai entropy introduced by Kolmogorov in 1958. Kolmogorov, who was familiar with the Shanonn entropy for random process, designed this concept and used it in order to solve the isomorphism problem of Bernoulli systems . In 1959, Sinai" modified and extended the ideas and the results of Kolmogorov to any dynamical system (DS) with an invariant probability measure (also called measurable DS). It is important to note that the measure theoretical entropy is a number that characterizes the family of isomorphic dynamical systems. It is one of the main tools to classify all measurable dynamical systems . Although this theory provided considerable information about their structure, many problems are still open . On the other hand, the non-equilibrium entropy, introduced by Boltzmann in kinetic theory of gases, can be defined in the case of measurable DS. Recall that the Boltzmann H-theorem defines the entropy for the one particle probability distribu tion!r(x) as

S(!r(x)) = -

J

!r(x)log(!r(x)dx.

(7.1)

Boltzmann showed that this quantity is monotonically increasing for all solutions of his celebrated equation. During many years until the beginnings of the twentieth century the Boltzmann H -theorem was the object of many discussions and controversies. Later on Ehrenfest proposed the urn Markov chain model for the approach to equilibrium with an Htheorem . The model consists of n = 2N balls distributed inside two halves of a box : left and right. On account of collisions between particles, Ehrenfest postulated that at regular time interval a particle can leave the right half or to join it . So if the state space of the system is described by the number X of particles in the right hand side, the dynamics of the system would be a Markov chain where the only allowed transitions are from X = m to X = m - I, with probability m /2N, or from X = m to X = m + I with the complementary probability. Mark Kac gave an exhau stive solution of this model in his book (Kac, 1959). Briefly speaking, it is possible to find a unique stationary probability distribution {)1i}, i = 0, I, ... , n = 2N; such that any initial distribution {Vi(t)} converges to {)1i}' The non-equilibrium entropy of the distribution {Vi(t)} with density !r = is given by the Boltzmann like formula :

vj;l

279

7 Entropy and Transport in Billiards

J

- it log itdu = -

L Vi(t) l oVi(t) g- . i

(7 .2)

J.!i

The variable X is a "macroscopic variable" which means that a given value of X corresponds to a region in the phase space of 2N dimensions. Distinct values of X correspond to distinct regions, {.9'i} . The set of .9'/ s form a partition of the phase space . However, it is obvious that strictly speaking the process X (z ) is not Markovian although some time it is claimed to be approximately Markovian. Independently, Gibbs imagined the dynamical mixing property as a mechanism of the approach to equilibrium for systems out of equilibrium. His ideas are based on the consideration of the phase space of an isolated system of N particles where the equilibrium is described by the microcanonical ensemble as an invariant measure. The system will approach the equilibrium if any initial probability distribution will converge to equilibrium under the hamiltonian flow. According to Gibbs this will happen if the shape of any subset will change boldly under the flow, although conserving a constant volume, winding as a twisted filament filling, proportionally, any other small subset of the phase space . The famous image of this mechanism is the mixing of a drop of ink in a glass of water. Later on, Hopf found a whole class of mixing DS: the differentiable hyperbolic OS where to each trajectory is attached two manifolds expanding and dilating in transversal directions. So, any domain of the phase space will be squeezed and folded filling densely any region of the phase space. For example in the baker transformation, the expanding and contracting manifolds are horizontal and vertical respectively, so that any small horizontal segments will be uniformly distributed in the phase space after few iterations of the transformation (see Fig. 7.1). The importance of mixing and exponential instability of trajectories for obtaining H -theorem has been discussed by (Krylov, 1950). The H-theorem for measurable dynamical systems describes the approach to equilibrium, the irreversibility and entropy increase for measurable deterministic evolutions. That is dynamical transformation T on a phase space X with some probability measure J.!, invariant under T, i.e. J.!(T - 1E) = J.!(E) for all measurable subsets E of X . Suppose also that there is some mixing type mechanism of the approach to equilibrium for T, i.e. there is a sufficiently large family of non-equilibrium measures v such that (7 .3)

Then, the H -theorem means the existence of a negative entropy functional S( VI) which increases monotonically with t to zero, being attained only for v = J.!. The existence of such functional in conservative dynamical systems has been the object of several investigations during last decades see (Courbage, 1983; Courbage and Prigogine, 1983; Garrido et al., 2004 ; Goldstein and Penrose, 1981; Misra et aI., 1979; Courbage and Misra, 1980; Goldstein et aI., 1981; Sinai', 1994). Here we study this problem for the Lorentz gas and hard disks . Starting from the non-equilibrium initial distribution v, and denoting by .9' a partition of the phase space formed by cells (.9'1,.9'2, ..., .9'11 ) and by Vi(t) = V O T - I (.9'i), the probability at time t for the system to be in the cell .9'i and such that V(.9'i) :f. J.!( .9'i) for some i, the approach to equilibrium implies that Vi(t) ---+ J.!i as

280

M. Courbage and S.M. Saberi Fathi

1.0

1.0

0.8

0.8

"

~ 0.6

'"

' ;(

< 0.4

~

0.2

0.2

0.0

0.0 0.0

0.2

0.4

0.6

0.8

0.0

1.0

0.2

X Axis Title

0.4

0.6

0.8

1.0

0.8

1.0

X Axis Title

1.0

1.0 0.8

0.8

~" 0.6

" ~0.6

'" ' ;(

'" ';(

< 0.4

<0.4

s,

~

0.2

0.2

0.0

0.0 0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

X Axis Title

0.4

0.6

X Axis Title

Fig.7.1 Evolution of an expanding fiber under baker transformation.

t

--+

00

for any i. The entropy functional will be defined by: (7.4)

which we simply denote here after Y' (t ). The H-functional (7.4) is maximal when the initial distribution is concentrated on only one cell and minimal if and only if vl (9 i ) = f.1( 9 i) ,'Vi. These properties are shown straightforwardly. This formula describes the relative entropy of the non-equilibrium measure VI with respect to f.1 for the observation associated to 9 . It coincides with the information theoretical concept of relative entropy of a probability vector (Pi) with respect to another probability vector (qi) defined as follows : -In Pi being the information of the jth issue under the first distribution, - L Pi In( ~), is equal to the average uncertainty gain of the experience (Pi) relatively to (qi)' A condition under which formula (7.4) shows a monotonic increase with respect to t is that the process VI (9 i ) = V 0 T - I (9 i ) verifies the Chapman-Kolmogorov

7 Entropy and Transport in Billiards

281

equation valid for Markov chains and other infinite memory chains (Courbage and Nicolis , 1990; Courbage and Hamdan, 1994). For a dynamical system, this condition is hardly verified for given partition 9. However, the very rapid mixing leads to a monotonic increase of the above entropy, at least during some initial stage, which can be compared with the relaxation stage in gas theory. In this Chapter, we will first compute the entropy increase for some remarkable non-equilibrium distributions over the phase space of the Sinai" billiard . The dynamical and stochastic properties of the Lorentz gas in two dimensions which we consider here was investigated by Sinai" and Bunimovich as an ergodic dynamical system (Sinai, 1970; Bunimovich et al., 1990; Chernov and Young, 2000b) . Other transport properties have been also studied numerically (Gaspard and Beijeren, 2002; Zaslavsky and Edelman, 2004) . This is a system of non interacting particles moving with constant velocity and being elastically reflected from periodically distributed scatterers. The scatterers are fixed disks . On account of the absence of interactions between particles the system is reduced to the motion of one billiard ball. We shall investigate the entropy increase under the effect of collisions of the particles with the obstacles. For this purpose, we consider the map T which associates to an ingoing state of a colliding particle the next ingoing collid ing state. The particle moves on an infinite plane, periodically divided into squares of side D called "cells", on the center of which are fixed the scatterers of radius a (Fig. 7.2). The ingoing colliding state is described by an ingoing unitary velocity arrow at some point of the disk. To a colliding arrow V I (PI) at point Pion the boundary of the disk the map associates the next colliding arrow V2(P2) according to elastic reflection

(I)

50 /2 1--- - -+- - - +-- - -+- - - +-- - ---1

30 /2

0 /2

1:::(1,...[["", ) '---+----+-----t---I-----;

L..-_ _- ' -_ _----L

30/2

50/2

Fig. 7.2 The motion of the particle on a toric billiard.

..L.<::._ _--'-_ _----'

282

M. Courbage and S.M. Saberi Fathi

law. Thus , the collision map does not take into account the free evolution between successive collisions. The billiard system is a hyperbolic system (with many singularity lines) and, in order to have a rapid mixing, we will consider initial distributions supported by the expanding fibers. Such initial measures have been used in (Courbage, 1983; Courbage and Prigogine, 1983; Sinai, 1994). For the billiard the expanding fibers are well approximated by particles with parallel arrows velocity. We call this class of initial ensemble beams of particles. We first compute the entropy increase under the collision map for these initial distributions. We will consider finite uniform partitions of the phase space as explained below. The entropy functional will be defined through (7.4). For this purpose, the phase space of the collision map is described using two angles ({3, ljI), where {3 is the angle between the outer normal at P and the incoming arrows V(P), {3 E [0, ~ [, and ljI E [0,1r] is the angle between x-axis and the outer normal at P. Thus , the coll ision map induces a map : ({31 , ljI,) ---+ ({32 , ljI2) (see Fig. 7.4) and we shall first use a uniform partition of the ({3 , ljI) space. The computation shows that whatever is the coarsening of these partitions the entropy has the monotonic property in the initial stage. It is clear that, along mixing proce ss, the initial distribution will spread over all cells almost reaching the equilibrium value. Physically, this process is directed by the strong instability, that is expressed by the positive Lyapounov exponent. It induces a relation between the rate of increase of the entropy functionals and Lyapounov exponents of the Lorentz gas . Our computation shows that this relation is expressed by an inequality

L Ai

max(9(n + I) - 9 (n)) = ~9 <

(7.5)

Ai:;'O

where the "max" is taken over n, which means that the K-S entropy is an upper bound of the rate of increase of this functional. Next, we shall consider the entropy increase of a system of N hard disks . Here the space in which moves a particle is a large torus divided into rectangular cells. Denoting the total number of cells by n and the number of particles initially distributed in only one region, by N, and following them until each executes t collisions with obstacles, we compute the probability that a particle is located in the ith cell as given by: Number of particles in cell i having made t collisions Pi () t = -----"------------===------N

The equi-distribution of the cells leads to take, as equilibrium measure, u, = ~ , so that this "space entropy" is defined by: 11

~,p (t)

= - LPi(t) In(Pi(t)n ).

(7.6)

i=l

The maximum of absolute value of this entropy is equal to -Inn. So we normalize as follows :

283

7 Entropy and Transport in Billiards

ssp(t) = SCI'P(t ) , Inn

(7.7)

We shall also do some comparisons of the H-theorem with the sum of normalized positive Lyapounov exponents.

7.2.1 Entropy in the Lorentz gas 7.2.1.1 Collision Map We shall give the formula of the collision map. We consider a particle which undergoes the first collision with the disk of center 0, with velocity V, (P,) and the second collision with the disk of center 02 with velocity V, (P2) . Two cases are possible . First, we consider non-crossing of the centers line as in the Fig. 7.3. In this figure the angle P(P;M is (X2 - f32 = -((Xl - f3I), where M is such that MP2 is parallel to 0,02 . We can write (7 .8)

and

PzM = P,P2sin(f3, - (X,)

= asin(X, -asin(X2 ,

(7 .9)

if we eliminate (xz between these equations we arrive at

f32

= arcsin [~ sin(f3, - (X,) + sinf3d · a

In crossing case which we present in Fig . 7.4 we see that the angle to (Xz - f32 = (X, - f3" and the length of PzM is changed to:

(7.10)

Ji;PJJ is equal (7 .11)

then, we have

f32

= arcsin [~ sin(f3, - (X,) -

(7.12) sinf3d · a To obtain f32 in the first collision between particle and obstacle Fig. 7.5, we take d = OP" f3, = 0 and (x, = ~ in the collision map.

7.2.1.2 Entropy for collision map The entropy for the collision map is computed for a beam of N particles on a toric checkerboard with n cells. We start to calculate the entropy, just after all particles have executed the first collision. In this computation, all particles have the same initial velocity and are distributed in a small part of one cell. For each particle we

284

M. Courbage and S.M. Saberi Fathi

Fig.7.3 A non-crossing collision.

n

Fig.7.4 A crossing collision.

Fig.7.5 Particle obstacle collision.

7 Entropy and Transport in Billiards

285

determine the first obstacle and the angles (131 , IfIl) of the velocity incoming vector V I (PI) ( see the figures given in the last Section). For a uniform partition fYJ of the space of the variables (13 , 1fI), the entropy Y'(t) is computed iteratively just after all particles have executed the tth collision. We use the formula (7.4) where (7.13) is the invariant measure (Sinai", 1970) of the cell fYJi = [f3i, f3i+ I [ X [lfIi, lfIi+ I [ and is the probability that a particle is located after t collisions in fYJi computed

VI (fYJi )

as

Number of particles in fYJj having made t collisions N

The velocity after the collision is computed from the following equation: V(P2)

= V(PI) -

2(V(PI ).0)0

(7.14)

where 0 is the normal vector at the collision point. We explained in the last section the main geometric formula used for this computation. This entropy increase is shown in the Fig . 7.6 for various partitions and various initial distributions. The absolute value of the entropy of a distribution of particles, that we call its amount of entropy, represents in fact its distance to equilibrium. This is illustrated in the examples of randomly distributed initial velocity of particles having small amount of entropy (see Fig. 7.7) comparatively with beams of particles . It is to be noted that the amount of entropy increase under one collision is remarkably greater for the few first ones (more or less 2-4 collisions) which corresponds to an exponential type increase (Fig . 7.8). In order to calculate Lyapounov exponents by using the method of Benettin et al (Bennetin et al., 1980), first we calculate the Jacobian matrix in the tangent space of the collision map:

df32 df32) df31 dal da2 da2 ( --df31 dal Now, comparing !iY' = max(Y'(t + I) - Y' (t )) (where the "max" is taken over t) with the positive Lyapounov exponent, I., of the collision map we verify the inequality : (7.15) as shown in Fig. 7.9, where this exponent is rv 3.2. The maximal entropy increase by collision for the distribution computed in this figure is not far from this value. So it could be conjectured that in some suitable refinement limit, the entropy increase of a beam tends to the positive Lyapounov exponent. The rate of the approach to equilibrium is thus related to the positive Lyapounov exponent.

286

M . Co urbage and S.M. Saberi Fathi

0

(~

-I c, oj

E <::

---.-.-.---.---.-.---.

-2

0

]

'0u Q)

•

-3

-5 0 """e-, c, g

-4

<::

~

-5

• o

4

2

6

8

10

12

14

16

Number of collisions (a)

0.0 c,

~ _. -. -. -. -. -. -. -. -. -. -.

-0.5

I I

oj

E

§ -1.0

]

"8

- 1.5

Q)

-5

'0 »

ce

-2.0 -2.5

~

-3 .0

•

-3.5

o

2

4

6

8

10

12

14

16

Number of collisions (b) Fig. 7.6 Entropy of the colli sio n map versus number of co llisions for (a) a beam of 640 particl es for a radi us a = 0 .2, neighbori ng disks ce nte rs distance I and a parti tion of (f3 , lJf) space into 25 x 25 cell s, (b) a beam of 5 12 particles for the ob stacles of rad ius 0.2, neigh boring disks cent ers dis tance I and a partition of (f3 , lJf) space into 9 x 9 cells .

287

7 En tropy and Transport in Billia rds

- 0.4

- 0.5

a~

.______-

/"

- 0.6

4-<

o

» Q.

g

---

~W"/'

------....

--------

- 0.7

&i - 0.8

- 0.9

o

2

4 6 Number of collisions (a)

8

10

2

4 6 Number of collisions (b)

8

10

0.0

- 0.1

.,0.. a

- 0.2

4-<

0

» - 0.3

0.. 0

~

~

- 0.4

- 0.5

-

- 0.6 0

Fig. 7.7 (a) and (b) are the entropy of the collision map with random initial conditions versu s numb er of colli sion s for the system of particl es of the Fig. 7.6, respecti vely.

288

M. Courbage and S.M. Saberi Fathi

2.0 » c..

:=

c "c ,

1.5

E

1.0

• Y =A+B*X

oj

c 0

]

"0 o

A

2.08466

B

- 0.60266

0.5

" -5

"0

E

~

0.0

•

oj

OIl 0

...J

•

- 0.5

•

- 1.0 0

2

4

10 6 8 Numbers of collisions (a)

12

14

16

0.5 e-,

c,

g

1.0

" §'

0.5

•

c

E c 0

]

"0 u

" -5

"0

E

Y =A +B*X

0.0 - 0.5

A

1.79045

B

-0.78351

- 1.0

-5 'C

oj

OIl 0

- 1.5

•

...J

- 2.0

•

- 2.5 0

2

4

6

8

10

12

14

16

Numbers of collisions (b) Fig. 7.8 Logarithm of the colIision map entropy versus number of colIisions for the system of particles of the Fig. 7.6.

289

7 Entropy and Transport in Billiards

6.0

•

\

5.5

E ., Vl

5.0

0

~

">0

•

4.5

\

<=I

;:l

0

§' »

-l

4.0

'-",

. ------.--.. .

3.5

----.--.

3.0 2

0

4

6

8

---------. 10

12

14

16

Number of collisions (a)

0.0 -0.5

§'

a<=I

.9

~~.-.-.-.-.-.-.-.-.-.-. -.

-1.0 -1.5

~

-2.0 '0 u

"

-;S

......

-2.5

0

e-, 0..

~

~

-3.0 -3.5 -4.0

•

-4.5

o

2

4

6 8 10 Number of collisions (b)

12

14

16

Fig. 7.9 (a) Lyapo unov exponent and (b) entropy of the colli sion map, versus of number of collisions for each part icle . We see that the maximum of the entropy increase between two collis ion s is less than of the value of the Lyapounov exponent.

290

M. Courbage and S.M . Saberi Fathi

7.2.1.3 Spatially extended Lorentz gas entropy The computation of the normalized space entropy equation by using (7.7) versus the number of collisions shows a remarkable exponential increase both for beams and for random initial distributions (Fig. 7.10). The computation of sum of the two positive Lyapounov exponents of the flow of one particle is equal to 1.046. Thus, we observe that the inequality between the normalized increase of the density of the space entropy and this sum is verified.

7.2.2 Some dynamical properties ofthe barrier billiard model A particle moves on an infinite plane among periodically distributed plate obstacles oflength f3 < I separated by slits, the spatial period in both (x,y) directions is I (see Fig . 7.11 , in the case f3 = 1/2 ). The particle reflects elastically at each collision with the obstacle. Let be the absolute value of the angle between the ingoing velocity vector and the obstacle. The particle will move either always to the right or always to the left undergoing reflections, according to the value of the angle e. Let us suppose that 0 < < ~. At time t = 0, the coordinates of the particle are given by its position XQ and the direction of its ingoing velocity. The position of the particle will increase between time t = 0 and time t = I by a along x-direction and by ± I along y-direction where a = (1 / tan e). The time unit is the time of the free motion of the particle in one cell of the lattice. The trajectory of the particle will be specified at times t = n, by its x-coordinate X/1 and the projection of the ingoing velocity vector along the y-direction, e/1, which will change under a collision. Thus, e; = ± I according as the velocity is upward or downward. We call e; the " velocity direction of the particle". From now on we suppose f3 = 1/2 . Let us introduce the function : x E 'f = [0, 1[--+ G = {± I} as follows (Courbage, 2005):

e

e

X

(x) =

{-I+ x E

if [0 , 1/2[, I if x E [1/2, 1[.

The transformation T maps the coordinates of the particle (x,e) at time t the coordinates of the particle at time t = I according to the formula:

T(x, e) = (x+a(modl),x(x)e)

(7 .16)

=0

into

(7.17)

So the dynamics for the x-coordinate is a translation on 'f and T maps an ingoing arrow to the next ingoing arrow. The map T has an invariant measure d,u (x,e) which is the product of the Lebesgue measure me on the x-coordinates and the (~, ~) measure on the e coordinate. The study of relaxation to equilibrium can be done using the spectral theory of the the Koopman unitary operator associated to T defined on the Hilbert space S) =L2(X ,,u) by : (7 .18) Uj(x) =j(Tx) .

7 Entropy and Transport in Billiards

29 1

~ ~.-.-.-.-.-.-.-.-.-.-.-.

-0.2 ssp -0.4

- 0.6

- 0.8

•. . .-.

- 1.0

2

0

In (ssp)

0.2

\.,

0.0

\

4

10 12 6 8 Number of collisions (a)

- 0.4 - 0,6

16

18

•

\ \ \

- 0.2

14

Y =A+B*X

\

\ \ \\

- 0,8

.

- 1.0 - 1.2

A

0.55988

B

- 0.48447

\

\

-1 .4

.\

- 1.6

\-

- 1.8 0

2

4

6 8 10 12 Number of collisions (b)

14

16

18

Fig. 7.10 (a) Normalize d space entropy of the Lorentz gas versus number of collisions for a beam of 640 particles for obstacles of radi us a=0.2, neighboring disks centers distance I and a partition of (x,y) space into 25 x 25 cells , (b) Loga rithm of the space entropy versus number of collisions for this system.

292

M. Courbage and S.M. Saberi Fathi

--'x

-110-_ _

Fig.7.11 Trajectory of a particle in the barrier billiard.

For a review see (Cornfeld et As well known, the space .I) = basis {I , e} of L2(G,mL), into

al., 1981; Courbage, 2005) . Here X denotes l' x G. L2(1' x G,)1) decomposes, by using the orthonormal a direct sum.l) =.1) 1 EEl .l)2 where :

.I)]

= {g(x ) E L2(1',mL)}, 2(

.1)2 = {eh(y),h E L 1',mL)}.

(7.19) (7.20)

In other words , any j(x,e) decomposes uniquely into j(x,e) = g(x) + eh(x), g,h E L2(1',)1) . The action of U restricted to .I) 1 (resp . .1)2) is given by: Ug(x) = g(x+ a) (resp. U(eh(x)) = ex(x)h(x+ a)) . It shows that n 1 and.l)2 are invariant. Therefore, the spectrum of U restricted to .I) 1 is reduced to the spectrum of the the Koopman operator of the rotation on .I) I . The spectral measure associated to any function from .1)2 is given through the Fourier coefficients: < U" (eh),eh) >. It is convenient to introduce the unitary operator Vx on L2( 1',mL) given by:

VXh(x)

= X(x)h(x+ a)

Now, using the relation U"(eh) = eV;((h) we obtain :

(U" (eh,eh)) = (eV;((h,eh)) = (V;((h,h)),

(7.21)

293

7 Entropy and Transport in Billiards

which means that the spectrum of U restricted to S)2 is reduced to the spectrum of Vx on L 2('f ,mL). The spectrum of this operator has been studied (Oseledets, 1966) and it is shown that for any irrational a ,Vx has no eigenfunction in L2 ('f, J1), which means that U restricted to S)2 has only continuous spectrum. This implies that the e;(x, e)(= U" e) has continuous spectrum . This also implies that T is ergodic for irrational a . The Wiener lemma implies therefore a weak decay of correlations of the velocity direction e: l"-I (7.22) lim l(ell , £o )12 = 0, Il ~ OO n k=O

L

where

(Xl ,el) = (x+ a,x(x)e) (x2 ,e2) = (x+ 2a,X(x+ a)x(x)e) (xll ,ell) = (x+na ,x(x+(n-I)a) · · ·x(x)c) .

(7.23)

As well known, the above decay of correlations is equivalently true for the average of the absolute value. In our simulation we study the relaxation of e; using the entropy functional. It appears that a relaxation of this variable is true not only in time average , but also for the probability distribution of ell = ± I. In the first part of this section, we shall present some results on the entropy increase problem, more exhaustively studied in (Courbage and Saberi Fathi, to appear) . In this system there is no relaxation to equilibrium even in the coarse-graining sense. However, inspired by the continuous spectrum associated to the relaxation of ell , we consider the relaxation in the time average :

7.2.2.1 Time averaged coarse-grained entropy for the barrier billiard We define a partition fYJ of the phase space as the product of e = ± 1 and a subdivision of the interval of X E [0, 1] in k-parts of equal size. Probability of a cell is defined as

fYJ+ (n) = The number of particles with velocity positive in cell i after nth second I N (Number of Particles) (7.24)

and

fYJ-:- (n) = The number of particles with velocity negative in cell i after nth second I N (Number of Particles) (7.25)

Then, the coarse- grained entropy is given by

294

M. Courbage and S.M. Saberi Fathi Sk(n)

0.0001

e

0.0000 -0.0001 -0.0002 -0.0003 -0.0004 -0.0005 -0.0006 -0.0007 -0.0008

o

200000

400000

600000

800000

1000000

Time

Fig. 7.12 Time averaged coarse- grained entropy for barrier billiard with a = tan number.

e=

e irrational

0.0000

-0.0001

-0.0002

-0.0003

-0.0004

-0.0005

o

200000

400000

600000

800000

1000000

Time

Fig.7.13 Time averaged coarse- grained entropy for barrier billiard with

a

= tan

e=

1+2V5.

295

7 Entropy and Tran sport in Billi ards e

0.0000 -0.0001 -0.0002 -0.0003

•

-0.0004 -0.0005 -0.0006 -0.0007 -0.0008

• o

10000 20000 30000

40000

50000 60000

70000

Time

Fig.7.14 Zoom for the time averaged coarse- grained entropy for barrier billiard with ex = tan e = e irrational numb er.

I-_-B I

-

0.00000

i · I,F1r'tr ~ I \

•

-0.00005

\I

• -0.00010

•

•

....I

~

-0.00015

-0.00020

• -0.00025 +--.-----,---.------,.-------,---r-----,---.--.-------.---.-----. o 20000 40000 60000 80000 100000 Time

Fig.7.15 Zoom for the time averaged coarse- grained entropy for barrier billiard with ex = tan e = I+ VS

- 2-'

M. Courb age and S.M. Saberi Fathi

296 - 3.0

"--!.. .,"---------

e ,----------

.

-3.5

•

"-----...

- 4.0

,--." ~" .,--,~

.

.........

'

- 4.5

•

. . . . .. . . . . ... . .

•

• -,.111..., • •

;" " ""~

•

- 5.0

"'--"',,-

= -0.43 B = -0.84

A

- 5.5 -6.0 - 6.5 -7.0 3.2

3.4

3.6

3.8

4.0

4.2

4.4

4.6

4.8

5.0

Fig. 7.16 Log-log fit for the time averaged coarse- grained entropy for barrier billiard with ex tan e = e irrational number, logSk(n ) = A + B logn.

=

-3 .2 - 3.4 - 3.6

A = - 1.02

- 3.8

B = - 0.71

- 4.0 - 4.2 - 4.4 -4.6 - 4.8 -5 .0

•

-5 .2 3.0

3.5

4.0

4.5

5.0

5.5

Fig . 7.17 Log-log fit for the time averaged coarse- grained entropy for barrier billiard with ex = tan B = 1+/'5 , log Sk(n ) = A+ B log n.

(fJ+ = )1- = 1/ 2)

(7.26)

7 Entropy and Transport in Billiards

297

where k is the number of elements of the subdivisions of the x-axis . For our experiment we choose k = 5. It is inspired by the weak mixing property of E (7.22) we define a time averaged probability as: (7.27) and then the corresponding entropy is given by

-

Sk(n)

k

=~

(

-Pi+ (n) - -Pi- (n) ) - + -Pi (n)log~ -Pi (n)log~ ,

(7.28) the results are shown in Figs. 7.12 and 7.13. In Figs. 7.14 and 7.15 we show a zoom of time averaged coarse- grained entropy : it decreases to zero in an oscillating way, therefore looking to the better fit for the local minima points , we found in Figs . 7.16 and 7.17 that the fits display a dependence on the irrational nature of a.

7.3 Transport In this part, we first review a joint paper with G. Zaslavsky and M. Edelman (Courbage et al., 2008) on transport in the Sinai' billiard with infinite horizon . The considered model of scattering billiards was initiated in (Krylov, 1950) to demonstrate an importance of mixing and dispersion of trajectories for foundat ion of statistical physics. Rigorous consideration of the problem , based on the dynamical chaos theory, started from the paper (Sinai', 1963; Sinai', 1970). The problem considered in (Sinai, 1963) is known now as Sinai billiard with infinite horizon (SBIH). After about 40 years statistical properties of this model and, particularly, the problem of particles transport, are still unclear being the subject of numerous publications. Sinai billiard is a square billiard table with a circular scatterer in the center and absolutely elastic collisions of a point particle (ball) with walls of the table and the scatterer. The adjoint model, known as the Lorentz gas, is the periodic continuation of scatterers in both directions, that forms a square lattice (Fig. 7.18). There are two different approaches to the problem with well distinguished results : (a) the coordinates of the ball can be considered as functions of time r(t) = (x(t),y(t)) ; (b) the same coordinates r(n) as a function of the number n of collisions with scatterers; (c) angle 1JI" of the velocity of a ball v" ' (Ivl = I) and the angle coordinate ell on the circle of the point of collision as function of n. The main difficulty in studying the SBIH is the existence of the corridors within which a ball can propagate infinite time without scattering . Such corridors make scars of zero measure on the phase plane (see for example in (Zaslavsky and Edelman, 200 I» raising a question how these scars influence the long term characteristics of dynamical and statistical processes of trajectories that do exhibit scattering . Some important results on the studying of the problem are as follows . It was mentioned in (Friedman and Martin,

298

M. Courbage and S.M. Saberi Fathi 9

8

7

6

5

4

3

2

I -5

-4

-3

-2

-I

0

Fig. 7.18 A sample of a trajectory in the periodic Lorentz gas.

1984; Bunimovic h, 1985) that the presence of corr idors leads to an algebraic decay of the velocity correlation (v(O), v(t )) rv

co~st.

(7.29)

that gives for the second moment of the displacement (Ir (t) - r (0)1 2) rv const. x (tint) .

(7 .30)

Similar estimates were confirmed in (Zacheri et al., 1986) although the simulation performed in (Zacherl et al., 1986) was not able to estab lish the presence of Int in 7.30. It was claimed in (Bleher, 1992) that the limit distribution of the particles displacement in the plane is Gaussian , but the normalization factor is (t Int)I /2 and not t l / 2 as in classical case . Simulations performed in (Artuso et al., 1996) confirmed 7.29 for the SB IH and the exponential decay of the velocity correlation for the finite horizo n when the corridors are absent. It is worthwhile to mention that while the map T :

(If/,,+I , 8"+1)

= T" (If/,,8,,)

(7 .3 1)

possesses expo nential decay of corre lations for finite horizon when n --+ 00 (Bunimovic h and Sinai, 1981; Chernov and Young, 2000a) , the case of the infinite hori-

299

7 Entropy and Transport in Billiards

zan creates severe difficulties for theoretical analysis , probably because of nonexponential decay of distribution of Poincare recurrences (Artuso et al., 1996; Zaslavsky and Edelman, 1997). Numeri cal simulations in (Zaslavsky and Edelman , 1997) show that

(IR(t ) 12m)

rv

const.

X

(t Int )J.l(m) ,

m EN

(7.32)

.u

and that ( I) = 1, while for m > I it doesn 't follow the Gaussian law . Some rigorous results of (Szasz and Varju , 2004 ; Szasz and Varju, 2006) don 't shed a light on the type of diffusion for the SBIH in the most important cases for the applic ations that are related to the mentioned above approaches (a) and (b).

7.3.1 Transport in Lorentz gas Here n is the number of collisions The last comment was studied separately in order to understand it does or does not exist a convergence to the Gaussian distribution of the particle displacements. For this goal the centr al moments were studied in an independent calculations and different code with averaging over N = 500 trajectories indexed by k below. Let us define the central moments as (7 .33)

where xk(n) is displacement along x of the kth trajectory after n collisions, with n ::; 106 . For n max = 106 , define

.u - M, -

_

_

I

_

1 ~ x(nm ax ) - x(O) - I-. N k=' J n max In n max

(7 .34)

and (7.35) where M~ (n) and (M; (n)) are the raw moment, the mth raw moment is defined as (7 .36)

Similar expressions can be introduced for y-coordinate. The results of the calculations are presented in the Table 7.1 and Table 7.2 for three different cases of the period D and radius of the scatterer a. While the values of M" are close to zero as it should be the central Gaussian moment, the value s for M3 are too far from zero. Moreover the fourth centr al Gauss ian moments should take the values M4 = 3 while the values in the tables deviate, sometimes significantly,

300

M . Co urbage and S.M . Sab eri Fathi

from that. More comments could be do from e plots in Fig. 7. 19 and Fig. 7.20 They display jumps and they don't display a good Convergence as it is seen especia lly from the Fig. 7.20. Table 7.1 X (n) mom ent s values at n = 1, 000, 000

D/ a 2./0.405 1./ 0.405 1./ 0.473

3.9x 10 6 2.5 X 10- 5 5.1 X 10- 5

1.7 - 0. 15 0.0013

1.00 1.00 1.00

4.17 3.67 3.37

Table 7.2 Y(n) moments values at n = 1,000,000

2./0.405 1./0.405 1./0.473

0.1

~,

u,

- 1.03 x 10 6 2.4 x 10- 5 7.0 1 x 10- 5

5.33 2.86 3.73

100

.

\,

M3

~\

0.0

0. 122 0.006 - 0. 13

1.00 0.9999 1.00 1

50

~

!

:h !

0

- 0.1 ;. i

\~~

. 1;' t-

/ - 0.2

0

- 50 500000

1 000 000

0

X-coordinates

500000

1 000 000

X-coordinates

M2

M4

8000

8

4000

4

"i

.~~ 0

0 0

500000 X-coordinates

1 000 000

0

500000

1 000 000

X-coordinates

Fig. 7.19 Dependenc e of the cent ral mom en ts on (n Inn ) for D = 2, a = 0.405 after 106 coll ision s.

7 Entropy and Transport in Billiards

301

-0.08 +-~~~~--,-~-~----.--~ o 500000 1000000 X-coordinates

o

500000 X-coordinates

1000000

500000 X-coordinates

1000000

Mz 1.6

20

1.4 ~

1.2

v>: ~J

.~

10

~

\-....

1.0 t-~-~~-,.--~-~--.-~

o

500 000 X-coordinates

I 000 000

Fig.7.20 Dependence of the central moments on (nlnn) for D

=

I, a = 0.473 after 106 collisions.

7.3.2 Transport in the barrier billiard The goal of this section is to provide renewal results of massive computations and to show that some well defined transport properties of particles do not follow the Gaussian law, exhibit superdiffusion, and, for the time of observation, do not displaya "normal" approach to the limit distribution. We consider two models: SBIH (periodic Lorentz gas) and stadium (Bunimovich) billiard (Bunimovich, 1979). We interpret these observations as a result of particles long "flights" in the corridors and, as a result of the flights, persistent fluctuations (Zaslavsky and Edelman, 2004) that do not have a finite time of relaxation as it exists for the Gaussian fluctuations. It is now established that the exponential instability of trajectories leads to strong ergodic and stochastic properties, diffusion and H-theorem . Recently, numerical computations have confirmed a relation between the rate of entropy increase in the H-theorem and the positive Lyapunov exponents (Courbage and Saberi Fathi, 2008) . Yet the transport properties of these systems are still unclear (see references in (Courbage et al., 2008». The situation is worst as to such properties in dynamical systems which does not exhibit exponential instability . The problem of a kind of mixing and stochastic properties of systems with zero Lyapounov exponents is a challenging problem that has been the object of many publications (see (Gutkin and

302

M. Courbage and S.M. Saberi Fathi

Katok, 1989; Horvat et al., 2009) and references therein) . Here we consider a simple model of a particle colliding elastically with horizontal bars periodically distributed on the plane . This model, some times called a barrier billiard , was studied by several authors. It has been used as a model of a pseudo-integrable system (Richens and Berry, 1981; Hannay and McCraw, 1990; Wiersig, 2000) and a model of free particle dynamics along field surfaces in plasma physics (Zaslavsky and Edelman, 2004). It is also a model of a gas of non-interacting particles moving with constant velocity on the plane and undergoing collisions only with those scatterers. The nonequilibrium statistical description is thus reduced to the dynamics of the one particle probability distribution. The motion of the particle along x-coordinate is simply a uniform translation, while the motion along y-coordinate is apparently a random walk as seen in the Fig. 7.11. The transport properties have been studied (Zaslavsky and Edelman, 2004) in considering the motion of bunch of trajectories with closed initial conditions. It was shown that the transport is anomalous and seems of the following form : (7 .37)

with ,11(1) > 1. The superdiffusion comes , as in the case the Sinai Billiard with infinite horizon , from the existence of corridors between the arrays of scatterers allowing arbitrarily long free trajectories (Courbage et al., 2008) . In this section we provide new computations in order to test the convergence of the displacement along y-coordinate to a gaussian distribution. We will study the reduced central moments of order I, 2, 3 and 4. A possible divergence with respect to the gaussian moment is then clearly displayed. The displacement along y-direction is given by n

y(n,x,c) = LCi(X,C).

(7 .38)

i= l

The mean value of y(n) with respect to the invariant measure is zero and the second moment for a rational a has been studied in (Courbage, 2005) where it is shown that

f (~~(x,e))

2 dp

(x,e) '" en'.

(7 .39)

So the motion is ballistic . In the next section we will study the case of irrational Note that, on account of the symmetry, all odd moments are zeros.

a.

Here n is the time scale We first simulate the convergence of the moments of the y-displacement. We shall take the first moment equal to O. The m th raw moments for an arbitrary variable y at the time n is defined by: (7.40)

7 Entropy and Transport in Billiards

303

a=:;r

3.0

2.5

2.0

1.5

1.0

o

500000

1000000 Time

1500000

2000000

Fig.7.21 The ratio of second and fourth moment s, M,~(II) , versus time for a = M2 (II)

7L

where Yk(n ) is the kth-experiment at time n, N is the total number of exper iments (trajectories). Let us now test the convergence of the norma lized moments to Gaussian moments . This can be done throug h the fourt h moment of the normalized displacement, MY}'(' l ) which should converge to 3. This should be the asymptotic of 2

Il

Fig. 7.21 shows

M,~

M2

M~ .

M,-

versus time . Table 7.3. shows that the limit is not Gaussian . -

Looking now for the second moment, we will compute o and B such that (7.41)

By drawing Log-Log diagram of M~ (n), we obtain B by linear fitting of the following relation (7.42) logM~(n) = Blogn +A, (A = log o ").

Table 7.3 The values of B, A , second and fourth moments and their ratio for different values of at n max = 106. The value of (j2 is equal to I(YI

a

B

A

M2(n max)

M4(n max)

V2

1.51 1.69 1.13 2.00

- 3.3 1 - 1.49

1.07 1.00 0.97 1.00

1.68 1.90 2.28 1.18

e 7r

7/ 3

1.69 - 1.03

M~ (nmax )

M;2(n max) 1.45 1.89 2.43 1.18

a

304

M. Courbage and S.M. Saberi Fathi

LogM;(n) IE11

-:

9E10 8E10 7E10

a

=

7/3

6E10 5EIO 4E10

M; =A +B Log n B =2.00 A = -1.026

3E10

2E10

500000

600000

700000

800000 900000 1000000

Logn LogM;(n)

600000 500000

400000

M;=A +B Log n B = 1.51 A = -3 .31

300000

200000 +--------.-------r---....----....-------r700000 800000 900000 1000000 500000 600000 Logn Fig. 7.22 Log-Log diagra m of MH n ) versus time, by linear fitting for y-variable, a = 7/3 and a = V2 and number of trajectorie s is N = 1000. The fit was done for time between 5.3 x 105 and 106 . The results for other values of a are given in Table 7.1.

7 Entro py and Transport in Billiards

305

M2 (n) 3.0 a =e

2.5 &

2.0

\

•

1.5

1.0 0

200000

400000

600000

800000 1000000

Time M4(n) 12 a =e

10 8 6 4 2 0 0

200000

400000

600000

Time Fig. 7.23 The moment M2(n ) and M4(n ) versus time for a

=

e.

800000 1000000

306

M. Courbage and S.M. Saberi Fathi

Fig. 7.22 shows a fitting for the case ex = v2. In the Table 7.3. we see some values of A and B. For ex = 7/3 we check that our result for B = 2.00 is adequate with the theoretical theorem stating that limll-.oo B = 2 (Courbage, 2005). Now, the m th reduced central moments is defined by (7.43) where N is the number of trajectory. For second central moments we have obviously limll-. ooM2(n) = I. We test again if the transport is" Gaussian" , then M4 should be equal to "3" . Table 7.3 shows the values of central moments for M2(n max ) and M4(n max ) . These values show that the transport is not Gaussian . This table shows that the B-value is I < B :::; 2. So we have a super diffusion . Fig. 7.23 represents the second and fourth normalized moments for ex = e and their zooms which show that they do not converge.

7.4 Concluding remarks The computations of the evolution of the entropy amount of some given nonequilibrium initial distributions in the Lorentz gas show an exponential type increase during initial stage after which the entropy increases slowly and fluctuates near its maximal value. These computations confirm the existence of a relaxation time generally assumed in the derivation of kinetic equations (Balescu, 1975) which is at the origin of the rapid increase of the entropy as due to the number of collisions. The dispersive nature of the obstacles is responsible of the exponential type increase. This exponential type increase has been demonstrated for the Sinai" entropy functional (Sinai", 1994) in hyperbolic automorphisms of the torus. On the other hand, the relation of the entropy increase to Lyapounov exponents can be understood through Pesin relation and Ruelle inequality. In fact, the rate of entropy increase should be bounded by the Kolmogorov-Sinai entropy and such bound have been found by Goldstein and Penrose for measure-theoretical dynamical systems under some assumptions (Goldstein and Penrose, 1981). An open question is to characterize the initial invariant probability measures reaching the upper bound. New simulations of transport of the barrier billiard confirm previous Zaslavsky and Edelman simulations showing a superdiffusion (Zaslavsky and Edelman, 2001). Our results show moreover the non gaussian nature of the transport. The strong dependence on the irrational velocity angle ex show however that the transport in this billiard is still unclear. The entropy results for irrational ex show mixing for the relaxation of the velocity direction. But, the coarse- graining entropy has relaxation only in time average reflecting some weak mixing for some distinguished observabies . The Gibbs coarse-graining entropy is not a completely monotonic function of time . A completely monotonic entropy functional has been obtained when the map

7 Entropy and Transport in Billiards

307

T on the space X is a Bernoulli system or, slightly more generally, a K-system (Courbage, 1983; Misra et al., 1979; Courbage and Misra, 1980; Goldstein et aI., 1981). That is to say : there is an invariant measure /l and some partition ~o of X such that T ~o becomes finer than ~o ( we denote it: T ~o ?: ~o). Using the notation : T"~o = ~Il' we obtain a family of increasingly refined partitions, in the sense of the above order of the partitions. Moreover, ~11 tends, as n --+ 00, to the finest partition of X into points, and ~Il tends , as n --+ - 0 0 , to the most coarse partition, into one set of measure I and another set of measure zero . A physical prototype of a Bernoulli and a K-system is the above Sinai' billiard (Sinai', 1970; Gallavotti et aI., 1974) . A geometric prototype of a Bernoulli and a K-system is uniformly hyperbolic system with Sinai' invariant measure (Sinai', 1972). Heuristically, the monotonic entropy increase corresponds to the process of dilation of expanding fibers.

References Artuso R., Casati G. and Guarneri 1., 1996, Numerical experiments on billiards, J. Stat. Phys., 8, 145-166. Balescu R., 1975, Equilibrium and Nonequilibrium Statistical Mechanics, John Wiley, New York. Bennetin G., Galgani L., Giorogilli A. and Strelcyn J.M., 1980, Lyapounov characteristic Exponents for smooth dynamical systems and for Hamiltonian systems; a method for all off them, Part I and 2, Meccanica, 15,9-30. Bleher P.M., 1992, Statistical properties of two-dimensional periodic Lorentz gas with infinite horizon, J.Stat. Phys. , 66, 315-373. Bunimovich L.A., 1979, On the ergodic properties of nowhere dispercing billiards, Commun. Math. Phys., 65, 295-312. Bunimovich L.A., 1985, Decay of correlations in dynamical systems with chaotic behavior, Sov. Phys. JETP, 62, 842-852. Bunimovich L.A ., Sinai' Ya.G. and Chernov N.1., 1990, Markov partitions for twodimensional hyperbolic billiards, Russian Math. Surveys, 45, 105-152. Bunimovich L. and Sinai' Ya.G., 1981, Statistical properties of Lorentz gas with periodic configuration of scatterers, Commun. Math. Phys., 78, 479-497. Chernov N. and Young L.S ., 2000a, Decay of correlations for Lorentz gases and hard balls.In "Encyclopedia of Mathematical Sciences" Ed. D. Szasz, 101,5188. Chernov N.!. and Young L.S ., 2000b, Decay of correlations for Lorentz gases and hard balls . Hard ball systems and the Lorentz gas , Encyclopaedia Math. Sci. 101, 89-120. Cornfeld 1.P., Fomin S.V. and Sinai' Ya.G., 1981, Ergodic theory, Springer, New York. Courbage M., 1983, Intrinsic Irreversibility in Kolmogorov Dynamical Systems, Physica A, 122 ,459.

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M. Courbage and S.M. Saberi Fathi

Courbage M., 2005, Notes on the spectral theory, mixing and transport, in Proceeding of Chaotic Dyanmi cs and Transport in Classical and Quantum Systems, P Collet et al (eds.), Kluwer Academi c Publishing, Netherlands. Courb age M. and Hamdan D., 1994, Chapman-Kolmogorov equation for nonMarkovian shift invariant measure, Ann. Prob. 22, 1662-1677. Courbage M. and Misra B.,1980, On the equivalence between Bernoulli systems and stochastic Markov proce sses, Physica A , 104, 359-377. Courbage M. and Nicolis G., 1990, Markov evolution and H-theorem under finite coarse-graining in con servative dynamical systems, Europhysics Letters, 11, 1-6. Courbage M. and Prigogine L, 1983, Intrinsic randomness and intrinsic irreversibility in classical dynamical systems, Proc. Natl. Acad. Sci. USA , 80, 2412-2416. Courbage M. and Saber i Fathi S.M., 2008, Computation of Entrop y increase in Lorent z gas and hard disks, Comm unications in Nonlinear Science and Numerical Simulations , 13, 100-104. Courbage M. and Saber i Fathi S.M., to appear, Transport and Ent ropy in the barrier billiard. Courb age M., Edelm an M., Saberi Fathi S.M. and Zaslavsky G.M., 2008, Problem of transport in billiards with infinite horizon, Phys. Rev. E, 77, I. Friedm an B. and Martin R.E Jr., 1984, Decay of the velocity autocorrelation function for the periodic Lorentz gas, Phys. Lett. A, 105,23-26. Gallavotti G. and Ornstein D.S., 1974, Billiards and Bernoulli schemes, Commun. Math .Phys., 38,83-101 . Garrido P.L., Goldstein S. and Lebowitz J.L., 2004, Boltzmann Entrop y for dense fluids not in local equil ibrium, Phys. Rev. Lett., 92,050602. Gaspard P. and Beijeren H., 2002 , When do tracer particles dominate the Lyapounov spectrum, J. Stat. Phys., 314, 671-704. Gold stein S. and Penro se 0 ., 1981, A nonequil ibrium entrop y for dynamical systems, J. Stat. Phys., 22 325-343. Gold stein S., Misra B. and Courb age M., 1981, On Intrinsic Randomness of Dynamical Systems, J. Stat. Phys., 25, 11-126. Gutkin E. and Katok A., 1989, Weakly mixing billiards, in "Holomorphic Dynamics", Lecture notes in Mathem atics 1345, Springer, Berlin. Hannay J.H. and McCraw R.J., 1990, Barrier Billiards - a simple pseudo integrable system, J. Phys A, 23, 887-889. Horvat M., Degli Esposti M., Isola S., Prosen T. and Bunimovich L., 2009, On ergodic and mixing properties of triangle map, Physica D, 238, 395--415. Kac M., 1959, Probability and Related Topics in Physical Sciences, Interscience Pub, New York. Krylov N.S., 1950, Works on the Foundation of Statistical Physics, in russian, English translation, Princeton Univ. Press , Princeton , NJ, 1979. Misra B., Prigog ine I. and Courb age M., 1979, From the Determin istic Dynamic s to Probabilistic Description s, Physica A, 98, 1-26. Oseledet s VI. , 1966, The spectrum of ergodic automorphisms, Dokl. Akad. Nauk USSR, 168, (Russian) - English Soviet Math . Dok!., 7, 776-779.

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309

Richens P.1. and Berry M. v., 1981, Pseudo integrable systems in in classical an quantum mechanics, Physica D, 2,495-512. Sinai" Y.G., 1963, Dokl. Akad. Nauk SSSR, 153, 1264. [SOy. Math . Dok!. 4, 1818 51963] Sinai" YAG., 1970, Dynamical systems with elastic reflections . Ergodic properties of dispersing billiards, Russ. Math. Survey , 25, 137-189. Sinai" YAG., Gibbs measures in ergodic theory, 1972, Russian Math. Surveys, 27, 21-69. Sinai" YAG., 1994, Topics in Ergodic Theory, Princeton University Press, Princeton. Szasz D. and Varju T., 2004 , Markov Towers and Stochast ic Propertiesoj Billiards, 433-445, Cambridge Univ. Press, Cambridge. Szasz D. and Varju T., 2006, Limit Laws and Recurrences for the Planar Lorentz Process with Infinite Horizon, Preprint. Wiersig J., 2000 , Singular continuous spectra in a pseudo integrable billiard, Phys. Rev E, 62, 21-24. Zacher! A, Geisel T., Nierwetberg J. and Radons G., 1986, Power spectra for anomalous diffusion in the extended billiard, Phys. Lett. A, 114, 317-321 . Zaslavsky G.M. and Edelman M., 1997, Maxwell 's Demon as a dynamical model , Phys. Rev. E, 56, 5310-5320. Zaslavsky G.M . and Edelman E.A, 200 I, Weak mixing and anomalous kinetics along filamented surfaces, Chaos, 11,295-305. Zaslavsky G.M. and Edelman M.A., 2004, Fractional kinetics : from pseudochaotic dyanamics to Maxwell's Demon, Physica D, 193,128-147.

Index

Barrier biliards Billi ards Brain dynamic s Brownian motion Chaotic motion s Coarse-graining Diffu sion Dilut ed networks Discrete map with memory Emotion -cogni tion interaction Entropy Ergodicit y Finite- size effects Fractional derivatives Fractional equations Fractional proces ses Fractionally stable distributions Henon map HMFmodel Inequivalence of ensembles Inhib ition Vlasov equation Kink K-S entropy Levy motion Long living eddies in crossed-field systems Long-range interactions Lorentz gas

Lump Markov processes Ment al disorder Mixing Non-equil ibrium entropy Non-equilibrium ensembles Normal hyperbo licit y Phase transitions Quasi-stationary states Random graphs Relaxation Repli ca method Sinai Billiard s Soliton Soliton chains Soliton interaction Stable heteroclinic channel Stochastic counterparts of nonlinear dynamics Stochasticity Structural stability Symmetry of the crossed-field system Transient behavior Transport Wave turbulen ce Weak mixing Winne rless competition Zasl avsky map

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