Springer Complexity Springer Complexity is a publication program, cutting across all traditional disciplines of sciences as well as engineering, economics, medicine, psychology and computer sciences, which is aimed at researchers, students and practitioners working in the field of complex systems Complex Systems are systems that comprise many interacting parts with the ability to generate a new quality of macroscopic collective behavior through self-organization, e.g., the spontaneous formation of temporal, spatial or functional structures This recognition, that the collective behavior of the whole system cannot be simply inferred from the understanding of the behavior of the individual components, has led to various new concepts and sophisticated tools of complexity The main concepts and tools – with sometimes overlapping contents and methodologies – are the theories of self-organization, complex systems, synergetics, dynamical systems, turbulence, catastrophes, instabilities, nonlinearity, stochastic processes, chaos, neural networks, cellular automata, adaptive systems, and genetic algorithms The topics treated within Springer Complexity are as diverse as lasers or fluids in physics, machine cutting phenomena of workpieces or electric circuits with feedback in engineering, growth of crystals or pattern formation in chemistry, morphogenesis in biology, brain function in neurology, behavior of stock exchange rates in economics, or the formation of public opinion in sociology All these seemingly quite different kinds of structure formation have a number of important features and underlying structures in common These deep structural similarities can be exploited to transfer analytical methods and understanding from one field to another The Springer Complexity program therefore seeks to foster crossfertilization between the disciplines and a dialogue between theoreticians and experimentalists for a deeper understanding of the general structure and behavior of complex systems The program consists of individual books, books series such as “Springer Series in Synergetics", “Institute of Nonlinear Science", “Physics of Neural Networks", and “Understanding Complex Systems", as well as various journals Springer Series in Synergetics Series Editor Hermann Haken Institut făur Theoretische Physik und Synergetik der Universităat Stuttgart 70550 Stuttgart, Germany and Center for Complex Systems Florida Atlantic University Boca Raton, FL 33431, USA Members of the Editorial Board Åke Andersson, Stockholm, Sweden Gerhard Ertl, Berlin, Germany Bernold Fiedler, Berlin, Germany Yoshiki Kuramoto, Sapporo, Japan Jăurgen Kurths, Potsdam, Germany Luigi Lugiato, Milan, Italy Jăurgen Parisi, Oldenburg, Germany Peter Schuster, Wien, Austria Frank Schweitzer, Zăurich, Switzerland Didier Sornette, Los Angeles, CA, USA, and Nice, France Manuel G Velarde, Madrid, Spain SSSyn – An Interdisciplinary Series on Complex Systems The success of the Springer Series in Synergetics has been made possible by the contributions of outstanding authors who presented their quite often pioneering results to the science community well beyond the borders of a special discipline Indeed, interdisciplinarity is one of the main features of this series But interdisciplinarity is not enough: The main goal is the search for common features of self-organizing systems in a great variety of seemingly quite different systems, or, still more precisely speaking, the search for general principles underlying the spontaneous formation of spatial, temporal or functional structures The topics treated may be as diverse as lasers and fluids in physics, pattern formation in chemistry, morphogenesis in biology, brain functions in neurology or self-organization in a city As is witnessed by several volumes, great attention is being paid to the pivotal interplay between deterministic and stochastic processes, as well as to the dialogue between theoreticians and experimentalists All this has contributed to a remarkable cross-fertilization between disciplines and to a deeper understanding of complex systems The timeliness and potential of such an approach are also mirrored – among other indicators – by numerous interdisciplinary workshops and conferences all over the world Till Daniel Frank Nonlinear Fokker–Planck Equations Fundamentals and Applications With 86 Figures and 18 Tables 123 Dr Till Daniel Frank Universităat Măunster Institut făur Theoretische Physik Wilhelm-Klemm-Strasse 48149 Măunster, Germany ISSN 0172-7389 ISBN 3-540-21264-7 Springer Berlin Heidelberg New York This 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, reproduction 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 permission for use must always be obtained from Springer Violations are liable to prosecution under the German Copyright Law Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2005 Printed in Germany 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 regulations and therefore free for general use Typesetting: by the author Cover design: design & production, Heidelberg Printed on acid-free paper 55/3141/ts 543210 This book is dedicated to my parents, wife and daughter Preface Nonlinear Fokker–Planck equations have found applications in various fields such as plasma physics, surface physics, astrophysics, the physics of polymer fluids and particle beams, nonlinear hydrodynamics, theory of electronic circuitry and laser arrays, engineering, biophysics, population dynamics, human movement sciences, neurophysics, psychology and marketing In spite of the diversity of these research fields, many phenomena addressed therein have a fundamental physical mechanism in common They arise due to cooperative interactions between the subsystems of many-body systems These cooperative interactions result in a reduction of the large number of degrees of freedom of many-body systems and, in doing so, bind the subunits of manybody systems by means of self-organization into synergetic entities These synergetic many-body systems admit low dimensional descriptions in terms of nonlinear Fokker–Planck equations that capture and uncover the essential dynamics underlying the observed phenomena The phenomena that will be addressed in this book range from equilibrium and nonequilibrium phase transitions and the multistability of systems to the emergence of power law and cut-off distributions and the distortion of Boltzmann distributions We will study possible asymptotic behaviors of systems such as the approach to stationary distributions and the emergence of nonstationary traveling wave distributions We will be concerned with normal and anomalous diffusion and we will examine how correlation functions evolve with time in these kinds of synergetic systems We will discuss a Fokker–Planck approach to quantum statistics, linear nonequilibrium thermodynamics, and generalized extensive and nonextensive thermostatistics The aim of this book is to provide an introduction to the theory of nonlinear Fokker–Planck equations and to highlight what systems described by nonlinear Fokker–Planck equations have in common Theoretical considerations and concepts will be illustrated by various examples and applications Due to the ramifications of the theory of nonlinear Fokker–Planck equations in various scientific fields, this book is designed for graduate students and researchers in physics and related fields such as biology, neurophysics, human movement sciences, and psychology I hope that this book will make graduate students interested in the topic of nonlinear Fokker–Planck equa- VIII Preface tions and will make researchers aware of the connections between the different areas in which nonlinear Fokker–Planck equations have been applied so far Acknowledgments I am indebted to Professor H Haken for inviting me to write a book on the topic of nonlinear Fokker–Planck equations for the Springer Series of Synergetics I am grateful to Professor P.J Beek and Professor R Friedrich for supporting my studies on nonlinear Fokker-Planck equations I would like to thank Dr A Daffertshofer for many helpful discussions Finally, I wish to thank Professor W Beiglbă ock and Ms B Reichel-Mayer of the SpringerVerlag for their help in preparing this manuscript Mă unster, September 2004 Till Daniel Frank Contents Introduction 1.1 Fokker–Planck Equations 1.1.1 Brownian Particles and Langevin Equations 1.1.2 Many-Body Systems and Mean Field Theory 1.2 Phase Transitions and Self-Organization 1.3 Stochastic Feedback 1.4 Applications 1.4.1 Collective Phenomena 1.4.2 Multistable Systems 1.4.3 Power Law and Cut-Off Distributions 1.4.4 Free Energy Systems 1.4.5 Anomalous Diffusion 1.5 Overview 1 7 10 12 15 16 Fundamentals 2.1 Stochastic Processes 2.2 Nonlinear Fokker–Planck Equation 2.2.1 Notation 2.2.2 Stratonovich Form 2.2.3 Transient Solutions 2.2.4 Continuity Equation 2.2.5 Boundary Conditions 2.2.6 Stationary Solutions 2.3 Self-Consistency Equations 2.4 Multistability and Basins of Attraction 2.5 Nonlinearity Dimension 2.6 Classifications 2.7 Derivations 2.8 Numerics 2.8.1 Path Integral Solutions 2.8.2 Fourier and Moment Expansions 2.8.3 Finite Difference Schemes 2.8.4 Distributed Approximating Functionals 19 19 20 21 21 22 22 23 23 24 25 25 26 28 28 28 29 29 30 X Contents Strongly Nonlinear Fokker–Planck Equations 3.1 Transformation to a Linear Problem 3.2 What Are Strongly Nonlinear Fokker–Planck Equations? 3.3 Correlation Functions 3.4 Langevin Equations 3.4.1 Two-Layered Langevin Equations 3.4.2 Self-Consistent Langevin Equations 3.4.3 Hierarchies and Correlation Functions 3.4.4 Numerics 3.5 Stationary Solutions 3.6 H-Theorem for Stochastic Processes 3.7 Nonlinear Families of Markov Processes∗ 3.7.1 Linear Versus Nonlinear Families of Markov Processes 3.7.2 Linear Families of Markov Processes 3.7.3 Nonlinear Families of Markov Diffusion Processes 3.7.4 Markov Embedding 3.7.5 Hitchhiker Processes 3.8 Top-Down Versus Bottom-Up Approaches∗ 3.9 Transient Solutions and Transition Probability Densities 3.9.1 Nonequivalence of Transient Solutions and Transition Probability Densities 3.9.2 Gaussian Distributions∗ 3.9.3 Purely Random Processes 3.9.4 Wiener Processes 3.9.5 Ornstein–Uhlenbeck Processes 3.9.6 Transient Solutions: Two Examples 3.10 Shimizu–Yamada Model – Transient Solutions 3.11 Fluctuation–Dissipation Theorem 31 31 33 36 36 36 38 39 39 42 43 46 46 47 48 50 50 52 55 Free Energy Fokker–Planck Equations 4.1 Free Energy Principle 4.2 Maximum Entropy Principle and Relationship between Noise Amplitude and Temperature 4.3 H-Theorem for Free Energy Fokker–Planck Equations 4.4 Boltzmann Statistics 4.5 Linear Nonequilibrium Thermodynamics 4.5.1 Derivation of Free Energy Fokker–Planck Equations 4.5.2 Drift and Diffusion Coefficients 4.5.3 Transition Probability Densities and Langevin Equations 4.5.4 Density Functions 4.5.5 Entropy Production and Conservative Force 4.5.6 Stationary Solutions 4.5.7 H-Theorem for Systems with Conservative Forces and Nontrivial Mobility Coefficients 73 75 55 57 61 62 62 63 66 70 76 77 79 80 80 84 86 87 87 88 89 Contents 4.6 Canonical-Dissipative Systems 4.6.1 Linear Case 4.6.2 Nonlinear Case 4.7 Boundedness of Free Energy Functionals∗ 4.7.1 Distortion Functionals 4.7.2 Kullback Measure and Entropy Inequality 4.7.3 Generic Cases and Schlă ogls Decomposition of Kullback Measures 4.8 First, Second, and Third Choice Thermostatistics Free Energy Fokker–Planck Equations with Boltzmann Statistics 5.1 Stability Analysis 5.1.1 Lyapunov’s Direct Method 5.1.2 Linear Stability Analysis 5.1.3 Self-Consistency Equation Analysis 5.1.4 Shiino’s Decomposition of Perturbations 5.1.5 Generic Cases 5.1.6 Higher-Dimensional Nonlinearities 5.1.7 Multiplicative Noise 5.1.8 Norm for Perturbations∗ 5.2 Natural Boundary Conditions 5.2.1 Shimizu–Yamada Model – Stationary Solutions 5.2.2 Dynamical Takatsuji Model – Basins of Attraction 5.2.3 Desai–Zwanzig Model 5.2.4 Bounded B(M1 )-Model 5.3 Periodic Boundary Conditions 5.3.1 Cluster Amplitude and Cluster Phase 5.3.2 KSS Model – Cluster Amplitude Dynamics 5.3.3 Mean Field HKB Model – Cluster Phase Dynamics 5.4 Characteristics of Bifurcations 5.4.1 Stability and Disorder 5.4.2 Emergence of Collective Behavior 5.4.3 Multistability and Symmetry 5.4.4 Continuous and Discontinuous Phase Transitions 5.5 Applications 5.5.1 Ferromagnetism 5.5.2 Synchronization 5.5.3 Isotropic-nematic Phase Transitions and Maier–Saupe Model 5.5.4 Muscular Contraction 5.5.5 Network Models for Group Behavior 5.5.6 Multistable Perception-Action Systems XI 90 90 92 96 97 97 100 107 109 110 112 113 115 118 120 126 130 132 134 135 140 148 152 155 156 157 167 183 183 184 186 187 188 188 191 195 202 206 209 ... equilibrium systems with constant temperatures T , the distributions of stationary states (stationary distributions) correspond to the distributions that minimize the free energy F = U − T GS, where... is distributed like u(x) at an initial time t0 Then, the stochastic trajectory X (t) provides us with a complete description of the process for t ≥ t0 To see this, we note that from X (t) we... denoted by Pst and satisfy ∂Pst /? ?t = ⇒ Pst (x, t) = Pst (x) Note that we will also refer to Pst as stationary distributions or stationary probability densities From (2.18) we read off that stationary