Statistical Physics of Complex Systems: A Concise Introduction (Springer Series in Synergetics)

This third edition of Statistical Physics of Complex Systems has been expanded to provide more examples of applications of concepts and methods from statistical physics to the modeling of complex systems. These include avalanche dynamics in materials, models of social agents like road traffic or wealth repartition, the real space aspects of biological evolution dynamics, propagation phenomena on complex networks, formal neural networks and their connection to constraint satisfaction problems. This course-tested textbook provides graduate students and non-specialists with a basic understanding of the concepts and methods of statistical physics and demonstrates their wide range of applications to interdisciplinary topics in the field of complex system sciences, including selected aspects of theoretical modeling in biology and the social sciences. It covers topics such as non-conserved particles, evolutionary population dynamics, networks, properties of both individual and coupled simple dynamical systems, and convergence theorems, as well as short appendices that offer helpful hints on how to perform simple stochastic simulations in practice. The original spirit of the book is to remain accessible to a broad, non-specialized readership. The format is a set of concise, modular, and self-contained topical chapters, avoiding technicalities and jargon as much as possible, and complemented by a wealth of worked-out examples, so as to make this work useful as a self-study text or as textbook for short courses. Preface to the Third Edition 7 Preface to the Second Edition 9 Preface to the First Edition 11 Contents 12 1 Equilibrium Statistical Physics 17 1.1 Microscopic Dynamics of a Physical System 17 1.1.1 Conservative Dynamics 17 1.1.2 Properties of the Hamiltonian Formulation 19 1.1.3 Many-Particle System 21 1.1.4 Case of Discrete Variables: Spin Models 22 1.2 Statistical Description of an Isolated System at Equilibrium 22 1.2.1 Notion of Statistical Description: A Toy Model 22 1.2.2 Fundamental Postulate of Equilibrium Statistical Physics 23 1.2.3 Computation of Ω(E) and S(E): Some Simple Examples 25 1.2.4 Distribution of Energy Over Subsystems and Statistical Temperature 27 1.3 Equilibrium System in Contact with Its Environment 29 1.3.1 Exchanges of Energy 29 1.3.2 Canonical Entropy 32 1.3.3 Exchanges of Particles with a Reservoir: The Grand-Canonical Ensemble 33 1.4 Phase Transitions and Ising Model 34 1.4.1 Ising Model in Fully Connected Geometry 35 1.4.2 Ising Model with Finite Connectivity 37 1.4.3 Renormalization Group Approach 39 1.5 Disordered Systems and Glass Transition 45 1.5.1 Theoretical Spin-Glass Models 46 1.5.2 A Toy Model for Spin Glasses: The Mattis Model 46 1.5.3 The Random Energy Model 48 1.6 Exercises 51 References 53 2 Non-stationary Dynamics and Stochastic Formalism 54 2.1 Markovian Stochastic Processes and Master Equation 55 2.1.1 Definition of Markovian Stochastic Processes 55 2.1.2 Master Equation and Detailed Balance 56 2.1.3 A Simple Example: The One-Dimensional Random Walk 58 2.2 Langevin Equation 61 2.2.1 Phenomenological Approach 61 2.2.2 Basic Properties of the Linear Langevin Equation 63 2.2.3 More General Forms of the Langevin Equation 66 2.2.4 Relation to Random Walks 68 2.3 Fokker–Planck Equation 70 2.3.1 Continuous Limit of a Discrete Master Equation 70 2.3.2 Kramers–Moyal Expansion 73 2.3.3 More General forms of the Fokker–Planck Equation 74 2.3.4 Stochastic Calculus 76 2.4 Anomalous Diffusion: Scaling Arguments 78 2.4.1 Importance of the Largest Events 79 2.4.2 Superdiffusive Random Walks 81 2.4.3 Subdiffusive Random Walks 82 2.5 First Return Times, Intermittency, and Avalanches 84 2.5.1 Statistics of First Return Times to the Origin of a Random Walk 85 2.5.2 Application to Stochastic On–Off Intermittency 87 2.5.3 A Simple Model of Avalanche Dynamics 88 2.6 Fast and Slow Relaxation to Equilibrium 90 2.6.1 Relaxation to Canonical Equilibrium 90 2.6.2 Dynamical Increase of the Entropy 92 2.6.3 Slow Relaxation and Physical Aging 94 2.7 Exercises 98 References 99 3 Models of Particles Driven Out of Equilibrium 101 3.1 Driven Steady States of a Particle with Langevin Dynamics 102 3.1.1 Non-zero Flux Solution of the Fokker–Planck Equation 102 3.1.2 Ratchet Effect in a Time-Dependent Asymmetric Potential 104 3.1.3 Active Brownian Particle in a Potential 105 3.2 Dynamics with Creation and Annihilation of Particles 107 3.2.1 Birth–Death Processes and Queueing Theory 108 3.2.2 Reaction–Diffusion Processes and Absorbing Phase Transitions 109 3.2.3 Fluctuations in a Fully Connected Model with an Absorbing Phase Transition 112 3.3 Solvable Models of Interacting Driven Particles on a Lattice 115 3.3.1 Zero-Range Process and Condensation Phenomenon 116 3.3.2 Dissipative Zero-Range Process and Energy Cascade 118 3.3.3 Asymmetric Simple Exclusion Process 121 3.4 Approximate Description of Driven Frictional Systems 125 3.4.1 Edwards Postulate for the Statistics of Configurations 126 3.4.2 A Shaken Spring-Block Model 127 3.4.3 Long-Range Correlations for Strong Shaking 129 3.5 Collective Motion of Active Particles 130 3.5.1 Derivation of Continuous Equations 131 3.5.2 Phase Diagram and Instabilities 135 3.5.3 Varying the Symmetries of Particles 135 3.6 Exercices 137 References 139 4 Models of Social Agents 142 4.1 Dynamics of Residential Moves 143 4.1.1 A Simplified Version of the Schelling Model 144 4.1.2 Condition for Phase Separation 146 4.1.3 The ``True'' Schelling Model: Two Types of Agents 149 4.2 Traffic Congestion on a Single Lane Highway 150 4.2.1 Agent-Based Model and Statistical Description 151 4.2.2 Congestion as an Instability of the Homogeneous Flow 153 4.3 Symmetry-Breaking Transition in a Decision Model 155 4.3.1 Choosing Between Stores Selling Fresh Products 156 4.3.2 Mean-Field Description of the Model 156 4.3.3 Symmetry-Breaking Phase Transition 158 4.4 A Dynamical Model of Wealth Repartition 159 4.4.1 Stochastic Coupled Dynamics of Individual Wealths 159 4.4.2 Stationary Distribution of Relative Wealth 161 4.4.3 Effect of Taxes 162 4.5 Emerging Properties at the Agent Scale Due to Interactions 163 4.5.1 A Simple Model of Complex Agents 163 4.5.2 Collective Order for Interacting Standardized Agents 167 4.6 Exercises 169 References 170 5 Stochastic Population Dynamics and Biological Evolution 172 5.1 Motivation and Goal of a Statistical Description of Evolution 172 5.2 Selection Dynamics Without Mutations 174 5.2.1 Moran Model and Fisher's Theorem 174 5.2.2 Fixation Probability 176 5.2.3 Fitness Versus Population Size: How Do Cooperators Survive? 178 5.3 Effect of Mutations on Population Dynamics 179 5.3.1 Quasi-static Evolution Under Mutations 179 5.3.2 Notion of Fitness Landscape 181 5.3.3 Selection and Mutations on Comparable Time Scales 182 5.3.4 Biodiversity Under Neutral Mutations 184 5.4 Real Space Neutral Dynamics and Spatial Clustering 187 5.4.1 Local Population Fluctuations in the Absence of Diffusion 188 5.4.2 Can Diffusion Smooth Out Local Population Fluctuations? 189 5.5 Exercises 191 References 192 6 Complex Networks 194 6.1 Basic Types of Complex Networks 195 6.1.1 Random Networks 195 6.1.2 Small-World Networks 197 6.1.3 Preferential Attachment 198 6.2 Dynamics on Complex Networks 201 6.2.1 Basic Description of Epidemic Spreading: The SIR Model 202 6.2.2 Epidemic Spreading on Heterogeneous Networks 204 6.2.3 Rumor Propagation on Social Networks 207 6.3 Formal Neural Networks 209 6.3.1 Modeling a Network of Interacting Neurons 209 6.3.2 Asymmetric Diluted Hopfield Model 210 6.3.3 Perceptron and Constraint Satisfaction Problem 214 6.4 Exercices 216 References 217 7 Statistical Description of Dissipative Dynamical Systems 220 7.1 Basic Notions on Dissipative Dynamical Systems 220 7.1.1 Fixed Points and Simple Attractors 220 7.1.2 Bifurcations 223 7.1.3 Chaotic Dynamics 225 7.2 Deterministic Versus Stochastic Dynamics 227 7.2.1 Qualitative Differences and Similarities 227 7.2.2 Stochastic Coarse-Grained Description of a Chaotic Map 228 7.2.3 Statistical Description of Chaotic Systems 230 7.3 Globally Coupled Oscillators and Synchronization Transition 232 7.3.1 The Kuramoto Model of Coupled Oscillators 232 7.3.2 Synchronized Steady State 234 7.3.3 Coupled Non-linear Oscillators and ``Oscillator Death'' Phenomenon 237 7.4 A General Approach for Globally Coupled Dynamical Systems 240 7.4.1 Coupling Low-Dimensional Dynamical Systems 240 7.4.2 Description in Terms of Global Order Parameters 241 7.4.3 Stability of the Fixed Point of the Global System 243 7.5 Exercices 245 References 246 8 A Probabilistic Viewpoint on Fluctuations and Rare Events 247 8.1 Global Fluctuations as a Random Sum Problem 247 8.1.1 Law of Large Numbers and Central Limit Theorem 248 8.1.2 Generalization to Variables with Infinite Variances 249 8.1.3 Case of Non-identically Distributed Variables 252 8.1.4 Case of Correlated Variables 256 8.1.5 Coarse-Graining Procedures and Law of Large Numbers 257 8.2 Rare and Extreme Events 259 8.2.1 Different Types of Rare Events 259 8.2.2 Extreme Value Statistics 260 8.2.3 Statistics of Records 262 8.3 Large Deviation Functions 264 8.3.1 A Simple Example: The Ising Model in a Magnetic Field 265 8.3.2 Explicit Computations of Large Deviation Functions 266 8.3.3 A Natural Framework to Formulate Statistical Physics 267 8.4 Exercises 268 References 269 Appendix A Dirac Distributions 271 Appendix B Numerical Simulations of Markovian Stochastic Processes 273 B.1 Discrete-Time Processes 273 B.2 Continuous-Time Processes 274 Appendix C Drawing Random Variables with Prescribed Distributions 275 C.1 Method Based on a Change of Variable 275 C.2 Rejection Method 277 Appendix Solutions of the Exercises 279