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

Description

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 Preface to the Second Edition Preface to the First Edition Contents 1 Equilibrium Statistical Physics 1.1 Microscopic Dynamics of a Physical System 1.1.1 Conservative Dynamics 1.1.2 Properties of the Hamiltonian Formulation 1.1.3 Many-Particle System 1.1.4 Case of Discrete Variables: Spin Models 1.2 Statistical Description of an Isolated System at Equilibrium 1.2.1 Notion of Statistical Description: A Toy Model 1.2.2 Fundamental Postulate of Equilibrium Statistical Physics 1.2.3 Computation of Ω(E) and S(E): Some Simple Examples 1.2.4 Distribution of Energy Over Subsystems and Statistical Temperature 1.3 Equilibrium System in Contact with Its Environment 1.3.1 Exchanges of Energy 1.3.2 Canonical Entropy 1.3.3 Exchanges of Particles with a Reservoir: The Grand-Canonical Ensemble 1.4 Phase Transitions and Ising Model 1.4.1 Ising Model in Fully Connected Geometry 1.4.2 Ising Model with Finite Connectivity 1.4.3 Renormalization Group Approach 1.5 Disordered Systems and Glass Transition 1.5.1 Theoretical Spin-Glass Models 1.5.2 A Toy Model for Spin Glasses: The Mattis Model 1.5.3 The Random Energy Model 1.6 Exercises References 2 Non-stationary Dynamics and Stochastic Formalism 2.1 Markovian Stochastic Processes and Master Equation 2.1.1 Definition of Markovian Stochastic Processes 2.1.2 Master Equation and Detailed Balance 2.1.3 A Simple Example: The One-Dimensional Random Walk 2.2 Langevin Equation 2.2.1 Phenomenological Approach 2.2.2 Basic Properties of the Linear Langevin Equation 2.2.3 More General Forms of the Langevin Equation 2.2.4 Relation to Random Walks 2.3 Fokker–Planck Equation 2.3.1 Continuous Limit of a Discrete Master Equation 2.3.2 Kramers–Moyal Expansion 2.3.3 More General forms of the Fokker–Planck Equation 2.3.4 Stochastic Calculus 2.4 Anomalous Diffusion: Scaling Arguments 2.4.1 Importance of the Largest Events 2.4.2 Superdiffusive Random Walks 2.4.3 Subdiffusive Random Walks 2.5 First Return Times, Intermittency, and Avalanches 2.5.1 Statistics of First Return Times to the Origin of a Random Walk 2.5.2 Application to Stochastic On–Off Intermittency 2.5.3 A Simple Model of Avalanche Dynamics 2.6 Fast and Slow Relaxation to Equilibrium 2.6.1 Relaxation to Canonical Equilibrium 2.6.2 Dynamical Increase of the Entropy 2.6.3 Slow Relaxation and Physical Aging 2.7 Exercises References 3 Models of Particles Driven Out of Equilibrium 3.1 Driven Steady States of a Particle with Langevin Dynamics 3.1.1 Non-zero Flux Solution of the Fokker–Planck Equation 3.1.2 Ratchet Effect in a Time-Dependent Asymmetric Potential 3.1.3 Active Brownian Particle in a Potential 3.2 Dynamics with Creation and Annihilation of Particles 3.2.1 Birth–Death Processes and Queueing Theory 3.2.2 Reaction–Diffusion Processes and Absorbing Phase Transitions 3.2.3 Fluctuations in a Fully Connected Model with an Absorbing Phase Transition 3.3 Solvable Models of Interacting Driven Particles on a Lattice 3.3.1 Zero-Range Process and Condensation Phenomenon 3.3.2 Dissipative Zero-Range Process and Energy Cascade 3.3.3 Asymmetric Simple Exclusion Process 3.4 Approximate Description of Driven Frictional Systems 3.4.1 Edwards Postulate for the Statistics of Configurations 3.4.2 A Shaken Spring-Block Model 3.4.3 Long-Range Correlations for Strong Shaking 3.5 Collective Motion of Active Particles 3.5.1 Derivation of Continuous Equations 3.5.2 Phase Diagram and Instabilities 3.5.3 Varying the Symmetries of Particles 3.6 Exercices References 4 Models of Social Agents 4.1 Dynamics of Residential Moves 4.1.1 A Simplified Version of the Schelling Model 4.1.2 Condition for Phase Separation 4.1.3 The ``True'' Schelling Model: Two Types of Agents 4.2 Traffic Congestion on a Single Lane Highway 4.2.1 Agent-Based Model and Statistical Description 4.2.2 Congestion as an Instability of the Homogeneous Flow 4.3 Symmetry-Breaking Transition in a Decision Model 4.3.1 Choosing Between Stores Selling Fresh Products 4.3.2 Mean-Field Description of the Model 4.3.3 Symmetry-Breaking Phase Transition 4.4 A Dynamical Model of Wealth Repartition 4.4.1 Stochastic Coupled Dynamics of Individual Wealths 4.4.2 Stationary Distribution of Relative Wealth 4.4.3 Effect of Taxes 4.5 Emerging Properties at the Agent Scale Due to Interactions 4.5.1 A Simple Model of Complex Agents 4.5.2 Collective Order for Interacting Standardized Agents 4.6 Exercises References 5 Stochastic Population Dynamics and Biological Evolution 5.1 Motivation and Goal of a Statistical Description of Evolution 5.2 Selection Dynamics Without Mutations 5.2.1 Moran Model and Fisher's Theorem 5.2.2 Fixation Probability 5.2.3 Fitness Versus Population Size: How Do Cooperators Survive? 5.3 Effect of Mutations on Population Dynamics 5.3.1 Quasi-static Evolution Under Mutations 5.3.2 Notion of Fitness Landscape 5.3.3 Selection and Mutations on Comparable Time Scales 5.3.4 Biodiversity Under Neutral Mutations 5.4 Real Space Neutral Dynamics and Spatial Clustering 5.4.1 Local Population Fluctuations in the Absence of Diffusion 5.4.2 Can Diffusion Smooth Out Local Population Fluctuations? 5.5 Exercises References 6 Complex Networks 6.1 Basic Types of Complex Networks 6.1.1 Random Networks 6.1.2 Small-World Networks 6.1.3 Preferential Attachment 6.2 Dynamics on Complex Networks 6.2.1 Basic Description of Epidemic Spreading: The SIR Model 6.2.2 Epidemic Spreading on Heterogeneous Networks 6.2.3 Rumor Propagation on Social Networks 6.3 Formal Neural Networks 6.3.1 Modeling a Network of Interacting Neurons 6.3.2 Asymmetric Diluted Hopfield Model 6.3.3 Perceptron and Constraint Satisfaction Problem 6.4 Exercices References 7 Statistical Description of Dissipative Dynamical Systems 7.1 Basic Notions on Dissipative Dynamical Systems 7.1.1 Fixed Points and Simple Attractors 7.1.2 Bifurcations 7.1.3 Chaotic Dynamics 7.2 Deterministic Versus Stochastic Dynamics 7.2.1 Qualitative Differences and Similarities 7.2.2 Stochastic Coarse-Grained Description of a Chaotic Map 7.2.3 Statistical Description of Chaotic Systems 7.3 Globally Coupled Oscillators and Synchronization Transition 7.3.1 The Kuramoto Model of Coupled Oscillators 7.3.2 Synchronized Steady State 7.3.3 Coupled Non-linear Oscillators and ``Oscillator Death'' Phenomenon 7.4 A General Approach for Globally Coupled Dynamical Systems 7.4.1 Coupling Low-Dimensional Dynamical Systems 7.4.2 Description in Terms of Global Order Parameters 7.4.3 Stability of the Fixed Point of the Global System 7.5 Exercices References 8 A Probabilistic Viewpoint on Fluctuations and Rare Events 8.1 Global Fluctuations as a Random Sum Problem 8.1.1 Law of Large Numbers and Central Limit Theorem 8.1.2 Generalization to Variables with Infinite Variances 8.1.3 Case of Non-identically Distributed Variables 8.1.4 Case of Correlated Variables 8.1.5 Coarse-Graining Procedures and Law of Large Numbers 8.2 Rare and Extreme Events 8.2.1 Different Types of Rare Events 8.2.2 Extreme Value Statistics 8.2.3 Statistics of Records 8.3 Large Deviation Functions 8.3.1 A Simple Example: The Ising Model in a Magnetic Field 8.3.2 Explicit Computations of Large Deviation Functions 8.3.3 A Natural Framework to Formulate Statistical Physics 8.4 Exercises References Appendix A Dirac Distributions Appendix B Numerical Simulations of Markovian Stochastic Processes B.1 Discrete-Time Processes B.2 Continuous-Time Processes Appendix C Drawing Random Variables with Prescribed Distributions C.1 Method Based on a Change of Variable C.2 Rejection Method Appendix Solutions of the Exercises