Statistical Physics of Complex Systems

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