Table of Contents
1 Brownian motion, Langevin and Fokker-Planck equations
1.1 Introduction
1.2 Kinetic theory
1.2.1 The ideal gas
1.2.2 Random Walk: a basic model of diffusion
1.3 Transport phenomena
1.4 Brownian Motion
1.4.1 The Langevin equation for the Brownian particle
1.4.2 The Fokker-Planck equation for the Brownian particle
1.5 Discrete time stochastic processes
1.5.1 Markov chains
1.5.2 Useful examples of Markov chains
1.5.3 Ergodic Markov chains
1.5.4 Master Equation and Detailed Balance
1.5.5 Monte Carlo Method
1.6 Continous time stochastic processes
1.6.1 Stochastic differential equations
1.6.2 General Fokker–Planck Equation
1.6.3 Physical applications of the Fokker–Planck Equation
1.6.4 A different pathway to the Fokker–Planck equation
1.6.5 The Langevin equation and detailed balance
1.7 Generalized Random Walks
1.7.1 Continuous time random walk
1.7.2 Lévy Walks
2 Linear response theory and transport phenomena
2.1 Introduction
2.2 The Kubo formula for the Brownian particle
2.3 Generalized Brownian motion
2.4 Linear response to a constant force
2.4.1 Linear response and fluctuation–dissipation relations
2.4.2 Work done by a time-dependent field
2.4.3 Simple applications of linear response theory
2.5 Hydrodynamics and the Green-Kubo relation
2.6 Generalized linear response function
2.6.1 Onsager regression relation and time reversal
2.7 Entropy production, fluxes and thermodynamic forces
2.7.1 Nonequilibrium conditions between macroscopic systems
2.7.2 Phenomenological equations
2.7.3 Variational principle
2.7.4 Nonequilibrium conditions in a continous system
2.8 Physical applications of Onsager reciprocity relations
2.8.1 Coupled transport of neutral particles
2.8.2 Onsager theorem and transport of charged particles
2.9 Linear response in Quantum Systems
2.10 Examples of linear response in quantum systems
2.10.1 Power dissipated by a perturbation field
2.10.2 Linear response in quantum field theory
2.11 Bibliographic notes
3 From Equilibrium to Out of equilibrium phase transitions
3.1 Introduction
3.2 Basic concepts and tools of equilibrium phase transitions
3.2.1 Phase transitions and thermodynamics
3.2.2 Phase transitions and statistical mechanics
3.2.3 Landau theory of critical phenomena
3.2.4 Critical exponents and scaling hypothesis
3.2.5 Phenomenological scaling theory
3.2.6 Scale invariance and renormalization group
3.3 Equilibrium states versus stationary states
3.4 The standard model
3.5 Phase transitions in systems with absorbing states
3.5.1 Directed percolation
3.5.2 The Domany-Kinzel model of cellular automata
3.5.3 Contact processes
3.6 The phase transition in DP-like systems
3.6.1 Control parameters, order parameters, and critical exponents
3.6.2 Phenomenological scaling theory
3.6.3 Mean-Field theory
4 Out of equilibrium critical phenomena
4.1 Introduction
4.2 Beyond the DP universality class
4.2.1 More absorbing states
4.2.2 Conservation laws
4.3 Self-organized critical models
4.3.1 The Bak-Tang-Wiesenfeld model
4.3.2 The Bak-Sneppen model
4.4 The TASEP model
4.4.1 Periodic boundary conditions
4.4.2 Open boundary conditions
4.5 Symmetry breaking: the bridge model
4.5.1 Mean-field solution
4.5.2 Exact solution for β ≪ 1
5 Stochastic dynamics of surfaces and interfaces
5.1 Introduction
5.2 Roughness: definition, scaling and exponents
5.3 Self-similarity and self-affinity
5.4 Continuum approach: towards Langevin type equations
5.4.1 Symmetries and power counting
5.4.2 Hydrodynamics
5.5 The random deposition model
5.6 The Edwards-Wilkinson equation
5.6.1 Dimensional analysis
5.6.2 The scaling functions
5.7 The Kardar-Parisi-Zhang equation
5.7.1 The Galilean (or tilt) transformation
5.7.2 Exact exponents in d = 1
5.7.3 Beyond the exponents
5.7.4 Results for d > 1
5.8 Experimental results
5.8.1 KPZ d = 1
5.8.2 KPZ d = 2
5.9 Nonlocal models
5.10 Bibliographic notes
6 Phase-ordering kinetics
6.1 Introduction
6.2 The coarsening law in d = 1 Ising-like systems
6.2.1 The nonconserved case: spin-flip (Glauber) dynamics
6.2.2 The conserved case: spin-exchange (Kawasaki) dynamics
6.3 The coarsening law in d > 1 Ising-like systems
6.3.1 The nonconserved case
6.3.2 The conserved case
6.4 Beyond the coarsening laws
6.4.1 Quenching and phase-ordering
6.4.2 The Langevin approach
6.4.3 Correlation function and structure factor
6.4.4 Domain size distribution
6.4.5 Off-critical quenching
6.5 The coarsening law in non-scalar systems
6.6 The classical nucleation theory
6.6.1 The Becker-Döring theory
7 Highlights on pattern formation
7.1 Pattern formation in laboratory and real world
7.2 Linear stability analysis and bifurcation scenarios
7.3 The Turing instability
7.3.1 Linear stability analysis
7.3.2 The Brusselator model
7.4 Periodic steady states
7.5 Energetics
7.6 Nonlinear dynamics for pattern forming systems: The envelope equation
7.7 The Eckhaus instability
7.8 Phase dynamics
7.9 Back to experiments
7.10 Bibliographic notes
Appendix
Appendix A Central limit theorem and its limitations
Appendix B Spectral properties of stochastic matrices
Appendix C Reversibility and Ergodicity in a Markov chain
Appendix D Diffusion equation and random walk
D.1 The diffusion equation with drift: general solution
D.2 Gaussian integral
D.3 Diffusion Equation: Fourier-Laplace transform
D.4 Random walk and its momenta
D.5 Isotropic random walk with a trap
D.6 Anisotropic random walk with a trap
Appendix E Kramers-Moyal expansion
Appendix F Mathematical properties of response functions
Appendix G The van der Waals equation
Appendix H The Ising model
H.1 The Ising model in one dimension
H.2 The renormalization of the 2d Ising model
Appendix I Derivation of the Ginzburg-Landau free energy
Appendix J Kinetic Monte Carlo
Appendix K Mean-field phase diagram of the bridge model
K.1 Symmetric solutions
K.2 The asymmetric HD-LD phase
K.3 The asymmetric MC-LD phase
K.4 The asymmetric LD-LD phase
Appendix L The deterministic KPZ and the Burgers equation
L.1 The functional derivative
Appendix M The perturbative Renormalization Group for KPZ: a few details
Appendix N The Gibbs-Thomson relation
Appendix O The Allen-Cahn equation
Appendix P The Rayleigh-Bénard instability
Appendix Q General conditions for the Turing instability
Appendix R Steady states of the one-dimensional TDGL equation
Appendix S Multiscale analysis