Non-equilibrium Statistical Physics with Application to Disordered Systems

Chapter 1. Probability elements

    1.1 Introduction to random variables

    1.2 Axiomatic scheme

             1.2.1 Conditional probability

             1.2.2 Bayes' theorem

             1.2.3 Statistical Independence

             1.2.4 Random Variable

      1.3 Frequency scheme

             1.3.1 Probability density

             1.3.2 Properties of the probability density

      1.4 Characteristic function G(k)

             1.4.1 The simplest of random walks

             1.4.2 Examples of G(k) is not developable in a Taylor series

             1.4.3 Characteristic function in a toroidal network

             1.4.4 Function of characteristic function

      1.5 Cumulants development

      1.6 Central limit theorem

      1.7 Random variable transformation

      1.8 Correlations between random variables

             1.8.1 Statistical independence

      1.9 Fluctuations development

      1.10 Multidimensional characteristic function

             1.10.1 Diagrams development (many variables)

      1.11 Terwiel cumulants

      1.12 Gaussian distribution (many variables)

             1.12.1 Gaussian with odd null moments

             1.12.2 Novikov's theorem

      1.13 Transformation for n dimensional probability densities

             1.13.1 Marginal probability density

      1.14 Conditional probability density

      1.15 Problems and solutions

 

Chapter 2. Fluctuations around thermal equilibrium<

      2.1 Spatial correlations (Einstein's distribution)

             2.1.1 The Gaussian approximation

      2.2 Minimal work

             2.2.1 Fluctuations in terms of P,V,T variables

      2.3 Fluctuations of mechanical character

             2.3.1 Fluctuations of a tight rope

      2.4 Temporal correlations

      2.5 Problems and solutions

 

Chapter 3. Elements of stochastic processes

      3.1 Introduction

             3.1.1 Time dependent random variable

             3.1.2 Characteristic functional (ensemble representation)

             3.1.3 Kolmogorov hierarchy (multidimensional representation)

             3.1.4 Generalities  about the multidimensional representation

             3.1.5 Generalities about the ensemble representation

      3.2 Conditional Probability

      3.3 Markov  processes

             3.3.1 The Chapman-Kolmogorov  equations

      3.4 Stationary processes

      3.5 Non- stationary periodic processes

      3.6 Brownian motion (The Wiener process)

             3.6.1 Increments of the Wiener process

      3.7 Increments of an arbitrary stochastic process

      3.8 Convergence criteria

             3.8.1 Theorem of Markov (ergodicity)

             3.8.2 Continuity of the realizations

      3.9 Gaussiano white noise

      3.10 Gaussian processes

             3.10.1 The non-singular case

             3.10.2 The singular case (white correlation)

      3.11 Spectrum of the fluctuations of stochastic processes

      3.12 Markovians and Gaussian processes

             3.12.1 The non-stationary periodic case

             3.12.2 The Ornstein-Uhlenbeck  process

      3.13 The Einstein relation

      3.14 The generalized Ornstein-Uhlenbeck process

      3.15 Phase Diffusion

             3.15.1 Dielectric relaxation

      3.16 Stochastic realizations (eigenfunctions)

      3.17 Stochastic differential equations

             3.17.1 The Langevin equations

             3.17.2 Wiener's integrals in the  Stratonovich calculus

             3.17.3 Stochastic differential equations (Stratonovich)

      3.18 The Fokker-Planck equation

             3.18.1 Stochastic fronts

      3.19 The multidimensional Fokker-Planck equation

             3.19.1 Spherical Brownian motion

      3.20 Problems and solutions

 

Chapter 4. Irreversibility, the Fokker-Planck equation

      4.1 Onsager's symmetries

      4.2 Entropy production in the linear approximations

             4.2.1 Mechanic-caloric effect

      4.3 Onsager relations in an electric circuit

      4.4 The Ornstein-Uhlenbeck multidimensional process

             4.4.1 The first theorem of fluctuation-dissipation

      4.5 Canonical distribution in classical statistical mechanics

      4.6 The stationary Fokker-Planck equation

             4.6.1 The inverse problem

             4.6.2 Detail balance

      4.7 Probability current

             4.7.1 The 1-dimentional case

             4.7.2 The multidimensional case

             4.7.3 The Kramers equation

<             4.7.4 Generalized Onsager relations

             4.7.5 Comments on the calculus of the non-equilibrium potential

      4.8 Fokker-Planck non-stationary processes

             4.8.1 Theory of eigenvalues

             4.8.2 The Kolmogorov operator

             4.8.3 Evolution in one period of time

             4.8.4 Periodic Detail Balance

             4.8.5 Strong Mixing

      4.9 Problems and solutions

 

Chapter 5. Irreversibility, linear response

      5.1 Theorem of Wiener-Khinchin

      5.2 Linear response, susceptibility

             5.2.1 Kramers-Kroning's relations

             5.2.2 Relaxation against a discontinuity at t=0

             5.2.3 Energy dissipation

      5.3 Dissipation and  correlations

             5.3.1 A Brownian particle in an Harmonic potential

             5.3.2 Brownian particle in the presence of a magnetic field

      5.4 About the Fluctuation Dissipation Theorem

             5.4.1 Theorem II, the Green-Callen formulae

      5.5 Problems and solutions

 

Chapter 6. Introduction to the diffusive transport

      6.1 Markov's chains

             6.1.1 Properties of the Matrices T (positives)

      6.2 The Random Walk

             6.2.1 Generating functions

             6.2.2 Moments of a random walk

             6.2.3 Realizations of a random walk

      6.3 The Master equation (diffusion in the lattice)

             6.3.1 Formal solution (the Green function)

             6.3.2 Transition to the next neighbor

             6.3.3 Solution of the homogeneous problem in one dimension

             6.3.4 Density of states, Localized states

      6.4 Model of disorder

             6.4.1 Stationary solution

             6.4.2 Short times

             6.4.3 Long times

      6.5 Boundary Condition in the Master equation

             6.5.1 Introduction to the boundary condition problems

             6.5.2 The equivalent problem

             6.5.3 The limbo absorbing state

             6.5.4 Reflecting state

             6.5.5 Boundary Conditions (the method of the image)

             6.5.6 The generalized method of the image

      6.6 The random times of the first passage

             6.6.1 Survival probability

      6.7 Problem and solutions

 

Chapter 7. Diffusion in random media

      7.1 Disorder in the Master equation

      7.2 The effective medium Approximation

             7.2.1 The problem of one impurity

             7.2.2 Calculating the Green function of one impurity

             7.2.3 The effective medium

             7.2.4 The short time limits

             7.2.5 The long time limit

      7.3 Anomalous diffusion and the CTRW approximation

             7.3.1 Relation between the CTRW and the generalized Master equation

             7.3.2 Return to the origin

             7.3.3 Relations between the waiting function and the disorder

             7.3.4 The super diffusion

      7.4 Diffusion with internal states

             7.4.1 The ordered case

             7.4.2 The disordered case

             7.4.3 The no separable case

      7.5 Problems and solutions

 

Chapter 8. Electric conductivity

      8.1 Transport and the quantum mechanics

      8.2 Transport and the Kubo formula

             8.2.1 Theorem III (Kubo)<

             8.2.2 The Kubo formula

             8.2.3 Application to the electric conductivity

<      8.3 Conductivity in the classical limit

             8.3.1 Conductivity  using an exponential relaxation

      8.4 The Scher & Lax formula for the conductivity

             8.4.1 Susceptibility in a Lorentz gas

             8.4.2 The static limit (The Fick law)

      8.5 Anomalous diffusive transport

             8.5.1 The CTRW technique (final conclusion)

             8.5.2 The self-consistent approximation

             8.5.3 Diffusion in spherical coordinates

      8.6 About the mean value over the disorder

      8.7 Problems and solutions

 

Chapter 9. Metastable and instable states

      9.1. Decay rates in the small noise approximation

      9.2. The Kramers slow diffusion approach

             9.2.1. Kramers' activation rates and the mean first passage time

      9.3. Variational treatment for estimating the relaxation time

      9.4. Genesis of the first passage time theory in higher dimensions

      9.5. Unstable states

      9.6. Suzuki's scaling time in the small noise approximation

             9.6.1. The first passage time approach for nonlinear unstable states

             9.6.2. Stochastic paths perturbation approach and the scaling times

      9.7. Genesis of extended systems and their relaxation from unstable states

      9.8. Problems and solutions

 

Appendices <

A. Thermodynamics variables in statistical mechanics

      A.1 The Boltzmann principle

             A.1.1 System in contact

      A.2 First and second law of the thermodynamics

 

B. Relaxation to the stationary state

      B.1 Temporal evolution

      B.2 The Lyapunov function

 

C. The Green function and the one impurity problem

      C.1 The anisotropic and asymmetric case

      C.2 The anisotropic and symmetric case

 

D. The waiting function of the CTRW

 

E. Non-Markov effects against the irreversibility

      E.1 The nonlocal kernel and generalized differential calculus

 

F. The density matrix

      F.1 Properties of the density matrix

      F.2 The reduced density matrix

      F.3 The von Neumann equation

      F.4 Entropy of information

 

G. The Kubo formula and the susceptibility

      G.1 Alternative derivation of the Kubo formula

 

H. Fractals

      H.1 Self-similar Objects

                H.2 Statistical self-similar objects