Non-equilibrium Statistical Physics with Application to Disordered Systems, 1st ed. 2017

This textbook is the result of the enhancement of several courses on non-equilibrium statistics, stochastic processes, stochastic differential equations, anomalous diffusion and disorder. The target audience includes students of physics, mathematics, biology, chemistry, and engineering at undergraduate and graduate level with a grasp of the basic elements of mathematics and physics of the fourth year of a typical undergraduate course. The little-known physical and mathematical concepts are described in sections and specific exercises throughout the text, as well as in appendices. Physical-mathematical motivation is the main driving force for the development of this text.

It presents the academic topics of probability theory and stochastic processes as well as new educational aspects in the presentation of non-equilibrium statistical theory and stochastic differential equations.. In particular it discusses the problem of irreversibility in that context and the dynamics of Fokker-Planck. An introduction on fluctuations around metastable and unstable points are given. It also describes relaxation theory of non-stationary Markov periodic in time systems. The theory of finite and infinite transport in disordered networks, with a discussion of the issue of anomalous diffusion is introduced. Further, it provides the basis for establishing the relationship between quantum aspects of the theory of linear response and the calculation of diffusion coefficients in amorphous 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

      1.3 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

             1.4.3 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

      1.9 Fluctuations development

      1.10 Multidimensional characteristic function

             1.10.1 Diagrams development (many variables)

      1.11 Terwiel cumulants

      1.12 Gaussian distribution (many variables)

             1.12.2 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

      2.5 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)

             3.1.4 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

      3.5 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

             3.8.2 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

             3.12.1 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

      3.16 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)

             3.18.1 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

             4.6.1 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

<             4.7.4 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

             4.8.4 Periodic Detail Balance

             4.8.5 Strong Mixing

      4.9 Problems and solutions

Chapter 5. Irreversibility, linear response

      5.1 Theorem of Wiener-Khinchin

             5.2.1 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

      5.4 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

      6.2 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)

             6.3.2 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

             6.4.3 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

             6.5.5 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

             7.2.5 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

             7.3.4 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

             8.3.1 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

             8.5.2 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

             9.6.1. 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

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

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

H. Fractals

      H.1 Self-similar Objects

                H.2 Statistical self-similar objects

Prof. Dr. Manuel Osvaldo Caceres did his PhD in Physics, in anomalous transport in heterogeneous media in 1986 at the Instituto Balseiro, Argentina. He had several Post-Doc stays in Spain, Germany and Belgium, therefore getting himself into the new trends of non-equilibrium statistical mechanics. His main contributions were in: dynamical localization, nonlinear optics, stochastic liquid crystals, pattern formation in fluids, and explosive systems. In 2005, he got an Associate Professor position at the International Center of Theoretical Physics (ICTP), Trieste, Italy, after which he returned to his country to teach non-equilibrium statistical mechanics and disorder.

Dr. Manuel O. Caceres is currently Full Professor at the Instituto Balseiro, Univ. Nac. Cuyo, Argentina. Senior Researcher at Centro Atómico Bariloche, Comisión Nacional de Energía Atómica (CNEA), Argentina. Former Head of the Statistical Division at the Centro Atomico Bariloche, CNEA, Argentina.

He has published more than a hundred of peer reviewed papers in international journals in the fields of anomalous transport in solid state physics, diffusion-advection in random fluids, pattern formation in stochastic Rayleigh-Bénard experiments, phenomenology in quantum dissipation, and theory of extreme densities in random media. He also published a textbook on statistical non-equilibrium mechanics and random media, in Spanish.

Fills the gap in the literature for an introductory and didactical book on stochastic processes with different applications in physics and biology

Presents a student-friendly text on non-equilibrium statistical mechanics and non-stationary stochastic processes

Provides a concise and didactical text addressing the Fluctuation-Dissipation theorem in classical and quantum systems from first principles

Includes, in each chapter, numerous problems and their solutions