# Nonequilibrium statistical mechanics

## Informations pratiques

 Discipline : Physique Niveau : Master 2 Semestre : S3a Crédits ECTS : 6 Volume Horaire : 24h Cours10h TD
 Responsable : Ludovic Berthier Université Montpellier II, Laboratoire Charles Coulomb Intervenants : Ludovic BerthierOlivier Pierre-Louis

## Language of instruction

The course will be given in english if one student is non french-speaking (or more).

## Objectives

To provide the fundamental basis to nonequilibrium statistical mechanics that are used in many different fields (condensed matter, physical chemistry, soft and biological matter) to describe and interpret from a physical viewpoint transport phenomena and dynamical response in complex systems. The lectures contains fundamental developments and applications to specific physical situations drawn from modern research problems.

## Syllabus

Introduction
Dynamics of equilibrium systems, evolution of nonequilibrium systems using phenomenological and microscopic descriptions.

Chapter 1 - Correlations, responses, and the fluctuation-dissipation theorem
I-1 Phenomenological description: Density and current fields, diffusion equation and Fick's law, entropy production, Einstein relation.
I-2 Formal microscopic description: Time correlation functions (Wiener-Khintchine), Onsager principle, linear response, fluctuation-dissipation theorem, susceptibilities (Kramers-Kronig, Green-Kubo), nonequilibrium effective temperatures.

Chapter 2 - Langevin and Fokker-Planck equations
II-1 Langevin equation: Einstein's approach, Langevin model and equation.
II-2 Caldeira-Leggett model and generalized Langevin equations.
II-3 Fokker-Planck equation, Kramers equation, Smoluchowski equation, the Kramers escape problem, Arrhenius law and viscous liquids, classical nucleation theory, mapping to Schrodinger equation, Ito-Stratonovich rules for stochastic calculus.

Chapter 3 - Markov processes and random walks
III.1 Markov processes: Chapman-Kolmogorov equation, master equation, detailed balance, Monte Carlo simulations, population dynamics, Ising model, trap model (glass transition and aging).
III-2 Random walks: simple random walk, single file diffusion, return and capture times (Smoluchowski), continuous time random walk, anomalous diffusion (sub-diffusion and Levy flights), Sinai model (disordered system).

Chapter 4 - Thermodynamic fluctuations
IV-1 Equilibrium distribution of fluctuations, density fluctuations in fluids, structure factor and pair correlation function.
IV-2 Transport: linear relations between flux and affinities, Onsager symmetry, coupled effects, critical dynamics.
IV-3 Stochastic thermodynamics: path integrals representation of stochastic processes, time reversibility and fluctuation relations (Crooks, Jarzynski, fluctuation theorem).

## Prerequisites

Statistical mechanics L3-M1, Thermodynamics L2

Written exam

## Bibliography

D. Chandler, Introduction to modern statistical mechanics (Oxford Univ. Press).
J.-L. Barrat, J.-P. Hansen, Basic concepts for simple and complex fluids (Cambridge Univ. Press).
N. Pottier, Nonequilibrium Statistical Physics: Linear Irreversible Processes (Oxford Univ. Press).
L. Landau and E. Lifchitz, Physique statistique (Ellipses).

## Keywords

Dynamic relaxation, response, fluctuations, transport coefficients, dynamics, general principles out-of-equilibrium, microscopic models.