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Nonequilibrium statistical mechanics

Informations pratiques


Discipline :

Physique

Niveau :

Master 2

Semestre :

S3a

Crédits ECTS :

6

Volume Horaire :

24h Cours
10h TD

Responsable :

Ludovic Berthier

Université Montpellier II, Laboratoire Charles Coulomb

Intervenants :

Ludovic Berthier
Olivier 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

Exam

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.