Advanced statistical mechanics

Outline

Stochastic processes in nature, Gaussian processes, Stationary processes, Markov and non-Markov processes, Brownian motion and random walks, Langevin and Fokker-Planck equations, Equilibrium vs. Non-equilibrium, First-passage properties, Brownian functionals via Feynman-Kac approach–path integrals and relation to quantum mechanics.
Keywords: Gaussian processes, Brownian motion, Langevin equations, Fokker-Planck equations, Brownian function- als, Path integrals

Lecture I: Introduction to Gaussian stochastic processes, correlation functions (Wick’s theorem), various examples including white noise, Brownian motion, Ornstein-Uhlenbeck processes etc.
Lecture II: Brownian motion, Langevin equation, forward and backward Fokker-Planck equations, path integrals, relation between Fokker-Planck and Schrodinger equation, lattice random walks, Levy flights etc.
Lecture III: Equilibrium vs. Non-equilibrium stationary states, violation of detailed balance, relaxation to the stationary state, various examples including coarsening dynamics in Ising model—dynamical scaling.
Lecture IV: Systems with nonequilibrium steady states (dynamics violating detailed balance), as in the asymmetric exclusion process, zero-range process, fluctuating interfaces (such as the Kardar-Parisi-Zhang equation), diffusion with stochastic resetting etc.
Lecture V: Brownian functionals via Feynman-Kac equation, and its various applications in physics and computer science.
Lecture VI: First-passage and persistence problems–several applications, backward Fokker-Planck equation–a pow- erful method.

Pré-requis

Physique statistique L3-M1, thermodynamique L2

Language

English

Exam

written exam

Suggested references

Stochastic processes in physics and chemistry, N. G. van Kampen (Elsevier, Amsterdam)
An introduction to probability theory and its applications, vol I and II, W. Feller (Wiley, New York)

A Guide to first-passage processes S. Redner (Cambridge University Press)
Brownian functionals in physics and computer science, S. N. Majumdar, Current Science, 89, 2076 (2005), available at: http://lptms.u-psud.fr/satya-majumdar/reviews-and-lectures-notes/
A kinetic view of Statistical physics, P. L. Krapivsky, S. Redner, E. Ben-Naim (Cambridge University Press).
Persistence and first-passage properties in nonequilibrium systems, A. J. Bray, S. N. Majumdar, G. Schehr, Adv. in Phys. 62, 225 (2013), available at: https://arxiv.org/pdf/1304.1195.pdf

Mots clefs

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