# M2 iPCS: Out of Equilibrium Statistical Mechanics, Classical and Quantum

Presentation of the lecture series

**Exam: TBA. **All hand-written documents are allowed. No electronic devices allowed (not even the cell phone).

Lectures on Wednesdays and Thursdays starting Feb 2nd, 2022 (02/02, 03/02, 09/02, 10/02, 16/02, 17/02, 23/02, 24/02, 02/03, 03/03, 09/03, 10/03, 16/03, 17/03).

Some **useful problems** to work your way through the lectures (feel free to ask for help if you get stuck).

**Some fully written out solutions** by the 2017, 2018, 2019, 2020 , 2021 (and soon the 2022) graduates:

Differential calculus likes Stratonovich discretization (by Bruno V.); A solvable master equation and dynamical complexity (by Martin M.); Glauber dynamics and the FDT (by Kevin B.); Kawasaki dynamics (by Sebastian G.); Triplet annihilation (by Bruno V.); Playing around with stochastic calculus (by Mallory D.); Recipe for a Gaussian white noise (by Aigars L.); Quantum formulation of classical stochastic dynamics (by Ludovico C.); Path statistics, Crooks and Jarzynski theorems (by Arthur A.) or the same exercise whose solution is written up differently (by Andrea P.); Critical Dynamics of Model C (by Julien H.); Dean-Kawasaki Equation (by Marc B.); Dean-Kawasaki dynamics from a microscopic approach (Gianluca B.); Persistsence in pair annihilation (Brieuc B.); Macroscopic Fluctuation Theory of a weakly asymmetric exclusion process (Toni M); Lotka–Volterra prey-predator population dynamics (Beatriz A); Differential caluclus likes Stratonovich discretization (Alberto D., notes of the 04/02/2021 presentation); How "natural" is Stratonovich calculus? (Guillaume C.); Current distribution in a TASEP with periodic boundary conditions (F).

**Outline**

Chapter 1 Methods of stochastic dynamics

A review and some new things. Master Equation/Fokker-Planck and Langevin, and path integrals (Janssen-De Dominicis). Introducing trajectories and histories. Large deviations, Gallavotti-Cohen, Onsager and Green-Kubo relations. Gallavotti-Cohen, Onsager and Green-Kubo relations. Reversibility vs irreversibility. Roughly 3 lectures.

Chapter 2 Critical Dynamics

Dynamics in the 1d Ising model and quantum spin chains. Persistence, coarsening. Halperin and Hohenberg classification. Critical dynamics and the concept of dynamical scaling. Roughly 2 lectures.

Chapter 3 Driven Systems

Diffusive systems display long range correlations. Macroscopic Fluctuation Theory. Phase transitions are possible in d=1 (the open TASEP as a model for traffic). A taste of the matrix ansatz solution for the open TASEP by Derrida, Evans and Hakim. The failure of mean-field and superdiffusive dynamics. Phase transitions in DDS. The Katz-Lebowitz-Spohn standard model. Effect of a drive on a critical point. The difficulty of building up a phenomenological coarse-grained description. Roughly 4 lectures.

Chapter 4 Population Dynamics

Connections between a master equation and a many-body quantum problem. The Doi-Peliti path integral formulation. Application of the Doi-Peliti formalism to pair annihilation with one or two species. Segregation phenomena and reaction zones. Introduction to epidemic models (SIR and percolation, SIS). Epidemics: SIS, and the Contact Process. Front propagation into an unstable state. Networks as growth processes. Networks as growth processes. Epidemics on networks and the vanishing of the epidemic threshold. Preys and Predators. Roughly 4 lectures.

Chapter 5 Roughly 1 lecture. Choose from

Chapter 5 Active Matter One particle models. Signatures of the departure from Eqilibrium. Coarse-grained descriptions of active matter.

Chapter 5 Glasses

Chapter 5 Grains

Chapter 5 The Quantum Langevin Equation Coupling to a thermostat in a quantum system.