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

Presentation of the lecture series

**Written Exam: March 2021, TBA. **All hand-written documents are allowed. No electronic devices allowed (not even the cell phone).

Lectures on Wednesdays (14:00-16:00, Room 302A Condorcet bldg) and Thursdays (10:00-12:00, Zoom session) starting Jan 28th, 2021 (28/01, 03/02, 04/02, 10/02, 11/02, 17/02, 18/02, 24/02, 25/02, 03/03, 04/03, 10/03, 11/03, 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 and 2021 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).

**Lecture 1**

ch1: Methods of stochastic dynamics: A review and some new things. Master Equation/Fokker-Planck and Langevin, and path integrals (Janssen-De Dominicis).

28/01 :Master equation (where it comes from & basic properties). Introducing trajectories and histories. Some notes. Look at Exercise 7.

**Lecture 2**

ch1: cont'd. Langevin equations and path integrals.

Problems 1,2 3 cover chapter 1. A good thing is to go through Julien Tailleur's lectures. Textbooks on this topic are those by Risken, Van Kampen or Gardiner.

03/02: Some notes.

**Lecture 3**

ch1: cont'd. Equilibrium vs Nonequilibrium, Fluctuation-dissipation theorem,

04/02 : Some notes.

**Lecture 4**

ch1: large deviations, Gallavotti-Cohen, Onsager and Green-Kubo relations. Gallavotti-Cohen, Onsager and Green-Kubo relations. Reversibility vs irreversibility (by the example).

10/02: a full black-board lecture!

Problem 4 is about the Glauber dynamics and the fluctuation-dissipation theorem. It's a good exercise to go through (covers the master equation, detailed balance, linear response).

ch2: Critical Dynamics: Dynamics in the 1d Ising model and quantum spin chains. Persistence, coarsening.

Best textbook here is that by Uwe Täuber, which I warmly recommend. Else there is the 1977 review by Halperin and Hohenberg themselves. See also the exercise on the quantum representation of a master equation.

**Lecture 5**

ch2: cont'd. Halperin and Hohenberg classification.

See the exercise on Model B dynamics and the Kawasaki rules.

11/02: Some notes.

**Lecture 6**

ch2: critical dynamics and the concept of dynamical scaling

ch3: Driven systems. Diffusive systems display long range correlations.

**Lecture 7**

ch3: Driven systems. Macroscopic Fluctuation Theory. Phase transitions are possible in d=1 (the open TASEP as a model for traffic).

18/02: Some notes.

Suggested exercises: 14, 18, 12.

**Lecture 8**

ch3: Driven systems. A taste of the matrix ansatz solution for the open TASEP by Derrida, Evans and Hakim. Other methods, specific to dimension 1, can be found in the review by Gunter Schütz. The failure of mean-field and superdiffusive dynamics (noisy Burgers equation, see Schmittmann and Janssen's article). Phase transitions in DDS. The Katz-Lebowitz-Spohn standard model.

**Lecture 9**

ch3 : cont'd. Effect of a drive on a critical point. The difficulty of building up a phenomenological coarse-grained description.

25/02 : Some notes.

ch4: Population Dynamics: Connections between a master equation and a many-body quantum problem.

**Lecture 10 **

A tutorial on exercises 12, 14, 20, 21. Some hints for the solution coming up soon.

**Lecture 11**

ch4: Population Dynamics, cont'd. The Doi-Peliti path integral formulation.

It may be good to refresh your memory or learn about second quantization as seen in the quantum many-body problem. C. Mora's notes on the topic are very useful.

04/03 : Some notes.

**Lecture 12**

ch4: Population Dynamics, cont'd. 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).

**Lecture 13**

ch4: Population dynamics, cont'd. Epidemics: SIS, and the Contact Process. Front propagation into an unstable state. Networks as growth processes.

11/03: Some notes.

**Lecture 14**

ch4: Population dynamics, cont'd. Networks as growth processes. Epidemics on networks and the vanishing of the epidemic threshold. Coarse-grained descriptions of Active Matter.