# Beg Rohu, May 30-June 12, 2021

# Methods for Non-Equilibrium Statistical Mechanics (12h)

### 1. Master Equation and stochastic processes:

The physics behind a master equation

Fokker-Planck equation

Stochastic calculus

A breather with a few useful examples

One final tool in the toolbox: path integrals

### 2. In, or out of, equilibrium

Characterizing equilibrium

A brief overview of fluctuation theorems

How about a single active particle

An active particle in a potential

Actually measuring the distance to equilibrium

### 3. Rare events

Kramers as usual

How about an active Kramers formula

For a particle subjected to a white but non-Gaussian noise

A trapping problem

### 4. Collective phenomena in active systems

### Introduction, and the two-particle problem

### From 2 to N particles

### A detour via equilibrium

### Mesoscopic description of scalar active matter

### 5. A chemical reaction

### Mean-field and the importance of fluctuations

### Langevin description, and beyond

### Towards a quantum formulation

### Exploiting the Doi-Peliti formalism

### Back to the chemical reaction

The particular one-dimensional case

### 6. Epidemics

The SIR model and its connection to percolation

The SIS model, sandpiles, and the depinning transition

The SIS model and directed percolation

Connection to directed percolation

Dynamical scaling

Critical discussion of diffusive models. Epidemics on networks

Front propagation into an unstable state

Branching and annihilating random walks

7. Preys and Predators

Around Lotka and Volterra

Incorporating fluctuations, the two-species problem

Three-species population

A set of extended lecture notes