Summary:  Within Statistical Mechanics, it is possible to rationalize how the properties of macroscopic bodies are related to their microscopic structure. Macroscopic bodies are systems composed of many elementary units whose microscopic dynamics are very irregular. At first glance, many microscopic states are compatible with a single equilibrium macroscopic state. 

This course addresses the problem of describing the dynamics of these microscopic states as they evolve toward stationary (and sometimes equilibrium) configurations. It begins with a discussion of kinetic theory, leading to the Boltzmann equation and the associated H-theorem. The course then moves to stochastic dynamics, focusing on Brownian motion and the Langevin equation (particularly in the overdamped regime). The Fluctuation-Dissipation Theorem will be derived, along with other theorems and relations regarding fluctuations in systems out of equilibrium. The Fokker-Planck equation will also be derived using different approaches, with a discussion of its solutions and the connection to equilibrium in simple contexts. Additionally, the course introduces path integral techniques for stochastic dynamics, which can be used to compute, for example, instantontonic trajectories and the entropy production rate. In the final part of the course, applications to pattern formation (via linear stability analysis) and nonequilibrium phase transitions (particularly in active matter) will be explored.

Required preliminary skills: Knowledge of fundamental concepts of thermodynamics, statistical mechanics, probability theory, analytical mechanics, and mathematical methods.