Lecture notes on large deviations in non-equilibrium diffusive systems

TL;DR

This paper reviews theoretical tools and recent developments in understanding large deviations and steady states in non-equilibrium diffusive systems, emphasizing macroscopic fluctuation theory and its applications.

Contribution

It provides a comprehensive overview of classical and modern methods for analyzing non-equilibrium steady states, highlighting recent advances in large deviation functions.

Findings

Application of macroscopic fluctuation theory to diffusive systems

Derivation of large deviation functions for density and current

Connection between fluctuation theorems and transport properties

Abstract

These notes are a written version of lectures given in the 2024 Les Houches Summer School on {\it Large deviations and applications}. They are are based on a series of works published over the last 25 years on steady properties of non-equilibrium systems in contact with several heat baths at different temperatures or several reservoirs of particles at different densities. After recalling some classical tools to study non-equilibrium steady states, such as the use of tilted matrices, the Fluctuation theorem, the determination of transport coefficients, the Einstein relations or fluctuating hydrodynamics, they describe some of the basic ideas of the macroscopic fluctuation theory allowing to determine the large deviation functions of the density and of the current of diffusive systems.