Lecture notes on large deviations in non-equilibrium diffusive systems

TL;DR

This work surveys the framework of macroscopic fluctuation theory (MFT) for large deviations in non‑equilibrium diffusive systems. It connects microscopic Markov dynamics with macroscopic fluctuation descriptions via tilted generators, fluctuation theorems, and linear response relations, and shows how transport coefficients $D(\rho)$ and $\sigma(\rho)$ determine current fluctuations and density profiles. Key results include explicit $D$ and $\sigma$ for paradigmatic models such as the SSEP and zero‑range process, the Hamilton‑Jacobi structure of the density large deviation functional, and the additivity principle for current fluctuations, leading to phase transitions and nonlocal LDFs out of equilibrium. The notes also illuminate the links between microscopic calculations and macroscopic fluctuating hydrodynamics, the role of long‑range correlations, and the conditions under which equilibrium local functionals extend to non‑equilibrium steady states. Overall, the compilation provides a comprehensive toolbox for predicting current and density fluctuations in diffusive systems across a broad range of models and boundary conditions, with significant implications for non‑equilibrium statistical mechanics and related applications.

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.

Figures (10)

• Figure 1: A system in contact with two heat baths at different temperatures
• Figure 2: SSEP on $L$ sites with open boundary conditions
• Figure 3: A system in contact with two heat baths. $Q_t$ is the flux of particles from heat bath 1 to heat bath 2 during time $t$.
• Figure 4: A system in contact with two reservoirs of particles at different densities
• Figure 5: In diffusive systems, the transport coefficients $\widehat{\sigma}$ and $\widehat{D}$ decay like the inverse of the system size $L$