Dynamical large deviations for homogeneous systems with long range interactions and the Balescu--Guernsey--Lenard equation

TL;DR

This paper derives a large deviation principle for the trajectories of empirical densities in homogeneous long-range interacting particle systems, extending classical kinetic theory and providing explicit fluctuation probabilities.

Contribution

It introduces an explicit large deviation Hamiltonian for empirical density trajectories, linking fluctuations to time reversal symmetry and macrostate entropy increase.

Findings

Explicit large deviation Hamiltonian derived

Connects fluctuations with time reversal symmetry

Provides a comprehensive stochastic process description

Abstract

We establish a large deviation principle for time dependent trajectories (paths) of the empirical density of $N$ particles with long range interactions, for homogeneous systems. This result extends the classical kinetic theory that leads to the Balescu--Guernsey--Lenard kinetic equation, by the explicit computation of the probability of typical and large fluctuations. The large deviation principle for the paths of the empirical density is obtained through explicit computations of a large deviation Hamiltonian. This Hamiltonian encodes all the cumulants for the fluctuations of the empirical density, after time averaging of the fast fluctuations. It satisfies a time reversal symmetry, related to the detailed balance for the stochastic process of the empirical density. This explains in a very simple way the increase of the macrostate entropy for the most probable states, while the stochastic process is time reversible, and describes the complete stochastic process at the level of large deviations.