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

Ouassim Feliachi (1, 2) , Freddy Bouchet (2)

1 IDP - Institut Denis Poisson
2 Phys-ENS - Laboratoire de Physique de l'ENS Lyon

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.

Mots clés: Plasma; Balescu-Guernsey-Lenard equation; Large deviation theory; Macroscopic fluctuation theory; Widom theorem

Domaines: Mécanique statistique [cond-mat.stat-mech]; Physique des plasmas [physics.plasm-ph]; Dynamique Chaotique [nlin.CD]

Ouassim Feliachi, Freddy Bouchet. Dynamical large deviations for homogeneous systems with long range interactions and the Balescu-Guernsey-Lenard equation. 2021. ⟨hal-03225459v1⟩