Dynamical large deviations for plasmas below the Debye length and the Landau equation

TL;DR

This paper derives a large deviation principle for the empirical density of a plasma modeled by the Landau equation, capturing rare fluctuations and excursions in the large plasma parameter limit.

Contribution

It provides the first large deviation Hamiltonian for plasma fluctuations within the Landau approximation, extending classical kinetic theory to include rare event analysis.

Findings

Derived large deviation Hamiltonian from Boltzmann limit and dynamics

Extended kinetic theory to describe rare fluctuations

Clarified the difference between plasma and mean-field particle Hamiltonians

Abstract

We consider a homogeneous plasma composed of $N$ particles of the same electric charge which interact through a Coulomb potential. In the large plasma parameter limit, classical kinetic theories justify that the empirical density is the solution of the Balescu-Guernsey-Lenard equation, at leading order. This is a law of large numbers. The Balescu-Guernsey-Lenard equation is approximated by the Landau equation for scales much smaller than the Debye length. In order to describe typical and rare fluctuations, we compute for the first time a large deviation principle for dynamical paths of the empirical density, within the Landau approximation. We obtain a large deviation Hamiltonian that describes fluctuations and rare excursions of the empirical density, in the large plasma parameter limit. We obtain this large deviation Hamiltonian either from the Boltzmann large deviation Hamiltonian in the grazing collision limit, or directly from the dynamics, extending the classical kinetic theory for plasmas within the Landau approximation. We also derive the large deviation Hamiltonian for the empirical density of $N$ particles, each of which is governed by a Markov process, and coupled in a mean field way. We explain that the plasma large deviation Hamiltonian is not the one of $N$ particles coupled in a mean-field way.