Dynamical Large Deviations for Plasmas Below the Debye Length and the Landau Equation

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.

Path large deviations for the kinetic theory of weak turbulence

“…Here, we go one step backward and define a mean-field system for finite L, for which the RPA approximation is broken. One can proceed in a similar way as for the original dynamics of the modes done previously [we refer to [10] where the detailed calculation is expounded]. The large deviations Hamiltonian (40) associated with the fluctuations of the empirical spectrum n in this mean-field system reads as…”

“…This derivation casts the large-deviations theory for wave kinetics in the same mathematical framework as that for the Boltzmann equation of low-density gases [9], for the Landau equation of weakly-coupled plasmas [10], or for the Lenard-Balescu equation of particle systems with long range interactions [11].…”

“…In three recently published papers, dynamical large deviation principles related to the main classical kinetic theories have been established, starting from Hamiltonian dynamics. The first one [9] dealt with the large deviations for dilute gases (associated with the Boltzmann equation), the second one dealt with large deviations for plasma fluctuations at scales much larger then the Debye length [10] (associated with the Landau equation), and the third one with large deviations for particles with mean field interactions [11] (associated with the Balescu-Guernsey-Lenard equation). The present paper gives a similar result for the kinetic theory of wave turbulence.…”

Dynamical large deviations for an inhomogeneous wave kinetic theory: linear wave scattering by a random medium

“…Our goal in this subsection is to show how one can derive a path large deviation theory for wave kinetics in the diffusive regime from the large deviation Hamiltonian (42). In a recent paper, a similar weak scattering limit has been considered to derive the path large deviation principle for plasma below the Debye length, related to the Landau equation, from the path large deviation principle for dilute gases, related to the Boltzmann equation [43].…”

“…Deriving such large deviation principles from deterministic microscopic dynamics is a fundamental endeavor in theoretical and mathematical physics. Recently, the large deviation principles for a number of classical kinetic theories, starting from first principles, have been uncovered: for discrete models that mimic dilute gases and with Boltzmann like behavior [38,39], for dilute gases related to the Boltzmann equation [1,40], for the Kac model [41,42], for plasma at length scales much smaller than the Debye length related to the Landau equation [43], for homogeneous systems with long range interactions related to the Balescu-Guernsey-Lenard equation [44], for weakly interacting waves in a homogeneous setup [2] related to the wave kinetic equation. The large deviation principles describe fluctuations but also uncover gradient structure for the deterministic kinetic equation, see [45] and a simple explanation in [1].…”