Linear Landau damping, Schr\"{o}dinger equation, and fluctuation theorem

TL;DR

This paper transforms the linearized Vlasov-Poisson system into a Schrödinger equation to demonstrate the fluctuation theorem's validity in Landau damping, providing new insights into collisionless plasma processes within nonequilibrium statistical mechanics.

Contribution

It introduces a novel approach by mapping Landau damping to a Schrödinger equation and validates the fluctuation theorem in this context.

Findings

Fluctuation theorem holds for the Landau damping process.

The energy and entropy perturbations form a time-independent invariant.

Exact solutions confirm the theoretical predictions.

Abstract

A linearized Vlasov-Poisson system of equations is transformed into a Schr\"{o}dinger equation, which is used to demonstrate that the fluctuation theorem holds for the relative stochastic entropy, defined in terms of the probability density functional of the particle velocity distribution function in the Landau damping process. The difference between the energy perturbation, normalized by the equilibrium temperature, and the entropy perturbation constitutes a time-independent invariant of the system. This invariant takes the quadratic form of the perturbed velocity distribution function and corresponds to the squared amplitude of the state vector that satisfies the Schr\"{o}dinger equation. Exact solutions, constructed from a discrete set of Hamiltonian eigenvectors, are employed to formulate and numerically validate the fluctuation theorem for the Landau damping process. The results offer new insights into the formulations of collisionless plasma processes within the framework of nonequilibrium statistical mechanics.