Time evolutions of information entropies in a one-dimensional Vlasov-Poisson system

TL;DR

This paper investigates how information entropies evolve in a one-dimensional Vlasov-Poisson system during Landau damping, combining analytical solutions and numerical simulations to understand phase-space dynamics and collisional relaxation.

Contribution

It provides analytical solutions for early and late time behaviors of the distribution function and electric field, and analyzes the evolution of information entropies and mutual information in the system.

Findings

Analytical solutions accurately describe Landau damping dynamics.

Information entropy increases while mutual information decreases during collisional relaxation.

The results confirm the validity of Boltzmann's H-theorem in the system.

Abstract

A one-dimensional Vlasov-Poisson system is considered to elucidate how the information entropies of the probability distribution functions of the electron position and velocity variables evolve in the Landau damping process. Considering the initial condition given by the Maxwellian velocity distribution with the spatial density perturbation in the form of the cosine function of the position, we derive linear and quasilinear analytical solutions that accurately describe both early and late time behaviors of the distribution function and the electric field. The validity of these solutions is confirmed by comparison with numerical simulations based on contour dynamics. Using the quasilinear analytical solution, the time evolutions of the velocity distribution function and its kurtosis indicating deviation from the Gaussian distribution are evaluated with the accuracy of the squared perturbation amplitude. We also determine the time evolutions of the information entropies of the electron position and velocity variables and their mutual information. We further consider Coulomb collisions which relax the state in the late-time limit in the collision less process to the thermal equilibrium state. In this collisional relaxation process, the mutual information of the position and velocity variables decreases to zero while the total information entropy of the phase-space distribution function increases by the decrease in the mutual information and demonstrates the validity of Boltzmann's H-theorem.