Spectrum Structure and Behaviors of the Vlasov-Maxwell-Boltzmann Systems

TL;DR

This paper analyzes the spectral properties and long-term behaviors of the Vlasov-Maxwell-Boltzmann systems, highlighting differences from classical models and establishing convergence rates to equilibrium for both one- and two-species cases.

Contribution

It provides a detailed spectral analysis of the VMB systems, revealing the effects of electromagnetic forces and differences between one- and two-species models.

Findings

Spectral structure differs from classical Boltzmann and Vlasov-Poisson-Boltzmann systems.

Established optimal convergence rates to equilibrium.

Identified phenomena of electric and magnetic field dominance.

Abstract

The spectrum structures and behaviors of the Vlasov-Maxwell-Boltzmann (VMB) systems for both two species and one species are studied in this paper. The analysis shows the effect of the Lorentz force induced by the electro-magnetic field leads to some different structure of spectrum from the classical Boltzmann equation and the closely related Vlasov-Poisson-Boltzmann system. And the significant difference between the two-species VMB model and one-species VMB model are given. The structure in high frequency illustrates the hyperbolic structure of the Maxwell equation. Furthermore, the long time behaviors and the optimal convergence rates to the equilibrium of the Vlasov-Maxwell-Boltzmann systems for both two species and one species are established based on the spectrum analysis, and in particular the phenomena of the electric field dominating and magnetic field dominating are observed for the one-species Vlasov-Maxwell-Boltzmann system.