Equilibration Rate for the Linear Inhomogeneous Relaxation-Time Boltzmann Equation for Charged Particles

Abstract

We study the long-time behavior of a linear inhomogeneous Boltzmann equation. The collision operator is modeled by a simple relaxation towards the Maxwellian distribution with zero mean and fixed lattice temperature. Particles are moving under the action of an external potential that confines particles, i.e., there exists a unique stationary probability density. Convergence rate towards global equilibrium is explicitly measured based on the entropy dissipation method and apriori time independent estimates on the solutions. We are able to prove that this convergence is faster than any algebraic time function, but we cannot achieve exponential convergence.

Keywords:

• Boltzmann equation
• Linear inhomogeneous equation
• Entropy functional

Acknowledgments

The authors were partially supported by the European IHP network “Hyperbolic and Kinetic Equations: Asymptotics, Numerics, Applications,” HPRN-CT-2002-00282. M.J.C. and J.A.C. acknowledge support from DGI-MCYT project BFM2002-01710. Part of this work was done while T.G. was visiting the Department of Applied Math. of the University of Granada. It is kindly acknowledged for its hospitality.