Three-dimensional Lorentz model in a magnetic field : exact and Chapman-Enskog solutions

TL;DR

This paper derives exact and Chapman-Enskog solutions for the 3D Lorentz model's Boltzmann equation in a magnetic field, revealing how electron velocity distributions evolve and lead to anisotropic diffusion.

Contribution

It provides the first exact solution for the Boltzmann equation in this context and details a systematic approach to Chapman-Enskog solutions in a magnetic field.

Findings

Velocity distribution exponentially relaxes to spherical symmetry.

Long-term diffusion is governed by an anisotropic tensor.

Systematic Chapman-Enskog solution method is established.

Abstract

We derive the exact solution of the Boltzmann kinetic equation for the three-dimensional Lorentz model in the presence of a constant and uniform magnetic field. The velocity distribution of the electrons reduces exponentially fast to its spherically symmetric component. In the long time hydrodynamic limit there remains only the diffusion process governed by an anisotropic diffusion tensor. The systematic way of building the Chapman-Enskog solutions is described.