Einstein Relation for Electrons in an Electric Field

Abstract

We study the diffusion of electrons moving through a monatomic gas. The two-term Legendre approximation for solving the Boltzmann equation is used to obtain an analytical expression for the isotropic part of the distribution function for hard spheres. We then obtain explicit expressions for the drift velocity, transversal diffusion coefficient, and mean kinetic energy in terms of the confluent hypergeometric function U. The longitudinal diffusion coefficient is obtained by direct numerical integration and the results are used to obtain generalized Einstein relations. The Langevin approach is analyzed and shown to be consistent with the results from the kinetic theory of gases if anisotropic friction is used. An example of the work fluctuation theorem is considered and the differences of the work fluctuation theorem obtained by using the Nernst–Townsend (Einstein) relation and the more accurate results from kinetic theory are calculated.