Time evolution of electron distributions to bimodal steady states for electrons dilutely dispersed in theinert gases Ar, Kr, and Xe with deep Ramsauer Townsend minima in themomentum transfer cross section

Abstract

Abstract The current paper considers the thermalization of an ensemble of electrons under the influence of an external electric field and dilutely dispersed in one of the inert gas moderators, Argon, Krypton or Xenon for which the electron momentum transfer cross sections have deep Ramsauer-Townsend minima. As a consequence, the steady state electron distribution functions are bimodal over a small range of external electric field strengths. The current work is directed towards the time evolution of the electron distribution function determined from the numerical solution of the Fokker-Planck equation. The kinetic theory of electrons dilutely dispersed in a heat bath of atoms at temperature T b has a very long history. The solution of the Fokker-Planck equation can be expressed as a sum of exponentials of the form e − λ n t where λ n are the eigenvalues of the Fokker-Planck operator. Alternatively, a finite difference algorithm is used to solve the time dependent Fokker-Planck equation to give the time dependent electron energy distribution function. We demonstrate the evolution of the initial Maxwellian into a nonequilibrium bimodal distribution which cannot be rationalized with either the Gibbs-Boltzmann entropy or the Tsallis nonextensive entropy. Instead, the time dependent approach of an initial Maxwellian to the bimodal distribution is described in terms of the Kullback-Leibler entropy. We also demonstrate the inapplicability of the Boltzmann entropy nor the Tsallis entropy for a model system with a power law momentum transfer cross section of the form, σ(x) = σ 0/x p , where x = m e v 2 / 2 k B T b is the reduced speed. This model with p = 2 is also employed to demonstrate a steady-state Kappa distribution which features prominently in space physics and other fields. For p > 2, we show distribution functions that increase without bound analogous to runaway electrons. The steady nonequilibrium distributions are interpreted as solutions of a Pearson ordinary differential equation.