Analytic expressions for electron-ion temperature equilibration rates from the Lenard-Balescu equation

TL;DR

This paper derives exact and approximate analytical formulas for electron-ion temperature equilibration rates in plasmas from the Lenard-Balescu equation, applicable across various temperature ratios and degeneracy conditions.

Contribution

It provides the first exact formula for the equilibration rate without needing a Coulomb logarithm, including corrections for large temperature ratios.

Findings

Exact formula valid for moderate temperature ratios and degeneracy

Efficient correction for coupled-mode effects at large temperature ratios

No input Coulomb logarithm required for the main formula

Abstract

In this work, we elucidate the mathematical structure of the integral that arises when computing the electron-ion temperature equilibration time for a homogeneous weakly-coupled plasma from the Lenard-Balescu equation. With some minor approximations, we derive an exact formula, requiring no input Coulomb logarithm, for the equilibration rate that is valid for moderate electron-ion temperature ratios and arbitrary electron degeneracy. For large temperature ratios, we derive the necessary correction to account for the coupled-mode effect, which can be evaluated very efficiently using ordinary Gaussian quadrature.