Numerical solution of the quantum Lenard-Balescu equation for a one-component plasma

TL;DR

This paper introduces a spectral numerical method using Laguerre polynomial expansion to solve the quantum Lenard-Balescu equation for a one-component plasma, conserving particles and energy without needing an input Coulomb logarithm.

Contribution

The paper presents a novel spectral approach that naturally incorporates Coulomb logarithms and can be extended to solve related plasma kinetic equations.

Findings

The method accurately models plasma equilibration for various initial conditions.

It naturally derives Coulomb logarithms without external input.

The approach compares favorably with traditional Landau and quantum Landau equations.

Abstract

We present a numerical solution of the quantum Lenard-Balescu equation using a spectral method, namely an expansion in Laguerre polynomials. This method exactly conserves both particles and energy and facilitates the integration over the dielectric function. To demonstrate the method, we solve the equilibration problem for a spatially homogeneous one-component plasma with various initial conditions. Unlike the more usual Landau/Fokker-Planck system, this method requires no input Coulomb logarithm; the logarithmic terms in the collision integral arise naturally from the equation along with the non-logarithmic order-unity terms. The spectral method can also be used to solve the Landau equation and a quantum version of the Landau equation in which the integration over the wavenumber requires only a lower cutoff. We solve these problems as well and compare them with the full Lenard-Balescu solution in the weak-coupling limit. Finally, we discuss the possible generalization of this method to include spatial inhomogeneity and velocity anisotropy.