BPS Explained III: The Leading Order Behavior of the BBGKY Hierarchy in a Plasma

TL;DR

This paper analyzes the leading order behavior of the BBGKY hierarchy in plasma physics using dimensional continuation, showing how it reduces to Boltzmann and Lenard-Balescu equations in different dimensions and clarifying the Coulomb logarithm calculation.

Contribution

It demonstrates the dimensional reduction of the BBGKY hierarchy to Boltzmann and Lenard-Balescu equations, providing an exact leading and next-to-leading order calculation of Coulomb interactions.

Findings

BBGKY hierarchy reduces to BE for dimensions > 3

BBGKY hierarchy reduces to LBE for dimensions < 3

Coulomb logarithm calculated without cut-off

Abstract

This is the third in a series of lectures on the technique of dimensional continuation, employed by Brown, Preston and Singleton (BPS), for calculating Coulomb energy exchange rates in a plasma. Two important examples of such processes are the charged particle stopping power and the temperature equilibration rate between different plasma species. The first lecture was devoted to understanding the machinery of dimensional continuation, and the second concentrated on calculating the electron-ion temperature equilibration rate in the extreme quantum limit. In this lecture, I will examine one of the main theoretical underpinnings of the BPS theory, namely, the dimensional reduction of the BBGKY hierarchy. I will prove that to leading order in the plasma coupling $g$, the BBGKY hierarchy reduces to the BE for dimensions greater then three and to the LBE for dimensions less than three. We must eventually return to three dimensions, and the BPS formalism shows that the simple poles associated with the BE and the LBE exactly cancel, rendering the limit in three dimensions finite. Furthermore, the leading order behavior of the LBE becomes next-to-leading order when the dimensions is analytically continued from less than three to greater than three. This provides the leading and next-to-leading order terms in $g$ exactly, which is equivalent to an exact calculation of the so-called Coulomb logarithm with no use of an integral cut-off. In this way, BPS takes all Coulomb interactions into account to leading and next-to-leading order in $g$.