A convergent kinetic equation for gravitational and Coulomb systems

TL;DR

This paper develops a new kinetic equation for gravitational and Coulomb systems by combining strong and weak interaction treatments, resulting in a fractional Laplacian term that explains long-tailed velocity distributions observed in simulations.

Contribution

It introduces a novel kinetic equation incorporating both strong and weak interactions, resolving divergence issues in traditional models.

Findings

The fractional Laplacian term leads to long-tailed velocity distributions.

Molecular dynamics simulations confirm the theoretical long tails.

The approach unifies treatment of strong and weak particle interactions.

Abstract

It is well known that due to its divergence at large impact parameters, the Boltzmann collision integral in the kinetic equation for 3D systems of particles interacting through a $1/r$ potential must be replaced by a Balescu-Lenard-like collision term. However, the latter diverges at small impact parameters. This comes from the fact that only weak interactions are considered while strong collisions between close particles are neglected in its derivation. We show that a solution to this dilemma exists in the framework of the BBGKY formulation of statistical mechanics. It is based on a separate treatment of the contribution of the strong interactions from that of the weak interactions. The strong interaction part leads to a new term that involves a fractional Laplacian operator in velocity space while the weak interaction component yields the Balescu-Lenard collision term with an explained lower cut-off at the Landau length. For spatially uniform initial conditions, the fractional Laplacian contribution leads to a long-tailed velocity distribution as long as the spatial inhomogeneity remains small. We present results from molecular dynamics simulations confirming the existence of such long tails.