Long velocity tails in plasmas and gravitational systems

TL;DR

This paper explains the universal origin of long velocity tails in plasmas and gravitational systems as a consequence of force fluctuations due to 1/r^2 interactions, leading to a modified kinetic equation with fractional derivatives.

Contribution

It introduces a new kinetic framework with a fractional Laplacian term to account for observed velocity tails, linking force fluctuations to distribution behavior.

Findings

Heavy tails follow a 1/v^(5/2) distribution in simulations.

The modified kinetic equation reproduces the observed tails.

Force fluctuations due to 1/r^2 interactions cause the tails.

Abstract

Long tails in the velocity distribution are observed in plasmas and gravitational systems. Some experiments and observations in far-from-equilibrium conditions show that these tails behave as 1/v^(5/2). We show here that such heavy tails are due to a universal mechanism related to the fluctuations of the total force field. Owing to the divergence in 1/r^2 of the binary interaction force, these fluctuations can be very large and their probability density exhibits a similar long tail. They induce large velocity fluctuations leading to the 1/v^(5/2) tail. We extract the mechanism causing these properties from the BBGKY hierarchy representation of Statistical Mechanics. This leads to a modification of the Vlasov equation by an additional term. The novel term involves a fractional power 3/4 of the Laplacian in velocity space and a fractional iterated time integral. Solving the new kinetic equation for a uniform system, we retrieve the observed 1/v^(5/2) tail for the velocity distribution. These results are confirmed by molecular dynamics simulations.