Relativistic transformation of phase-space distributions

TL;DR

This paper explores how phase-space distributions transform under Lorentz transformations in relativistic plasmas, deriving a Lorentz-invariant distribution function and analyzing its implications for plasma parameters.

Contribution

It establishes the Lorentz invariance of phase-space volume and derives a corrected relativistic thermal distribution, advancing understanding of relativistic plasma behavior.

Findings

Phase-space volume element is Lorentz invariant.

Derived a modified Jüttner distribution for relativistic thermal plasmas.

Discussed relativistic effects on plasma parameters.

Abstract

We investigate the transformation of the distribution function in the relativistic case, a problem of interest in plasma when particles with high (relativistic) velocities come into play as for instance in radiation belt physics, in the electron-cyclotron maser radiation theory, in the vicinity of high-Mach number shocks where particles are accelerated to high speeds, and generally in solar and astrophysical plasmas. We show that the phase-space volume element is a Lorentz constant and construct the general particle distribution function from first principles. Application to thermal equilibrium lets us derive a modified version of the isotropic relativistic thermal distribution, the modified J\"uttner distribution corrected for the Lorentz-invariant phase-space volume element. Finally, we discuss the relativistic modification of a number of plasma parameters. Keywords: Relativistic invariance of the one-particle distribution function, J\"uttner distribution, relativistic plasma parameters