We investigate the transformation of the distri- bution 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 vicin- ity of high-Mach number shocks where particles are acceler- ated 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 distri- bution function from first principles. Application to thermal equilibrium lets us derive a modified version of the isotropic relativisticthermaldistribution,themodifiedJu ̈ttnerdistribu- tion corrected for the Lorentz-invariant phase-space volume element. Finally, we discuss the relativistic modification of a number of plasma parameters.
Treumann, R A.; Nakamura, R; and Baumjohann, W, "Relativistic transformation of phase-space distributions" (2011). Open Dartmouth: Faculty Open Access Articles. 438.