Kinetic theory of a relativistic beam

TLDR: In this paper, a quasi-static Bennett equilibrium, with isothermal distribution of transverse momenta, is shown to be a similarity solution of the Fokker-Planck equation with scale radius increasing in accord with Nordsieck's formula.

Abstract: A Fokker–Planck equation is derived to study the evolution of a stable, low‐current beam propagating in a gas‐plasma medium. Small‐angle scattering of the beam particles by the medium causes diffusion in the phase space projected transverse to the direction of propagation. The projected components of dynamical friction vanish. As a result, there is a continued input of energy into the transverse particle motions, which is taken up in expansion against the pinch field. A quasi‐static Bennett equilibrium, with isothermal distribution of transverse momenta, is shown to be a similarity solution of the Fokker–Planck equation with scale radius increasing in accord with Nordsieck’s formula. An H theorem is proved and the Bennett distribution is shown to minimize both H and −dH/dt; hence, it is the time‐dependent asymptotic state. The predicted current profile and radius are shown to be in fair agreement with experiment.