Kinetic theory of a relativistic beam

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 Nordsiéck’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.