Kinetic approach to the envelope equation for a relativistic electron beam propagating through a scattering gas-plasma medium in the presence of the reverse plasma current with an arbitrary radial profile

A kinetic approach is applied to derive the transport equations, the virial equation, the dynamic-equilibrium equation, and the envelope equation for an axially symmetric paraxial relativistic electron beam propagating through a scattering gas-plasma medium in the presence of a reverse plasma current with a radial density profile that is generally different from the beam density profile. The equations obtained include additional terms that account for this difference.