Limiting density distribution for charged particle beams in free space

An analytic solution is given to the Vlasov equation describing the time evolution of an axially symmetric beam of charged particles spreading radially under the influence of self fields. The marginal densities as a function of radius and of radial velocity are found to be flat out to the expanding edges of the beam. This solution is shown to have the maximum entropy for a given phase‐space area. Thus an arbitrary initial distribution will evolve toward this solution as a limiting distribution.