One-dimensional inhomogeneous Vlasov equilibria

Bernstein, Greene, and Kruskal (1957) showed how, given the electrostatic potential as a function of position, the entire distribution function for ions, and the distribution function for untrapped electrons in a one-dimensional two-species Vlasov plasma, one could determine the distribution function for the trapped electrons as the solution of an Abel integral equation. This result is generalized by replacing the statements about the distribution functions with the specification of the velocity-space distribution functions of both species at a single position in the plasma. Limiting cases are shown to correspond to the earlier treatment, and the applicability of the theory to experiment is discussed.