Stationary voltage characteristics of a plasma

In this paper a one dimensional, relativistic model of a two component mono energetic plasma of electrons and ions is investigated. Given an applied electric potential φA the electron and ion currents je and ji are calculated. The analysis is based on the associated Lagrange functional. The model equation is a nonlinear ODE of order two in terms of the electric potential φ. Electron and ion current are unknown parameters. This ODE is subject to four boundary conditions. It is shown that infinitely many solutions (φ,Je,Ji) exist but only one distinguished solution minimizes the Lagrange functional. The asymptotic voltage-current-characteristics in non relativistic and ultra relativistic settings are computed and rigorously justified. (authors)