Formation of the distribution function for runaway electrons in strong fields of pulsed gas discharges

A simplified Boltzmann equation describing the escape of electrons in a weakly ionized gas is constructed. The electric fields are assumed to be so strong that all electrons are runaway electrons and the electron distribution function is strongly anisotropic. The equation is solved analytically, and it is shown that the electron density in relatively weak fields exponentially increases with time, while the momentum dependence of the distribution function exponentially decreases. In strong fields, the electron density increases with time logarithmically and the momentum dependence of the electron distribution function is nonmonotonic. The characteristic scales of time and energy, which determine different scenarios, are obtained.