Closed-Form Equilibrium Solution to Boltzmann Equation for Charged Particles in Electric Field

For threshold-level electrostatic discharge at atmospheric pressure, in the initial stages of the spark, ambient temperatures are moderate, and the rate at which electrons lose energy due to collisions with heavy species is such that the electron temperature (their kinetic energy due to random motion) is constrained. The direct acceleration of electrons by the electric field in a mean-free path can be comparable to or greater than the thermal velocity.

The electron velocity distribution function is significantly skewed due to this direct acceleration. With some assumptions about how collisions drive the distribution, the Boltzmann equation can be used to solve for equilibrium distribution functions. Assuming a constant relaxation rate to the Maxwell-Boltzmann distribution, a closed form solution (Krook distribution) can be found in the literature. We present a more realistic, closed form (albeit complicated) solution assuming the relaxation rate is a sum of a constant rate plus a rate proportional to the velocity in the direction of the electric field. The drift velocity (first moment of the distribution) has the correct limit in both the low electric field (high temperature/constant mobility) limit, and the high electric field (low temperature/ballistic) limit.

LA-UR-21-25744