A time-dependent multi-term approach is developed to model the electron kinetics under highly transient (nanosecond) electric field pulses. The method solves the time-dependent Boltzmann equation for the electron velocity distribution function (EVDF) using multi-term expansion using Legendre polynomials including the temporal evolution of the anisotropic distributions. A zero-dimensional analysis is presented for electrons in dry and humid air by comparing results from the non-stationary, multi-term approach with the conventional quasi-stationary, two-term approach. It is found that the quasi-stationary, two-term approach fails to accurately represent the EVDF when electric fields change on timescales comparable to electron momentum relaxation, even when the quasi-stationary anisotropy assumption is relaxed. In contrast, the non-stationary, multi-term approach produces excellent agreement with the results obtained from a Monte Carlo simulation. This agreement confirms that rapid field variations create conditions where anisotropic velocity distribution components become significant and cannot be captured by conventional, quasi-stationary, two-term methods.
Vialetto et al. (2026) studied this question.