Electron kinetics in pulsed and non-homogeneous gas discharges

Pulsed plasmas are widely used in industry for material processing, micro/nano-electronics fabrication, plasma-assisted combustion, and medicine applications. Many of these plasmas are highly transient sustained by short duration (nanosecond and sub-nanosecond) pulses. In these regimes, the peak electric field may exceed the DC breakdown threshold, generating high-energy electrons, while steep spatial gradients can cause significant variation in the electron energy distribution over a single mean free path. Electron transport under such extreme conditions is governed by transient, non-local effects, leading to highly non-Maxwellian and anisotropic electron velocity distribution functions (VDFs).

Current modeling efforts typically rely on the two-term approximation of the Boltzmann equation with quasi-stationary or stationary VDFs, which neglect both the temporal dynamics and non-local energy transport. Monte Carlo simulations, although more flexible, are computationally intensive and often suffer from poor statistics in the high-energy tail of the distribution.

In this work, a strictly non-stationary solution of the electron Boltzmann equation tailored for pulsed, non-homogeneous plasmas is presented. Our model, based on a multi-term spherical harmonic expansion with time-resolved treatment of the VDF, captures transient and non-local effects with high fidelity. We demonstrate its application to streamer-like conditions and compare results with kinetic simulations and conventional two-term solutions. The model results reveal strong anisotropy and rapid temporal evolution of the VDF, underscoring the importance of non-stationary modeling in predicting electron kinetics and energy deposition. These insights are essential for accurate prediction and optimization of modern low-temperature highly transient plasma systems.