A numerical technique, starting from the Boltzmann equation, for obtaining the time-dependent behavior of the electron-velocity distribution function in a gas is presented. A unique feature of this technique is that, unlike previously used procedures, it does not make use of any expansion of the distribution function. This allows the full anisotropy of the distribution function to be included in the solution. Furthermore, the problem associated with multiterm-expansion techniques of choosing a sufficient number of terms for convergence is completely avoided. The distribution function obtained by the present method is exact and, in principle, contains all of the expansion terms of the previous procedures. Details of the algorithm, including stability conditions, treatment of the boundaries, and evaluation of the collision integrals, are presented. This technique has been applied for obtaining the time-dependent behavior of electron swarms in gaseous argon and neon for various values of E/N (the ratio of the applied uniform dc field to the gas density), and the corresponding results are presented.
Drallos PJ, Wadehra JM. Exact time-dependent evolution of electron-velocity distribution functions in a gas using the Boltzmann equation. Phys. Rev. A. 1989;40(4):1967-1975. doi: 10.1103/PhysRevA.40.1967