The Mathematical Theory of Electrical Discharges in Gases

The Boltzmann theory treatment of electrical discharges in gases has been one of the undeveloped fields in physics. The aim of this paper is to show what type of molecular model makes the mathematical treatment of the problem possible.The fundamental processes of the molecular kinetic theory of electrical discharges concern the collisions between gas molecules and charged particles such as ions or electrons. The effective cross sections of the molecule for elastic, exciting, and ionizing collisions are the fundamental quantities regarding these phenomena. Two methods are considered, according to how these fundamental quantities are introduced, to establish the molecular kinetic theory.One method is to adopt a proper molecular model for collision processes. The other method is to adopt the "exact" (i.e., observed or quantum-mechanically calculated) values of collision cross sections. This paper concerns itself with the first method, and the principal objective is to attain mathematical simplicity for the theory. The second method is abandoned because the statistical calculation on each step is complicated, and furthermore, the calculation must be achieved separately for each kind of gas. This second method may be adequately adopted after the treatment based on a model is accomplished.