Microscopic Approach to Kinetic Theory

A microscopic kinetic theory is developed for a plasma by the use of approximate equations of motion for the microscopic ``exact'' distribution function $$\mathrm{f̂(}\mathrm{r},\mathrm{v}\mathrm{,t)=}{\displaystyle \sum _{\mathrm{i=l}}^N}\delta \mathbf{(}\mathbf{r}-{\mathrm{r}}_i\mathrm{(t)}\mathbf{)}\mathrm{\hspace{0.17em}\delta }\mathbf{(}\mathbf{v}-{\mathrm{v}}_i\mathrm{(t)}\mathbf{)}$$ . These equations can be solved to obtain asymptotic expressions for f̂(r, v, t) that are then used to calculate correlation functions. These approximate equations are also used to obtain the approximate equations for the correlation functions and the principal results of the test‐particle approach. In particular, two sets of equations are constructed. One set describes a homogeneous, slowly varying system and yields the Balescu‐Lenard equation. The other set describes a homogeneous, slowly varying system that contains small, inhomogeneous and quickly varying perturbations.