Different Ways of Describing Plasma Dynamics

Abstract

In order to realize which approximations that are made in the descriptions of plasmas that we generally use, it is instructive to start from the most general description which includes all individual particles and their correlations in the six dimensional phase space (r,v). I the absence of particle sources or sinks we must have a continuity equation for the delta function density N:

$$ N(X,t) = \sum\limits_{{i = 1}}^N {(X - {{X}_i}} (t))\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,X = ({\mathbf{r}},{\mathbf{v}}) $$

$$ \frac{\partial }{{\partial t}}N + \sum\limits_i {\frac{\partial }{{\partial {{r}_i}}}} \left( {N\frac{{\partial {{r}_i}}}{{\partial t}}} \right) + \sum\limits_i {\frac{\partial }{{\partial {{\hbox{v}}_i}}}} \left( {N\frac{{\partial {{\text{v}}_i}}}{{\partial t}}} \right) = 0, $$