Evolution Equation and Transport Coefficients of Swarms in Short Time Development of Initial Relaxation Processes

Abstract

An evolution equation and transport coefficient expressing the short time development of swarms were derived from the space-time dependent Boltzmann equation by introducing the “projection operator” which acts on the velocity distribution function. The evolution equation of the density $n(\vec r,t)$ $f(\vec r,\vec v,t = 0) = f0(\vec v)*n(\vec r,t = 0)$ can be generally written as follows: $\partial _t n(r.t) = \hfill \\ \sum\limits_{k = 1} {\int\limits_o^t {at\Omega ^k } } (t - z)0( - \nabla _r )^k n(r.z) + \sum\limits_{k = 1} {w_o^k } (t)\Theta ( - \nabla _r )^k n(r.t = 0) \hfill \\$