Exact Analytic Solution of Fokker-Planck Equation for Propagation of Particles Scattered Isotropically

Abstract

An analytic solution for a Fokker-Planck equation that describes propagation of energetic particles through a scattering medium is obtained. The solution is found in terms of an infinite series of mixed moments of particle distribution. The spatial dispersion of a particle cloud released at t=0 evolves through three phases, ballistic (t<Tc), transdiffusive (t~Tc) and diffusive (t>>Tc), where Tc is the collision time.The ballistic phase is characterized by a decelerating expansion of the initial point source in form of "box" distribution with broadening walls. The next, transdiffusive phase is marked by the box walls broadened to its size and a noticeable slow down of expansion. Finally, the evolution enters the conventional diffusion phase.