An Exact Solution of the Fokker-Planck Equation for Isotropic Scattering

TL;DR

This paper derives an exact analytic solution to the Fokker-Planck equation for isotropic scattering, describing how energetic particles propagate through a scattering medium across different phases of dispersion.

Contribution

It presents a novel exact solution in terms of mixed moments, capturing the transition from ballistic to diffusive phases in particle propagation.

Findings

The solution characterizes three phases: ballistic, transdiffusive, and diffusive.

The ballistic phase features a decelerating box-shaped expansion.

The transdiffusive phase shows thickened walls and slowed expansion.

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 a "box" distribution with thickening walls. The next, transdiffusive phase is marked by the box walls thickened to its size and a noticeable slow down of expansion. Finally, the evolution enters the conventional diffusion phase.