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

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.

15 pages, 3 Figures, improved numerics, to appear in PRD

References in corpus (4)

• Magnetic field amplification in nonlinear diffusive shock acceleration including resonant and non-resonant cosmic-ray driven instabilities
• Diffusion of Cosmic Rays in the Expanding Universe. II. Energy Spectra of Ultra-High Energy Cosmic Rays
• Probing Nearby CR Accelerators and ISM Turbulence with Milagro Hot Spots
• On the Structure and Scale of Cosmic Ray Modified Shocks

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