A Path Integral Solution to the Stochastic Differential Equation of the Markov Process for Cosmic Ray Transport

Abstract

Cosmic ray transport in interplanetary or interstellar magnetic fields can be viewed as a Markov stochastic process and the transport equation has therefore recently been reformulated with a set of stochastic differential equations that describe the guiding center and the momentum of individual charged particles. The Fokker-Planck diffusion equation for the cosmic ray flux can be derived from these stochastic differential equations. Alternatively, the Fokker-Planck equation, like the Schroedinger equation in quantum mechanics, can be solved with a path integral method. Both new methods enable us to solve modulation, propagation and acceleration problems for cosmic ray spectra. In addition, both can reveal insights into the physical processes behind the solutions to these problems since they follow the tra jectory and the momentum of individual particles. In this paper, we derive a path integral representation from the stochastic differential equations and thus prove that the two new methods are consistent with each other.