A Markov Stochastic Process Theory of Cosmic‐Ray Modulation

Abstract

This paper introduces the use of Markov stochastic process theory to study heliospheric modulation of cosmic rays. The basic cosmic-ray transport equation is reformulated with a set of stochastic differential equations that describe the guiding center and momentum of individual charged particles randomly walking in the heliospheric magnetic field. The Fokker-Planck diffusion equation for the cosmic-ray isotropic distribution function can be derived from these stochastic differential equations of basic cosmic-ray transport. General exact solutions to the initial-boundary value problems of the Fokker-Planck diffusion equation are obtained in terms of time-backward Markov stochastic processes. It is demonstrated with a few examples that modulated cosmic-ray spectra can be calculated with Monte-Carlo simulation of stochastic processes, and the results from the stochastic process simulation are completely consistent with those from calculations that numerically solve the Fokker-Planck diffusion equation. In addition to the capability of modulation spectrum calculation, the stochastic process simulation reveals new information about the behavior of individual particles during their transit through the heliospheric magnetic field, which includes the trajectory of particles, momentum-loss history, source particle distribution as functions of location and momentum, and distribution of transport time or path length, all of which are normally not available by just solving the Fokker-Planck diffusion equation. This method will allow us to tackle very complicated time-dependent cosmic-ray modulation problems that are impossible by numerically solving the Fokker-Planck diffusion equation. Diffusive particle acceleration and interstellar propagation of Galactic cosmic rays may also be studied with this stochastic process method.