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 di †eren-tial equations that describe the guiding center and momentum of individual charged particles randomly walking in the heliospheric magnetic Ðeld. The Fokker-Planck di †usion equation for the cosmic-ray isotropic distribution function can be derived from these stochastic di †erential equations of basic cosmicray transport. General exact solutions to the initial-boundary value problems of the Fokker-Planck di †usion 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 di †usion 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 Ðeld, 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 di †usion 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 di †usion equation. Di †usive particle acceleration and interstellar propagation of Galactic cosmic rays may also be studied with this stochastic process method. Subject headings : cosmic rays È di †usion È interplanetary medium

Modulation of galactic cosmic rays during the unusual solar minimum between cycles 23 and 24

“…In this work, we use the time-backward Markov stochastic process method proposed by Zhang [1999] to solve the Parker transport equation in 3-D spherical coordinate (1). As it is more versatile and less computationally intensive, this method has been successfully implemented with different cosmic ray transport models, such as Qin et al [2005] and Ball et al [2005].…”

“…The empirical and phenomenological GCR modulation models are derived from observations without considering the physical processes [e.g., Nymmik et al, 1992;Zhao and Qin, 2013]. But in order to understand the physical causes for such phenomenon, one needs to use theoretical models for GCR modulation.…”

“…More detailed theories were summarized in review papers such as Potgieter [1998], Jokipii and Kóta [2000], Heber et al [2006], andPotgieter [2013]. Finite difference method [Jokipii and Kopriva, 1979;Kóta and Jokipii, 1983] and stochastic method [Zhang, 1999;Ball et al, 2005;Pei et al, 2010] have been used to solve the 2-D or 3-D Parker's transport equation for GCR modulation. Calculation results were able to reproduce many observed features from measurements by spacecraft, balloon experiments, and NMs.…”

“…In this work a relatively simple spatial and momentum dependence of the diffusion coefficients is assumed following Zhang [1999] and Ferreira et al [2001]. First, parallel diffusion is set as [Zhang, 1999;Ferreira et al, 2001]…”

Effect of Adiabatic Focusing on the Diffusion of Energetic Charged Particles

“…Theoretically, the Fokker-Plank equation is equivalent to a set of stochastic differential equations (SDEs). As a result, the SDEs have been widely used to study the transport process of SEPs instead of directly solving the Fokker-Planck equation (Zhang 1999;Qin et al 2006Qin et al , 2013Dröge et al 2010;He et al 2011;Dresing et al 2012;Wang et al 2012). Recently, SDEs have been used to analyze the diffusion transport of SEPs in the presence of adiabatic focusing (Litvinenko 2012a(Litvinenko , 2012b.…”