A path integral approach to the theory of heliospheric cosmic-ray modulation

This paper introduces the path integral method, which has been widely used in quantum mechanics and statistical mechanics, into the field of cosmic-ra y modulation theory to solve the Fokker-Planck equation for cosmic-ray tran sport. The path integral approach recognizes that the motion of cosmic rays is a Markov stochastic process. The derivation of the path integral yields a Lagrangian, L, consisting of parameters characterizing particle diffusio n, drift, convection, adiabatic energy change, and Fermi acceleration. When its action functional integral integral L dt is minimized, it yields an Eu ler-Lagrange equation that describes the most probable trajectory of charge d particles randomly walking in heliospheric magnetic fields. The most prob able trajectory is equivalent to the classical trajectory of particles in q uantum mechanics. A general solution to the cosmic-ray modulation equation with an initial boundary value problem is also formulated in this paper. Th e path integral has been applied to an example case of steady-state, one-di mensional, spherically symmetric modulation with a boundary at 100 AU. The modulated cosmic-ray spectra obtained with the path integral method agree v ery well with those from other methods, even though a simple semiclassical approximation is used in the evaluation of the path integral in this calcul ation. In addition to being able to calculate the modulated spectrum, the p ath integral method reveals new information about the average behavior of i ndividual particles during their transit through the heliosphere, such as t he particle trajectories, energy-loss behavior, and source-particle distrib ution, all of which are normally not available through simply solving the F okker-Planck equation. It is expected that more complex modulation problems can also be dealt with by this method, since with the path integral approa ch, the mathematical problem of cosmic-ray modulation can be treated as a p roblem of quantum mechanics, for which many mathematical tools have been de veloped in the past five decades.