A Path Integral Approach to the Theory of Heliospheric Cosmic‐Ray Modulation

Abstract

This paper introduces the path integral method, which has been widely used in quantum mechanics and statistical mechanics, into the field of cosmic-ray modulation theory to solve the Fokker-Planck equation for cosmic-ray transport. 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 diffusion, drift, convection, adiabatic energy change, and Fermi acceleration. When its action functional integral ∫Ldt is minimized, it yields an Euler-Lagrange equation that describes the most probable trajectory of charged particles randomly walking in heliospheric magnetic fields. The most probable trajectory is equivalent to the classical trajectory of particles in quantum mechanics. A general solution to the cosmic-ray modulation equation with an initial boundary value problem is also formulated in this paper. The path integral has been applied to an example case of steady-state, one-dimensional, spherically symmetric modulation with a boundary at 100 AU. The modulated cosmic-ray spectra obtained with the path integral method agree very well with those from other methods, even though a simple semiclassical approximation is used in the evaluation of the path integral in this calculation. In addition to being able to calculate the modulated spectrum, the path integral method reveals new information about the average behavior of individual particles during their transit through the heliosphere, such as the particle trajectories, energy-loss behavior, and source-particle distribution, all of which are normally not available through simply solving the Fokker-Planck equation. It is expected that more complex modulation problems can also be dealt with by this method, since with the path integral approach, the mathematical problem of cosmic-ray modulation can be treated as a problem of quantum mechanics, for which many mathematical tools have been developed in the past five decades.