Diagrammatic computation of the random flight motion

Abstract

We present a perturbation theory by extending a prescription due to Feynman for computing the probability density function for the random flight motion. The method can be applied to a wide variety of otherwise difficult circumstances. The series for the exact moments, if not the distribution itself, for many important cases can be summed for arbitrary times. As expected, the behavior at early time regime, for the sample processes considered, deviate significantly from diffusion theory; a fact with important consequences in various applications such as financial physics. A half dozen sample problems are solved starting with one posed by Feynman who originally solved it only to first order. Another illustrative case is found to be a physically more plausible substitute for both the Uhlenbeck–Ornstein and the Wiener processes. The remaining cases we have solved are useful for applications in regime switching and isotropic scattering of light in turbid media. It is demonstrated that under isotropic random flight with invariant speed, the description of motion in higher dimensions is recursively related to either the one- or two-dimensional movement. Also for this motion, we show that the solution heretofore presumed correct in one dimension, applies only to the case we have named the “shooting gallery” problem; a special case of the full problem. A new class of functions dubbed “damped-exponential integrals” are also identified.z