EXACT AND ASYMPTOTIC PROPERTIES OF MULTISTATE RANDOM-WALKS

A method is presented which allows one to obtain explicit analytical expressions (both exact and asymptotic) for many of the physically interesting quantities related to a multistate random walk (MRW). The exact results include the Laplace Fourier-transformed probability distribution (continuous time) and generating function (discrete time), and closed evolution equations for the propagators related to each "internal" state of the walker. Analytical expressions for the scattering dynamical structure function and the frequency-dependent diffusion coefficient are given as illustrations. Asymptotic approximations to the single-state propagators are derived, allowing a detailed analysis of the longtime behavior and the calculation of asymptotic properties by single-state random walk standard methods. As an example, analytical expressions for the drift and diffusion coefficients are given.