Reaction-subdiffusion and reaction-superdiffusion equations for evanescent particles performing continuous time random walks

Abstract

Starting from a continuous time random walk (CTRW) model of particles that may evanesce as they walk, our goal is to arrive at macroscopic integro-differential equations for the probability density for a particle to be found at point r at time t given that it started its walk from r_0 at time t=0. The passage from the CTRW to an integro-differential equation is well understood when the particles are not evanescent. Depending on the distribution of stepping times and distances, one arrives at standard macroscopic equations that may be "normal" (diffusion) or "anomalous" (subdiffusion and/or superdiffusion). The macroscopic description becomes considerably more complicated and not particularly intuitive if the particles can die during their walk. While such equations have been derived for specific cases, e.g., for location-independent exponential evanescence, we present a more general derivation valid under less stringent constraints than those found in the current literature.