Continuous time random walks revisited: first passage time and spatial distributions

Abstract

We investigate continuous time random walk (CTRW) theory, which often assumes an algebraic decay for the single transition time probability density function (pdf) ψ( t)∼ t −1− β for large times t. In this form, β is a constant (0< β<2) defining the functional behavior of the transport. The use of algebraically decaying single transition time/distance distributions has been ubiquitous in the development of different transport models, as well as in construction of fractional derivative equations, which are a subset of the more general CTRW. We prove the need for and develop modified solutions for the first passage time distributions (FPTDs) and spatial concentration distributions for 0.5< β<1. Good agreement is found between our CTRW solutions and simulated distributions with an underlying lognormal single transition time pdf (that does not possess a constant β). Moreover, simulated FPTD distributions are observed to approximate closely different Lévy stable distributions with growing β as travel distance increases. The modifications of CTRW distributions also point to the limitations of fractional derivative equation (FDE) approaches appearing in the literature. We propose an alternative form of a FDE, corresponding to our CTRW distributions in the biased 1d case for all 0< β<2, β≠1.