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

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.