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) psi(t)similar tot(-1-beta) for large times t. In this form, beta is a constant (0 < beta < 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 < beta < 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 beta). Moreover, simulated FPTD distributions are observed to approximate closely different Levy stable distributions with growing beta 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 < beta < 2, beta not equal 1. (C) 2003 Elsevier B.V. All rights reserved.