Continuous time random walk with correlated waiting times

TL;DR

This paper generalizes the Continuous Time Random Walk model by introducing correlated waiting times, analyzing how different correlation types affect diffusion behavior and mean squared displacement over time.

Contribution

It presents a novel extension of CTRW with correlated waiting times and derives the resulting diffusion properties for exponential and power-law correlations.

Findings

Exponential correlations slow down short-time diffusion.

Power-law correlations lead to persistent subdiffusion.

Long-range correlations do not reduce subdiffusion to zero exponent.

Abstract

Based on the Langevin description of the Continuous Time Random Walk (CTRW), we consider a generalization of CTRW in which the waiting times between the subsequent jumps are correlated. We discuss the cases of exponential and slowly decaying persistent power-law correlations between the waiting times as two generic examples and obtain the corresponding mean squared displacements as functions of time. In the case of exponential-type correlations the (sub)diffusion at short times is slower than in the absence of correlations. At long times the behavior of the mean squared displacement is the same as in uncorrelated CTRW. For power-law correlations we find subdiffusion characterized by the same exponent at all times, which appears to be smaller than the one in uncorrelated CTRW. Interestingly, in the limiting case of an extremely long power-law correlations, the (sub)diffusion exponent does not tend to zero, but is bounded from below by the subdiffusion exponent corresponding to a short time behavior in the case of exponential correlations.