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Biased continuous-time random walks with asymmetric waiting-time distributions as a model of anomalous diffusion in the presence of drift

Guohua Li*

Andrey G. Cherstvy†

Wenya Guo, Hong Zhang, Ting Liu, and Wendong Pu

Ralf Metzler‡

  • *Contact author: liguohua13@cdut.edu.cn
  • †Contact author: a.cherstvy@gmail.com
  • ‡Contact author: rmetzler@uni-potsdam.de
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Phys. Rev. E 113, 064116 – Published 5 June, 2026

DOI: https://doi.org/10.1103/6c55-c9tp

Abstract

Mathematical models of diffusion based on stochastic processes of the type of continuous-time random walks (CTRWs) are widely used for the description of various transport characteristics in complex media. An external or internal bias in a given CTRW model is often attributed to an asymmetric distribution of jump lengths, while the waiting times are assumed to possess identical distributions in all directions. However, for a number of real-world transport phenomena the latter assumption is violated. To account for such situations, we examine the properties of such biased CTRWs in one dimension. We propose a CTRW-based model of diffusion with the waiting-time distributions (WTDs) of the tracers chosen unequal for right- and left-directed jumps. Below we name CTRWs with such asymmetric WTDs (AWTDs) as At-CTRWs. We derive the corresponding generalized Montroll-Weiss and master equations in Laplace and Z space. We consider both exponential and power-law WTDs as the standard and most useful functional choices for the purposes of data analysis. For these two WTD classes, we analyze the average displacement (generally nonzero) and the mean-squared displacement of the diffusing particles and analytically compute the variance as effective statistical measures. Our analytical predictions favorably agree with the results of the performed computer simulations. Specifically, for a CTRW with power-law WTDs we find that at long times the emerging anomalous diffusion depends not only on the smallest exponent of the left- and right-directed WTDs. Such diffusion models can, e.g., be used to describe the anomalous diffusion in various anisotropic media in the presence of a drift. One hydrological example is a gravity-affected spreading of contaminants in porous rocks, a system where the CTRW-based models are widely known to be applicable. A biological example of a biased-CTRW-based transport is a regulated driven bidirectional transport of cargoes between the center of a living biological cell and its periphery taking place along the microtubule filaments.

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Phys. Rev. E 113, 064116– Published 5 June, 2026
  • Received 26 November 2025
  • Accepted 24 April 2026
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