Stochastic calculus of run-and-tumble motion: an applied perspective
File(s)
Author(s)
Bressloff, Paul
Type
Journal Article
Abstract
The run-and-tumble particle (RTP) is one of the simplest examples of an active particle in which the direction of constant motion randomly switches. In the one-dimensional (1D) case, this means switching between rightward and leftward velocities. Most theoretical studies of RTPs are based on the analysis of the Chapman–Kolmogorov (CK) differential equation describing the evolution of the joint probability densities for particle position and velocity state. In this paper, we develop an alternative, probabilistic framework of 1D RTP motion based on the stochastic calculus of Poisson and diffusion processes. In particular, we show how a generalization of Itô’s lemma provides a direct link between sample paths of an RTP and the underlying CK equation. This allows us to incorporate various non-trivial extensions in a systematic fashion, including stochastic resetting and partially absorbing sticky boundaries. The velocity switching process and resetting process are represented by a pair of independent Poisson processes, whereas a sticky boundary is modelled using a boundary layer. We then use the probabilistic formulation to calculate stochastic entropy production along individual trajectories of an RTP and show how the corresponding Gibbs–Shannon (GS) entropy is recovered by averaging over the ensemble of sample paths. Finally, we extend the probabilistic framework to a population of RTPs and use this to explore the effects of global resetting.
Date Issued
2025-07-01
Date Acceptance
2025-06-02
Citation
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2025, 481 (2317)
ISSN
1364-5021
Publisher
The Royal Society
Journal / Book Title
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume
481
Issue
2317
Copyright Statement
© 2025 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.
License URL
Publication Status
Published
Article Number
20240815
Date Publish Online
2025-07-02