Stochastic calculus of run-and-tumble motion: an applied perspective

TL;DR

This paper introduces a probabilistic calculus approach to analyze one-dimensional run-and-tumble particles, enabling systematic inclusion of complex features like resetting and sticky boundaries, and linking individual trajectories to entropy production.

Contribution

It develops a novel probabilistic framework based on stochastic calculus for RTPs, providing a direct path from sample trajectories to the Chapman-Kolmogorov equation and extensions.

Findings

Probabilistic calculus links sample paths to CK equation.

Framework incorporates resetting and sticky boundaries.

Entropy production is computed along individual trajectories.

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 generalisation of It\^o'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 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.