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arXiv · 2312.13200

Relating absorbing and hard wall boundary conditions for a one-dimensional run-and-tumble particle

Mathis Guéneau·Léo Touzo
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Abstract

The connection between absorbing boundary conditions and hard walls is well established in the mathematical literature for a variety of stochastic models, including for instance the Brownian motion. In this paper we explore this duality for a different type of process which is of particular interest in physics and biology, namely the run-tumble-particle, a toy model of active particle. For a one-dimensional run-and-tumble particle subjected to an arbitrary external force, we provide a duality relation between the exit probability, i.e. the probability that the particle exits an interval from a given boundary before a certain time $t$, and the cumulative distribution of its position in the presence of hard walls at the same time $t$. We show this relation for a run-and-tumble particle in the stationary state by explicitly computing both quantities. At finite time, we provide a derivation using the Fokker-Planck equation. All the results are confirmed by numerical simulations.

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BibTeXRIS

Mathis Guéneau, Léo Touzo. 2023-12-20. Relating absorbing and hard wall boundary conditions for a one-dimensional run-and-tumble particle. https://doi.org/10.1088/1751-8121%2Fad4753

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