Relating absorbing and hard wall boundary conditions for a one-dimensional run-and-tumble particle
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| Format: | Preprint |
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2023
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| _version_ | 1866910456211308544 |
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| author | Guéneau, Mathis Touzo, Léo |
| author_facet | Guéneau, Mathis Touzo, Léo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_13200 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Relating absorbing and hard wall boundary conditions for a one-dimensional run-and-tumble particle Guéneau, Mathis Touzo, Léo Statistical Mechanics Soft Condensed Matter Mathematical Physics Probability 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. |
| title | Relating absorbing and hard wall boundary conditions for a one-dimensional run-and-tumble particle |
| topic | Statistical Mechanics Soft Condensed Matter Mathematical Physics Probability |
| url | https://arxiv.org/abs/2312.13200 |