Encounter-based model of a run-and-tumble particle II: absorption at sticky boundaries
File(s)
Accepted version
Author(s)
Bressloff, Paul C
Type
Journal Article
Abstract
In this paper we develop an encounter-based model of a run-and-tumble particle (RTP) confined to a finite interval $[0,L]$ with partially absorbing, sticky boundaries at both ends. We assume that the particle switches between two constant velocity states ±v at a rate α. Whenever the particle hits a boundary, it becomes stuck by pushing on the boundary until either a tumble event reverses the swimming direction or it is permanently absorbed. We formulate the absorption process by identifying the first passage time (FPT) for absorption with the event that the time A(t) spent attached to either wall up to time t (the occupation time) crosses some random threshold $\widehat{A}$. Taking $\Psi(a)\equiv \mathbb{P}[\widehat{A}\gt a]$ to be an exponential distribution, $\Psi[a] = {\mathrm e}^{-\kappa_0 a}$, we show that the joint probability density for particle position X(t) and velocity state $\sigma(t) = \pm $ satisfies a well-defined boundary value problem (BVP) with κ0 representing a constant absorption rate. The solution of this BVP determines the so-called occupation time propagator, which is the joint probability density for the triplet $(X(t),A(t),\sigma(t))$. The propagator is then used to incorporate more general models of absorption based on non-exponential (non-Markovian) distributions $\Psi(a)$. We illustrate the theory by calculating the mean FPT and splitting probabilities for absorption. We also show how our previous results for partially absorbing, non-sticky boundaries can be recovered in an appropriate limit. Absorption now depends on the number of collisions $\ell(t)$ of the RTP with the boundary. Finally, we extend the theory by taking absorption to depend on the individual occupation times at the two ends.
Date Issued
2023-04
Date Acceptance
2023-04-03
Citation
Journal of Statistical Mechanics: Theory and Experiment, 2023, 2023 (4)
ISSN
1742-5468
Publisher
IOP Publishing
Journal / Book Title
Journal of Statistical Mechanics: Theory and Experiment
Volume
2023
Issue
4
Copyright Statement
This is the Accepted Manuscript version of an article accepted for publication in Journal of Statistical Mechanics: Theory and Experiment, Volume 2023, April 2023. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at 10.1088/1742-5468/accce2
Identifier
http://dx.doi.org/10.1088/1742-5468/accce2
Subjects
0105 Mathematical Physics
0203 Classical Physics
Fluids & Plasmas
Publication Status
Published
Article Number
043208
Date Publish Online
2023-04-25