Encounter-based reaction-subdiffusion model II: partially absorbing traps and the occupation time propagator
File(s)
Accepted version
Author(s)
Bressloff, Paul C
Type
Journal Article
Abstract
In this paper we develop an encounter-based model of reaction-subdiffusion in a domain Ω with a partially absorbing interior trap ${\mathcal U}\subset \Omega$. We assume that the particle can freely enter and exit ${\mathcal U}$, but is only absorbed within ${\mathcal U}$. We take the probability of absorption to depend on the amount of time a particle spends within the trap, which is specified by a Brownian functional known as the occupation time A(t). The first passage time (FPT) for absorption is identified with the point at which the occupation time crosses a random threshold $\widehat{A}$ with probability density $\psi(a)$. Non-Markovian models of absorption can then be incorporated by taking $\psi(a)$ to be non-exponential. The marginal probability density for particle position $\mathbf{X}(t)$ prior to absorption depends on ψ and the joint probability density for the pair $(\mathbf{X}(t),A(t))$, also known as the occupation time propagator. In the case of normal diffusion one can use a Feynman–Kac formula to derive an evolution equation for the propagator. However, care must be taken when combining fractional diffusion with chemical reactions in the same medium. Therefore, we derive the occupation time propagator equation from first principles by taking the continuum limit of a heavy-tailed continuous-time random walk. We then use the solution of the propagator equation to investigate conditions under which the mean FPT for absorption within a trap is finite. We show that this depends on the choice of threshold density $\psi(a)$ and the subdiffusivity. Hence, as previously found for evanescent reaction-subdiffusion models, the processes of subdiffusion and absorption are intermingled.
Date Issued
2023-10-27
Date Acceptance
2023-09-25
Citation
Journal of Physics A: Mathematical and Theoretical, 2023, 56 (43)
ISSN
1751-8113
Publisher
IOP Publishing
Journal / Book Title
Journal of Physics A: Mathematical and Theoretical
Volume
56
Issue
43
Copyright Statement
Copyright © 2023 IOP Publishing Ltd. This is an author-created, un-copyedited version of an article published in Journal of Physics A: Mathematical and Theoretical,. 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 10.1088/1751-8121/acfcf4
Identifier
http://dx.doi.org/10.1088/1751-8121/acfcf4
Publication Status
Published
Article Number
435005
Date Publish Online
2023-10-06