Journal of Physics A: Mathematical and Theoretical
PAPER
Encounter-based reaction-subdiffusion model I: surface adsorption and the local time propagator
Paul C Bressloff1,2

Published 6 October 2023 • © 2023 IOP Publishing Ltd
Journal of Physics A: Mathematical and Theoretical, Volume 56, Number 43
Citation Paul C Bressloff 2023 J. Phys. A: Math. Theor. 56 435004
DOI 10.1088/1751-8121/acfcf3
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Author e-mails
bressloff@math.utah.edu

Author affiliations
1 Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom

2 Department of Mathematics, University of Utah 155 South 1400 East, Salt Lake City, UT 84112, United States of America

ORCID iDs
Paul C Bressloff https://orcid.org/0000-0002-7714-9853

Dates
Received 8 March 2023
Accepted 25 September 2023
Published 6 October 2023
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Peer review information
Method: Single-anonymous
Revisions: 1
Screened for originality? Yes

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Abstract
In this paper, we develop an encounter-based model of partial surface adsorption for fractional diffusion in a bounded domain. We take the probability of adsorption to depend on the amount of particle-surface contact time, as specified by a Brownian functional known as the boundary local time $\ell(t)$. If the rate of adsorption is state dependent, then the adsorption process is non-Markovian, reflecting the fact that surface activation/deactivation proceeds progressively by repeated particle encounters. The generalized adsorption event is identified as the first time that the local time crosses a randomly generated threshold. Different models of adsorption (Markovian and non-Markovian) then correspond to different choices for the random threshold probability density $\psi(\ell)$. The marginal probability density for particle position $\mathbf{X}(t)$ prior to absorption depends on ψ and the joint probability density for the pair $(\mathbf{X}(t),\ell(t))$, also known as the local time propagator. In the case of normal diffusion one can use a Feynman–Kac formula to derive an evolution equation for the propagator. Here we derive the local time propagator equation for fractional diffusion by taking a continuum limit of a heavy-tailed continuous-time random walk (CTRW). We begin by considering a CTRW on a one-dimensional lattice with a reflecting boundary at n = 0. We derive an evolution equation for the joint probability density of the particle location $N(t)\in \{n\in {\mathbb{Z}},n\unicode{x2A7E} 0\}$ and the amount of time $\chi(t)$ spent at the origin. The continuum limit involves rescaling $\chi(t)$ by a factor $1/\Delta x$, where $\Delta x$ is the lattice spacing. In the limit $\Delta x \rightarrow 0$, the rescaled functional $\chi(t)$ becomes the Brownian local time at x = 0. We use our encounter-based model to investigate the effects of subdiffusion and non-Markovian adsorption on the long-time behavior of the first passage time (FPT) density in a finite interval $[0,L]$ with a reflecting boundary at x = L. In particular, we determine how the choice of function ψ affects the large-t power law decay of the FPT density. Finally, we indicate how to extend the model to higher spatial dimensions.