Encounter-based reaction-subdiffusion model I: surface adsorption and the local time propagator

Abstract

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 ℓ ( 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 ψ ( ℓ ) . The marginal probability density for particle position X ( t ) prior to absorption depends on ψ and the joint probability density for the pair ( X ( t ) , ℓ ( 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 ) ∈ { n ∈ Z , n ⩾ 0 } and the amount of time χ ( t ) spent at the origin. The continuum limit involves rescaling χ ( t ) by a factor 1 / Δ x , where Δ x is the lattice spacing. In the limit Δ x → 0 , the rescaled functional χ ( 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.