Diffusion-mediated adsorption versus absorption at partially reactive targets: a renewal approach

Renewal theory is finding increasing applications in non-equilibrium statistical physics. One example relates the probability density and survival probability of a Brownian particle or an active run-and-tumble particle with stochastic resetting to the corresponding quantities without resetting. A second example is so-called snapping out Brownian motion, which sews together diffusions on either side of an impermeable interface to obtain the corresponding stochastic dynamics across a semi-permeable interface. A third example relates diffusion-mediated surface adsorption-desorption (reversible adsorption) to the case of irreversible adsorption. In this paper we apply renewal theory to diffusion-mediated adsorption processes in which an adsorbed particle may be permanently removed (absorbed) prior to desorption. We construct a pair of renewal equations that relate the probability density and first passage time (FPT) density for absorption to the corresponding quantities for irreversible adsorption. We first consider the example of diffusion in a finite interval with a partially reactive target at one end. We use the renewal equations together with an encounter-based formalism to explore the effects of non-Markovian adsorption/desorption on the moments and long-time behaviour of the FPT density for absorption. We then analyse the corresponding renewal equations for a partially reactive semi-infinite trap and show how the solutions can be expressed in terms of a Neumann series expansion. Finally, we construct higher-dimensional versions of the renewal equations and derive general expression for the FPT density using spectral decompositions.