Renewal theory for a run-and-tumble particle with stochastic resetting and a sticky boundary

TL;DR

This work analyzes a one-dimensional run-and-tumble particle on the half-line with a sticky boundary under stochastic bulk resetting. It develops a renewal-theory framework that decomposes trajectories into bulk excursions and wall-binding events to derive the propagator and first-passage statistics, including a nonequilibrium stationary state (NESS) when the boundary is non-absorbing and an MFPT for absorption when it is partially absorbing. The key finding is that the initial orientation $k_0$ and the desorption protocol (resetting after desorption versus no-reset after desorption) strongly influence both the NESS and MFPT, with resetting potentially reducing absorption times under certain conditions and desorption kinetics needing to be matched to bulk kinetics to optimize performance. The results highlight trade-offs between adsorption, desorption, and absorption times and reveal regimes where desorption-induced resetting is advantageous, offering insights for optimal search strategies in constrained active-particle systems.

Abstract

We consider a run-and-tumble particle (RTP) with stochastic resetting confined to the half line $[0,\infty)$ with a sticky boundary at $x=0$. In the bulk the RTP tumbles at a constant rate $α>0$ between velocity states $\pm v$ with $v>0$ and randomly resets to its initial position and orientation $(x_0,k_0)\in(\mathbb{R}^+,\pm)$. When the RTP reaches the target at $x=0$ it attaches to the boundary for a random waiting time before either detaching and continuing to navigate the bulk domain or (permanently) entering the target. These events are the analogs of adsorption, desorption, and absorption of a particle by a partially reactive surface in physical chemistry. We use renewal theory to characterize the particle trajectory in terms of successive binding events under two distinct desorption protocols: via resetting to $(x_0,k_0)$ and via continuous movement from $x=0$ with velocity $+v$. First we derive the nonequilibrum stationary state (NESS) in the case of no absorption and characterize the accumulation at the boundary. Second, we compute the mean first passage time (MFPT) statistics. In addition to observing the usual unimodal dependence of the MFPT on bulk resetting, both the NESS and MFPT strongly depend on the initial orientation $k_0$ and the desorption protocol. For instance, if the initial orientation is toward the boundary, we find that the desorption-induced resetting protocol can reduce the MFPT more effectively than the non-resetting desorption protocol. We also show how matching the desorption kinetics with the bulk resetting or tumbling rate introduces a trade-off between minimizing the adsorption and absorption times. In this setting we find that the desorption protocol which minimizes the absorption MFPT for a given set of parameters is almost always the opposite of that favored when desorption and bulk kinetics are not the same.

Figures (10)

• Figure 1: An RTP confined to the half-line $[0,\infty)$ by a non-absorbing sticky wall at $x=0$. (a) In the bulk domain the particle randomly switches between a left-moving and a right-moving constant velocity state at a Poisson rate $\alpha$. (b) Whenever the RTP collides with the wall in the left-moving state it binds to the wall (adsorbs). It subsequently unbinds (desorbs) at a constant rate $\gamma_0$ and reenters the bulk domain in the right-moving state.
• Figure 2: Example trajectory of an RTP with a (non-absorbing) sticky wall at $x=0$. Each time the particle collides with the wall in the left-moving state it is adsorbed (AD). After the $j$th collision, the particle remains attached or bound to the wall for a random waiting time $\tau_j$, after which it is desorbed (DE) and re-enters the bulk domain in the right-moving state.
• Figure 3: Example trajectories of an RTP on the half-line with a (non-absorbing) sticky wall at $x=0$ and instantaneous stochastic resetting at a rate $r$ to $(x_0,k_0)$. For the sake of illustration we take $k_0=-$. The sequence of waiting times in the bound state are given by $\{\tau_1,\tau_2,\ldots \}$. (a) The particle continues from $x=0$ in the right-moving state after each desorption event. (b) The particle resets to $x_0$ in the velocity state $k_0=-$ immediately after each desorption event.
• Figure 4: NESS of the bulk resetting RTP for (a,b) no resetting after desorption as considered in Sect. IIIA,B and (c,d) desorption-induced resetting to $(x_0,k_0)$ as considered in Sect. IIIC. In the first column $x_0=1$ and we vary the bulk resetting rate; in the second column $r=1$ and we vary the initial and resetting position. Other parameters are $\alpha=1$, $\langle \tau\rangle =1$, and $v=2$. Dark blue curves indicate $k_0=+$ and light green curves indicate $k_0=-$.
• Figure 5: Ratio of the bound time fraction (BTF) of the non-resetting desorption protocol to the desorption-induced resetting protocol for various mean waiting times $\langle\tau\rangle$ with (a,b) $v=1$ and (c,d) $v=2$. Data below unity (horizontal red line) implies that the desorption-induced resetting protocol yields a relatively higher BTF. In the first column $r=1$ and we vary the tumbling rate; in the second column $\alpha=1$ and we vary the bulk resetting rate. Other parameters are $x_0=1$ and $v=2$. Dark blue curves indicate $k_0=+$ and light green curves indicate $k_0=-$.