Diffusion with stochastic resetting on a lattice

TL;DR

This paper derives an exact solution for diffusion with stochastic resetting on a $d$-dimensional hypercubic lattice, obtaining a general renewal relation that expresses the MFPT in terms of the nonresetting propagator via $\langle T\rangle_r(\vec{R}_0)=\frac{1}{r}\left[\frac{\tilde{P}_0(\vec{0},\vec{0},r)}{\tilde{P}_0(\vec{0},\vec{R}_0,r)}-1\right]$. It then provides an explicit lattice MFPT formula using lattice Green's functions: $\langle T\rangle_r(\vec{R}_0)=\frac{1}{r}\left[\frac{\int_0^{\infty} dt\, e^{-t}\,[I_0(2t/(r+2d))]^d}{\int_0^{\infty} dt\, e^{-t}\prod_{i=1}^d I_{|m_i|}(2t/(r+2d))}-1\right]$, valid for starting point $\vec{R}_0=a\vec{m}$. The continuum limit is recovered in the appropriate scaling limit $a\to 0$, $r\to 0$ with $r=a^2\tilde{r}$ and $\vec{R}_0\to\infty$ with $\sqrt{r}\,R_0$ fixed, yielding the known $d$-dimensional results, though lattice theory reveals additional large-$r$ behavior absent in the continuum. The paper also presents a continuous-time event-driven algorithm for simulations, explicit results in $d=1$ and $d=2$ (including exact closed forms and asymptotics), and general $d$ asymptotics, as well as the nonequilibrium stationary state distribution in the absence of a target. Overall, the lattice framework exposes rich, dimension- and position-dependent MFPT behavior and stationary states that extend and sometimes qualitatively differ from continuum predictions, with potential implications for stochastic search and transport on discrete media.

Abstract

We provide an exact formula for the mean first-passage time (MFPT) to a target at the origin for a single particle diffusing on a $d$-dimensional hypercubic {\em lattice} starting from a fixed initial position $\vec R_0$ and resetting to $\vec R_0$ with a rate $r$. Previously known results in the continuous space are recovered in the scaling limit $r\to 0$, $R_0=|\vec R_0|\to \infty$ with the product $\sqrt{r}\, R_0$ fixed. However, our formula is valid for any $r$ and any $\vec R_0$ that enables us to explore a much wider region of the parameter space that is inaccessible in the continuum limit. For example, we have shown that the MFPT, as a function of $r$ for fixed $\vec R_0$, diverges in the two opposite limits $r\to 0$ and $r\to \infty$ with a unique minimum in between, provided the starting point is not a nearest neighbour of the target. In this case, the MFPT diverges as a power law $\sim r^φ$ as $r\to \infty$, but very interestingly with an exponent $φ= (|m_1|+|m_2|+\ldots +|m_d|)-1$ that depends on the starting point $\vec R_0= a\, (m_1,m_2,\ldots, m_d)$ where $a$ is the lattice spacing and $m_i$'s are integers. If, on the other hand, the starting point happens to be a nearest neighbour of the target, then the MFPT decreases monotonically with increasing $r$, approaching a universal limiting value $1$ as $r\to \infty$, indicating that the optimal resetting rate in this case is infinity. We provide a simple physical reason and a simple Markov-chain explanation behind this somewhat unexpected universal result. Our analytical predictions are verified in numerical simulations on lattices up to $50$ dimensions. Finally, in the absence of a target, we also compute exactly the position distribution of the walker in the nonequlibrium stationary state that also displays interesting lattice effects not captured by the continuum theory.

Figures (7)

• Figure 1: $\langle T\rangle_r(m_1)$ vs. $r$ in one dimension ($d=1$) for $m_1=1$, $m_1=2$ and $m_1=3$. While for any $m_1>1$, the curve exhibits a unique minimum at some $r=r^*(|m|)$, for $m_1=1$ it decays monotonically with increasing $r$, approaching the limiting value $1$ as $r\to \infty$. The solid lines correspond to the analytical formula in Eq. (\ref{['mfpt_d1.1']}).
• Figure 2: $\langle T\rangle_r(1,0)$ and $\langle T\rangle_r(1,1)$ vs. $r$ in two dimensions. The solid line shows the analytical results in Eqs. (\ref{['mfpt_d210.2']}) and (\ref{['mfpt_d211.2']}), while the symbols represent the numerical results. The agreement is excellent.
• Figure 3: $\langle T\rangle_r(1,1)$ vs. $r$ and $\langle T\rangle_r(2,1)$ vs. $r$ in two dimensions. The solid lines represent the asymptotic large $r$ behavior predicted analytically in Eq. (\ref{['large_r_asymp']}), while the symbols represent the numerical results. The agreement is excellent for large $r$.
• Figure 4: The simulation results $\langle T\rangle_r(2,0,0,\ldots, 0)$ vs. $r$ in dimensions $d=1$ to $d=6$. The minima represent the optimal resetting rate $r^*$. The inset shows $r^*$ as a function of $d$ for some values $d\le 50$ which seems to fit at larger value of $d$ well with the linear growth $r^*(d)=2 d$ predicted in Eq. (\ref{['r_opt_2']}).
• Figure 5: The number $n_{\rm final}(r^*)$ of steps the walker takes since the final, i.e. successful, reset until reaching the target, measured at the optimium reset rate $r^*$, shown as function of the lattice dimension $d$. Here the target is located at $m=(2,0,\ldots,0)$. The inset shows $n_{\rm final}$ as function of resetting rate $r$ for dimension $d=2$.