Optimal threshold resetting in collective diffusive search

Abstract

Stochastic resetting has attracted significant attention in recent years due to its wide-ranging applications across physics, biology, and search processes. In most existing studies, however, resetting events are governed by an external timer and remain decoupled from the system's intrinsic dynamics. In a recent Letter by Biswas et al, we introduced threshold resetting (TR) as an alternative, event-driven optimization strategy for target search problems. Under TR, the entire process is reset whenever any searcher reaches a prescribed threshold, thereby coupling the resetting mechanism directly to the internal dynamics. In this work, we study TR-enabled search by $N$ non-interacting diffusive searchers in a one-dimensional box $[0,L]$, with the target at the origin and the threshold at $L$. By optimally tuning the scaled threshold distance $u = x_0/L$, the mean first-passage time can be significantly reduced for $N \geq 2$. We identify a critical population size $N_c(u)$ below which TR outperforms reset-free dynamics. Furthermore, for fixed $u$, the mean first-passage time depends non-monotonically on $N$, attaining a minimum at $N_{\mathrm{opt}}(u)$. We also quantify the achievable speed-up and analyze the operational cost of TR, revealing a nontrivial optimization landscape. These findings highlight threshold resetting as an efficient and realistic optimization mechanism for complex stochastic search processes.

Figures (6)

• Figure 1: Schematic representation of threshold resetting with $N=4$ non-interacting diffusive searchers. If any one of them reaches the target at $x=0$ we mark the process as complete and note the associated first passage time $\mathcal{T}_N^{\text{TR}}(L,x_0)$. However, if any of them reaches the threshold at $L$ first, then all of them are simultaneously reset to $x_0$ from where they renew their search. Our aim is to compute the MFPT to the target by these diffusive searchers.
• Figure 2: Panel (a) shows the variation of the non-dimensionalized MFPT $\langle \overline{\mathcal{T}}_N^{\text{TR}}(u) \rangle$ as a function of $u=x_0/L$ for various values of $N$ as in Eq. (\ref{['mfpt-diff']}). For any values of $N>1$, the curves show non-monotonic behaviour with respect to $u$ with the optimal MFPT being at $u=u_{\text{opt}}$. Panel (b) shows the variation of the point $u_{\text{opt}}$ with respect to $N$.
• Figure 3: Panel (a) shows the variation of the non-dimensionalized MFPT with the number of searchers $N$ for various values of $u$. The solid line represents the analytical results (Eq. (\ref{['mfpt-diff']})) and the markers represent results from simulation. The dashed line corresponds to the underlying process (i.e.$u\to 0$). Note that, for a fixed $u$, when the solid curves with $u \ne 0$ lie below the dashed curve, MFPT with TR turns out to be a favourable strategy than the underlying process. The intersection point where these two curves cross each other is denoted by $N_c(u)$. This is the critical number of searchers, below which TR serves as a better strategy than the underlying process. In panel (b) we show the variation of $N_c(u)$ with $u$ (the solid line). Evidently, when $N \le N_c(u)$, TR helps to expedite the collective search process (shown by the shaded region).
• Figure 4: Variation of the optimal number of searchers $N_{\text{opt}}(u)$ where the MFPT with TR is the lowest with respect to $u$. As a representative case, in the inset, we show the $N_\text{opt}(u=0.6)$ for marked by the star. The critical $u_c \approx 0.8$, beyond which the collective search becomes detrimental, is marked by an open circle.
• Figure 5: Variation of the optimal speed-up gained under threshold resetting mechanism with $N$. For the function values $>1$ (marked on the $y$-axis), TR optimizes the search process. The blue cross shows the variation of $\mathcal{S}_1(N)$, which quantifies the speed-up with gained with multiple searchers compared to a single searcher. The quantity $\mathcal{S}_2(N)$ shown by the red squares amounts to the speed-up gained with TR in comparison to the underlying reset-free search process.