Target search optimization by threshold resetting

TL;DR

A unified framework to compute mean search times for these correlated stochastic processes is developed, with ballistic and diffusive searchers as key examples uncovering diverse optimization behaviors.

Abstract

We introduce a new class of first passage time optimization driven by threshold resetting, inspired by many natural processes where crossing a critical limit triggers failure, degradation or transition. In here, search agents are collectively reset when a threshold is reached, creating event-driven, system-coupled simultaneous resets that induce long-range interactions. We develop a unified framework to compute search times for these correlated stochastic processes, with ballistic- and diffusive- searchers as key examples uncovering diverse optimization behaviors. A cost function, akin to breakdown penalties, reveals that optimal resetting can forestall larger losses. This formalism generalizes to broader stochastic systems with multiple degrees of freedom.

Figures (4)

• Figure 1: Schematic representation of the TR mechanism with $N$ searchers (here, $N=3$) starting from $x_0$ and undergoing generic stochastic dynamics. The process is considered complete when any of the searchers reaches the target located at the origin, and the corresponding first passage time is denoted by $\mathcal{T}_N^\text{TR}(L,x_0)$. However, if any searcher reaches the resetting threshold at position $L$ before the target is found, all searchers are collectively returned to their initial positions, and the search process is restarted.
• Figure 2: Panel (a): Variation of the scaled MFPT (\ref{['scaling']}) with respect to $u$ and $N$. The plot shows a global minima in the MFPT at $u=1$ for any $N>1$. A cross section for $N=7$ is shown in panel (b). Panel (b) - central figure: Optimization of $\langle \overline{\mathcal{T}}_N^\text{TR}(u) \rangle$ as a function of $u$ (shown by the solid blue curve) -- superimposed with individual components as given by the RHS of (\ref{['fp-2']}) namely $\langle {\mathcal{N}}_{\text{TR}} (u,N) \rangle$ (shown by orange dotted-dashed line), $\langle \overline{\mathbb{T}}_N^{\text{TR}}(u) \rangle$ (shown by green dashed line) and $\langle \overline{\mathbb{T}}_N^{\text{FP}}(u) \rangle$ (shown by red dotted line). Simulation results are shown by the cross markers demonstrating an excellent match. The accompanying subpanels showcase typical stochastic trajectories (with $x$ and $t$ standing for the position and time, respectively)
• Figure 3: Behavior of the cost function with respect to $u=x_0/L$ for $N=3$ and $\beta=1$. The cost attains a minimum at an optimal $u^*$ -- the variation of which as a function of $N$ is shown in the inset panel. The squares represent data from simulations.
• Figure 4: Diffusive searchers under TR: Mean first-passage time (MFPT) as a function of the threshold parameter $u$ for different values of $N$ as in Eq. (\ref{['mfpt-diff']}). The horizontal dashed lines (for $N \geq 3$) represent the MFPT in the absence of threshold resetting, corresponding to the $u \to 0$ limit (note that the MFPTs for $N=1,2$ in this limit are diverging). The fact that the MFPT curves fall below these baselines highlights the enhanced search efficiency achieved through the TR mechanism, also in the case of diffusive searchers. The markers represent the simulation results.