Optimization and Growth in First-Passage Resetting

TL;DR

This paper introduces first-passage resetting, analyzing its effects on diffusion, optimizing system operation under reset constraints, and exploring growth dynamics of domains influenced by resetting events.

Contribution

It defines and analyzes first-passage resetting, providing analytical solutions, optimization frameworks, and growth models across different geometries.

Findings

The spatial distribution of particles under first-passage resetting is derived analytically.

Optimal reset strategies balance system performance and failure penalties.

Domain growth rates depend on boundary recession rules and resetting dynamics.

Abstract

We combine the processes of resetting and first-passage to define \emph{first-passage resetting}, where the resetting of a random walk to a fixed position is triggered by a first-passage event of the walk itself. In an infinite domain, first-passage resetting of isotropic diffusion is non-stationary, with the number of resetting events growing with time as $\sqrt{t}$. We calculate the resulting spatial probability distribution of the particle analytically, and also obtain this distribution by a geometric path decomposition. In a finite interval, we define an optimization problem that is controlled by first-passage resetting; this scenario is motivated by reliability theory. The goal is to operate a system close to its maximum capacity without experiencing too many breakdowns. However, when a breakdown occurs the system is reset to its minimal operating point. We define and optimize an objective function that maximizes the reward (being close to maximum operation) minus a penalty for each breakdown. We also investigate extensions of this basic model to include delay after each reset and to two dimensions. Finally, we study the growth dynamics of a domain in which the domain boundary recedes by a specified amount whenever the diffusing particle reaches the boundary after which a resetting event occurs. We determine the growth rate of the domain for the semi-infinite line and the finite interval and find a wide range of behaviors that depend on how much the recession occurs when the particle hits the boundary.