The many facets of stochastic resetting in first-passage and non-equilibrium processes [HBNI Th281]

Abstract

The focus of this thesis was threefold: i) application and effects of stochastic resetting on physical in a variety of processes, namely, chemical reactions, operations research and viscoelastic media, ii) a unified renewal approach to experimental realisation of resetting, and iii) proposing a novel FP optimization strategy based on threshold resetting. In the following, we summarise the primary results emanated out of this thesis.

The Chapters 2–4 are dedicated to the understanding of applications of resetting in several fields. In Chapter 2, we study how a gated chemical reaction can be facilitated with resetting. Resetting a chemical reaction refers to the unbinding of the enzyme-substrate complex, after which the reaction is started afresh. As a primary result, we find, a certain unbinding rate can indeed expedite the product formation of a chemical reaction, especially when the target is ‘gated’ in nature, which models the active-inactive phase of the substrate. We modelled the chemical reaction with a one-dimensional drift-diffusion process with a gated target. By doing so, we find a parameter space in terms of diffusion and drift coefficients where the reaction rate can be enhanced with stochastic resetting.

In Chapter 3, we discuss how service resetting can reduce the queue length. Considering the vast fields of applications of queuing theory, strategies to effectively manage a queue remain a pivotal challenge. With analytical formulations, we explicitly find the mean queue length of an M/G/1 queue subjected to the service resetting strategy, where resetting consumes a random period of overhead time. We show that depending on the fluctuations of the overhead time, one can suitably choose an optimal resetting rate that can reduce the mean queue length to its best. Furthermore, we find that periodic resetting does a better optimization in contrast to resetting at random times.

In Chapter 4 we study the effects of resetting in a particle embedded in a viscoelastic medium. Viscoelastic medium are different from the traditional viscous medium since the collisions between the particles are not forgotten instantaneously. Due to the memory dependence of the particle’s trajectory, the system is highly non-Markovian in nature. Our study reveals that, resetting a particle in such a medium generates non-trivial non-equilibrium steady states and correlations. It also gives rise to a resetting rate-dependent timescale in the system that can help the external agent to gain control over the dynamics of the particle in such a highly non-Markovian system.

In Chapter 5, we propose an experimentally amenable approach to stochastic resetting. Although the idea of resetting being a subject to extensive studies, a majority of them (along with the models considered in the previous section) consider resetting to be an instantaneous event. In reality, a resetting event is space-time coupled so that a finite time is consumed when the particle heads back to the starting position. We propose a universal framework of stochastic resetting where the agent returns to the starting position in a space-time coupled non-instantaneous fashion. By considering the paradigmatic example of diffusion, we show that a stochastic component in the return motion of the particle can indeed enhance the search efficiency in a FP process. Furthermore, we corroborate our theory with experimental parameters and identify suitable parameter regimes where stochastic return can indeed be beneficial than the classical instantaneous return. By adding a drift to the particle dynamics, we also show that stochastic return can outperform the underlying reset-free process as well as the optimal instantaneous return protocol. In summary, we demonstrate that search with stochastic returns provides not only a physically realizable approach to resetting but also can be more advantageous.

In Chapter 6, we propose a novel approach to the collective search process optimization based on threshold resetting. Under this set-up, the system resets only when any one of many searchers hits a pre-defined threshold value. Although the resetting epochs are still stochastic in time, it is now governed by some space-dependent threshold, leading to an event-driven process resetting strategy. We explicitly analyse the effect of such a resetting strategy in a multi-agent search process. Note that all the search agents are reset to the initial position at each resetting epoch, which introduces long-range correlations among the searchers. We show that the mean FPT of the process shows rich optimisation features with respect to the threshold distance and the number of searchers. We also quantify the cost of maintaining such a threshold-dependent search process and find an interesting optimisation of the cost function.

Despite the extensive progress achieved through the studies presented in this thesis, stochastic resetting continues to be a fertile ground for exploration. Its remarkable ability to induce nonequilibrium steady states, optimise search and reaction efficiencies, and generate novel dynamical correlations underscores its fundamental and applied importance. Future studies could explore resetting in systems with spatial or temporal disorder, active matter, and biological networks, where feedback-controlled or adaptive resetting rules may yield richer dynamics. Moreover, extending resetting concepts to quantum systems, machine learning algorithms, or complex supply-chain models could open exciting interdisciplinary avenues. The insights obtained here not only advance our understanding of resetting as a universal control mechanism but also lay the groundwork for its potential exploitation in experimental and technological contexts—from microscale transport processes to large-scale optimisation problems.