We study a one-dimensional gas of $N$ Brownian particles that diffuse independently, but are {\it simultaneously} reset to the origin at a constant rate $r$. The system approaches a non-equilibrium stationary state (NESS) with long-range interactions induced by the simultaneous resetting. Despite the presence of strong correlations, we show that several observables can be computed exactly, which include the global average density, the distribution of the position of the $k$-th rightmost particle and the spacing distribution between two successive particles. Our analytical results are confirmed by numerical simulations. We also discuss a possible experimental realisation of this resetting gas using optical traps. 

/pdf/extreme-statistics-and-spacing-distribution-in-a-brownian-2n0bderx.pdf

Extreme Statistics and Spacing Distribution in a Brownian Gas Correlated by Resetting

How do near-bankruptcy events in the past affect the dynamics of stock-market prices in the future? Specifically, what are the long-time properties of a time-local exponential growth of stock-market prices under the influence of stochastically occurring economic crashes? Here, we derive the ensemble- and time-averaged properties of the respective "economic" or geometric Brownian motion (GBM) with a nonzero drift exposed to a Poissonian constant-rate price-restarting process of "resetting." We examine-based both on thorough analytical calculations and on findings from systematic stochastic computer simulations-the general situation of reset GBM with a nonzero [positive] drift and for all special cases emerging for varying parameters of drift, volatility, and reset rate in the model. We derive and summarize all short- and long-time dependencies for the mean-squared displacement (MSD), the variance, and the mean time-averaged MSD (TAMSD) of the process of Poisson-reset GBM under the conditions of both rare and frequent resetting. We consider three main regions of model parameters and categorize the crossovers between different functional behaviors of the statistical quantifiers of this process. The analytical relations are fully supported by the results of computer simulations. In particular, we obtain that Poisson-reset GBM is a nonergodic stochastic process, with generally MSD(Δ)≠TAMSD(Δ) and Variance(Δ)≠TAMSD(Δ) at short lag times Δ and for long trajectory lengths T. We investigate the behavior of the ergodicity-breaking parameter in each of the three regions of parameters and examine its dependence on the rate of reset at Δ/T≪1. Applications of these theoretical results to the analysis of prices of reset-containing options are pertinent.

Time-averaging and nonergodicity of reset geometric Brownian motion with drift.

We review recent work on systems with multiple interacting-particles having the dynamical feature of stochastic resetting. The interplay of time scales related to inter-particle interactions and resetting leads to a rich behaviour, both static and dynamic. The presence of multiple particles also opens up a new possibility for the resetting dynamics itself, namely, that of different particles resetting all together (global resetting) or independently (local resetting). We divide the review on the basis of specifics of reset dynamics (global versus local resetting), and further, on the basis of number (two versus a large number) of interacting particles. We will primarily be dealing with classical systems, and only briefly discuss resetting in quantum systems.

/pdf/stochastic-resetting-in-interacting-particle-systems-a-2vplzf5r.pdf

Stochastic resetting in interacting particle systems: a review

During a random search, resetting the searcher’s position from time to time to the starting point often reduces the mean completion time of the process. Although many different resetting models have been studied over the past ten years, only a few can be physically implemented. Here we study theoretically a protocol that can be realised experimentally and which exhibits unusual optimization properties. A Brownian particle is subject to an arbitrary confining potential v(x) that is switched on and off intermittently at fixed rates. Motion is constrained between an absorbing wall located at the origin and a reflective wall. When the walls are sufficiently far apart, the interplay between free diffusion during the ‘off’ phases and attraction toward the potential minimum during the ‘on’ phases give rise to rich behaviours, not observed in ideal resetting models. For potentials of the form v(x) = k|x − x 0| n /n, with n > 0, the switch-on and switch-off rates that minimise the mean first passage time (MFPT) to the origin undergo a continuous phase transition as the potential stiffness k is varied. When k is above a critical value k c, potential intermittency enhances the target encounter: the minimal MFPT is lower than the Kramer time and is attained for a non-vanishing pair of switching rates. We focus on the harmonic case n = 2, extending previous results for the piecewise linear potential (n = 1) in unbounded domains. We also study the non-equilibrium stationary states emerging in this process.

/pdf/reducing-mean-first-passage-times-with-intermittent-1e71w74c.pdf

Reducing mean first passage times with intermittent confining potentials: a realization of resetting processes

We propose a universal framework to compute record age statistics of a stochastic time series that undergoes random restarts. The proposed framework makes minimal assumptions on the underlying process and is furthermore suited to treat generic restart protocols going beyond the Markovian setting. After benchmarking the framework for classical random walks on the 1D lattice, we derive a universal criterion underpinning the impact of restart on the age of the nth record for generic time series with nearest-neighbor transitions. Crucially, the criterion contains a penalty of order n that puts strong constraints on restart expediting the creation of records, as compared to the simple first-passage completion. The applicability of our approach is further demonstrated on an aggregation-shattering process where we compute the typical growth rates of aggregate sizes. This unified framework paves the way to explore record statistics of time series under restart in a wide range of complex systems. 

/pdf/universal-framework-for-record-ages-under-restart-ez9c3smo.pdf

Universal Framework for Record Ages under Restart

Stochastic processes offer a fundamentally different paradigm of dynamics than deterministic processes, the most prominent example of the latter being Newton's laws of motion. Here, we discuss in a pedagogical manner a simple and illustrative example of stochastic processes in the form of a particle undergoing standard Brownian diffusion, with the additional feature of the particle resetting repeatedly and at random times to its initial condition. Over the years, many different variants of this simple setting have been studied, all of which serve as illustrations of non-trivial and interesting static and dynamic features that characterize stochastic dynamics at long times. We will provide in this work a brief overview of this active and rapidly evolving field by considering the arguably simplest example of Brownian diffusion in one dimension. Along the way, we will learn about some of the general techniques that a physicist employs to study stochastic processes. Relevant to the special issue, we will discuss in detail how introducing resetting in an otherwise diffusive dynamics provides an explicit optimization of the time to locate a target through a special choice of the resetting protocol. We also discuss thermodynamics of resetting, and provide a bird's eye view of some of the recent work in the field of resetting. 

/pdf/stochastic-resetting-a-very-brief-review-1971jqub.pdf

Stochastic Resetting: A (Very) Brief Review

In the context of unitary evolution of a generic quantum system interrupted at random times with non-unitary evolution due to interactions with either the external environment or a measuring apparatus, we adduce a general theoretical framework to obtain the average density operator of the system at any time during the dynamical evolution. The average is with respect to the classical randomness associated with the random time intervals between successive interactions, which we consider to be independent and identically-distributed random variables. We provide two explicit applications of the formalism in the context of the so-called tight-binding model relevant in various contexts in solid-state physics. In one dimension, the corresponding tight-binding chain models the motion of a charged particle between the sites of a lattice, wherein the particle is for most times localized on the sites, but which owing to spontaneous quantum fluctuations tunnels between nearest-neighbour sites. We consider two representative forms of interactions: stochastic reset of quantum dynamics, in which the density operator is at random times reset to its initial form, and projective measurements performed on the system at random times. In the former case, we demonstrate with our exact results how the particle is localized on the sites at long times, leading to a time-independent mean-squared displacement of the particle about its initial location. In the case of projective measurements at random times, we show that repeated projection to the initial state of the particle results in an effective suppression of the temporal decay in the probability of the particle to be found on the initial state. The amount of suppression is comparable to the one in conventional Zeno effect scenarios, but which however does not require performing measurements at exactly regular intervals that are hallmarks of such scenarios. 

Quantum unitary evolution interspersed with repeated non-unitary interactions at random times: the method of stochastic Liouville equation, and two examples of interactions in the context of a tight-binding chain

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Stochastic Resetting: A (Very) Brief Review

Quantum unitary evolution interspersed with repeated non-unitary interactions at random times: the method of stochastic Liouville equation, and two examples of interactions in the context of a tight-binding chain