Stochastic resetting and large deviations

We consider closed quantum many-body systems subject to stochastic resetting. This means that their unitary time evolution is interrupted by resets at randomly selected times. When a reset takes place the system is reinitialized to a state chosen from a set of reset states conditionally on the outcome of a measurement taken immediately before resetting. We construct analytically the resulting non-equilibrium stationary state, thereby establishing an explicit connection between quantum quenches in closed systems and the emergent open system dynamics induced by stochastic resetting. We discuss as an application the paradigmatic transverse-field quantum Ising chain. We show that signatures of its ground-state quantum phase transition are visible in the steady state of the reset dynamics as a sharp crossover. Our findings show that a controlled stochastic resetting dynamics allows to design non-equilibrium stationary states of quantum many-body systems, where uncontrolled dissipation and heating can be prevented. These states can thus be created on demand and exploited, e.g., as a resource for quantum enhanced sensing on quantum simulator platforms.

Markov processes with stochastic resetting towards the origin generically converge towards non-equilibrium steady-states. Long dynamical trajectories can be thus analyzed via the large deviations at level 2.5 for the joint probability of the empirical density and the empirical flows, or via the large deviations of semi-Markov processes for the empirical density of excursions between consecutive resets. The large deviations properties of general time-additive observables involving the position and the increments of the dynamical trajectory are then analyzed in terms of the appropriate Markov tilted processes and of the corresponding conditioned processes obtained via the generalization of Doob’s h-transform. This general formalism is described in detail for the three possible frameworks, namely discrete-time/discrete-space Markov chains, continuous-time/discrete-space Markov jump processes and continuous-time/continuous-space diffusion processes, and is illustrated with explicit results for the Sisyphus random walk and its variants, when the reset probabilities or reset rates are space-dependent.

In this Topical Review we consider stochastic processes under resetting, which have attracted a lot of attention in recent years. We begin with the simple example of a diffusive particle whose position is reset randomly in time with a constant rate $r$, which corresponds to Poissonian resetting, to some fixed point (e.g. its initial position). This simple system already exhibits the main features of interest induced by resetting: (i) the system reaches a nontrivial nonequilibrium stationary state (ii) the mean time for the particle to reach a target is finite and has a minimum, optimal, value as a function of the resetting rate $r$. We then generalise to an arbitrary stochastic process (e.g. Levy flights or fractional Brownian motion) and non-Poissonian resetting (e.g. power-law waiting time distribution for intervals between resetting events). We go on to discuss multiparticle systems as well as extended systems, such as fluctuating interfaces, under resetting. We also consider resetting with memory which implies resetting the process to some randomly selected previous time. Finally we give an overview of recent developments and applications in the field.

Motion under stochastic resetting serves to model a myriad of processes in physics and beyond, but in most cases studied to date resetting to the origin was assumed to take zero time or a time decoupled from the spatial position at the resetting moment. However, in our world, getting from one place to another always takes time and places that are further away take more time to be reached. We thus set off to extend the theory of stochastic resetting such that it would account for this inherent spatio-temporal coupling. We consider a particle that starts at the origin and follows a certain law of stochastic motion until it is interrupted at some random time. The particle then returns to the origin via a prescribed protocol. We study this model and surprisingly discover that the shape of the steady-state distribution which governs the stochastic motion phase does not depend on the return protocol. This shape invariance then gives rise to a simple, and generic, recipe for the computation of the full steady state distribution. Several case studies are analyzed and a class of processes whose steady state is completely invariant with respect to the speed of return is highlighted. For processes in this class we recover the same steady-state obtained for resetting with instantaneous returns—irrespective of whether the actual return speed is high or low. Our work significantly extends previous results on motion with stochastic resetting and is expected to find various applications in statistical, chemical, and biological physics.

In this work, we investigate a series of mathematical aspects for the fractional diffusion equation with stochastic resetting. The stochastic resetting process in Evans–Majumdar sense has several applications in science, with a particular emphasis on non-equilibrium physics and biological systems. We propose a version of the stochastic resetting theory for systems in which the reset point is in motion, so the walker does not return to the initial position as in the standard model, but returns to a point that moves in space. In addition, we investigate the proposed stochastic resetting model for diffusion with the fractional operator of Prabhakar. The derivative of Prabhakar consists of an integro-differential operator that has a Mittag–Leffler function with three parameters in the integration kernel, so it generalizes a series of fractional operators such as Riemann–Liouville–Caputo. We present how the generalized model of stochastic resetting for fractional diffusion implies a rich class of anomalous diffusive processes, i.e., ⟨ ( Δ x ) 2 ⟩ ∝ t α , which includes sub-super-hyper-diffusive regimes. In the sequence, we generalize these ideas to the fractional Fokker–Planck equation for quadratic potential U ( x ) = a x 2 + b x + c . This work aims to present the generalized model of Evans–Majumdar’s theory for stochastic resetting under a new perspective of non-static restart points.

Many chemical reactions can be formulated in terms of particle diffusion in a complex energy landscape. Transition path theory (TPT) is a theoretical framework for describing the direct (reaction) pathways from reactant to product states within this energy landscape, and calculating the effective reaction rate. It is now the standard method for analyzing rare events between long lived states. In this paper, we consider a completely different application of TPT, namely, a dual-aspect diffusive search process in which a particle alternates between collecting cargo from a source domain A and then delivering it to a target domain B. The rate of resource accumulation at the target, k AB , is determined by the statistics of direct (reactive or transport) paths from A to B. Rather than considering diffusion in a complex energy landscape, we focus on pure diffusion with stochastic resetting. Resetting introduces two non-trivial problems in the application of TPT. First, the process is not time-reversal invariant, which is reflected by the fact that there exists a unique non-equilibrium stationary state (NESS). Second, calculating k AB involves determining the total probability flux of direct transport paths across a dividing surface S between A and B. This requires taking into account discontinuous jumps across S due to resetting. We derive a general expression for k AB and show that it is independent of the choice of dividing surface. Finally, using the example of diffusion in a finite interval, we show that there exists an optimal resetting rate at which k AB is maximized. We explore how this feature depends on model parameters.

Properties of stochastic systems are defined by the noise type and deterministic forces acting on the system. In out-of-equilibrium setups, e.g., for motions under action of L\'evy noises, the existence of the stationary state is not only determined by the potential but also by the noise. Potential wells need to be steeper than parabolic in order to assure existence of stationary states. The existence of stationary states, in sub-harmonic potential wells, can be restored by stochastic resetting, which is the protocol of starting over at random times. Herein we demonstrate that the combined action of L\'evy noise and Poissonian stochastic resetting can result in the phase transition between non-equilibrium stationary states of various multimodality in the overdamped system in super-harmonic potentials. Fine-tuned resetting rates can increase the modality of stationary states, while for high resetting rates the multimodality is destroyed as the stochastic resetting limits the spread of particles.

Properties of stochastic systems are defined by the noise type and deterministic forces acting on the system. In out-of-equilibrium setups, e.g., for motions under action of Lévy noises, the existence of the stationary state is not only determined by the potential but also by the noise. Potential wells need to be steeper than parabolic in order to assure the existence of stationary states. The existence of stationary states, in sub-harmonic potential wells, can be restored by stochastic resetting, which is the protocol of starting over at random times. Herein, we demonstrate that the combined action of Lévy noise and Poissonian stochastic resetting can result in the phase transition between non-equilibrium stationary states of various multimodality in the overdamped system in super-harmonic potentials. Fine-tuned resetting rates can increase the modality of stationary states, while for high resetting rates, the multimodality is destroyed as the stochastic resetting limits the spread of particles.

It is known that the introduction of stochastic resetting in an uncorrelated random walk process can lead to the emergence of a stationary state, i.e. the diffusion evolves towards a saturation state, and a steady Laplace distribution is reached. In this paper, we turn to study the anomalous diffusion of the correlated continuous-time random walk considering stochastic resetting. Results reveal that it displays quite different diffusive behaviors from the uncorrelated one. For the weak correlation case, the stochastic resetting mechanism can slow down the diffusion. However, for the strong correlation case, we find that the stochastic resetting cannot compete with the space-time correlation, and the diffusion presents the same behaviors with the one without resetting. Meanwhile, a steady distribution is never reached.

This study examines the dynamics of a tracer particle diffusing in a nonequilibrium medium under stochastic resetting. The nonequilibrium state is induced by harmonic coupling between the tracer and bath particles, generating memory effects with an exponential decay in time. We explore the tracer's behavior under a Poissonian resetting protocol, where resetting does not disturb the bath environment, with a focus on key dynamical behavior and first-passage properties, both in the presence and absence of an external force. The interplay between coupling strength and diffusivity of bath particles significantly impacts both the tracer's relaxation dynamics and search time, with external forces further modulating these effects. Our analysis identifies distinct hot and cold bath particles based on their diffusivities, revealing that coupling to a hot particle facilitates the searching process, whereas coupling to a cold particle hinders it. Using a combination of numerical simulations and analytical methods, this study provides a comprehensive framework for understanding resetting mechanisms in non-Markovian systems, with potential applications to complex environments such as active and viscoelastic media, where memory-driven dynamics and nonequilibrium interactions are significant.

We consider a shear-driven anomalous diffusion by introducing a memory kernel in the Fokker–Planck equation, which results from the long-tailed waiting time of the particle. We analyze the probability density function and the corresponding moments in the framework of the subordination approach. The moments, obtained analytically, show that the system exhibits characteristic crossover anomalous dynamics. We also explore corresponding process under stochastic resetting, and we find that the system reaches a non-equilibrium stationary state in the long time limit that also results in saturation of the evolution of corresponding mean squared displacement, variance, skewness, and kurtosis.

In this paper, we introduce a new stochastic process of N interacting particles on the line that evolve via Dyson Brownian motion (DBM) with Dyson's index β>0 and undergo simultaneous resetting to their initial positions at a constant rate r. We call this process the resetting Dyson Brownian motion (RDBM) with a parameter β>0, in short the β-RDBM process. For β=1,2,4, the positions of the particles in the RDBM can be interpreted as the eigenvalues of a new random matrix ensemble where the entries of an N×N Gaussian matrix evolve as simultaneously resetting Brownian motions (with rate r) in the presence or absence of a harmonic trap. For r=0 and in the presence of a harmonic trap, this system reaches an equilibrium Gibbs-Boltzmann state of the so called Dyson log-gas. However, the stochastic resetting drives the system at long time to a nonequilibrium stationary state (NESS). We compute exactly the joint distribution of the positions of the particles in this NESS for all β>0 and calculate several macroscopic and microscopic observables in the large N limit. These include the average density profile of the gas, the extreme value statistics, the spacing between two consecutive particles and the full counting statistics, i.e., the distribution of the number of particles in an interval [-L,L]. We show that a nonzero resetting rate r>0 drastically changes the nature of the fluctuations in the stationary state: while the log-gas (r=0) is rather rigid, the β-RDBM in its NESS becomes fluffy, i.e., the fluctuations of different observables are of the same order as their mean. In the absence of a harmonic trap, our results for the β=2 RDBM can be related to nonintersecting Brownian motions (vicious walkers) in the presence of resetting. Our model demonstrates interesting effects arising from the interplay between the eigenvalue repulsion and the all-to-all attraction (generated by stochastic resetting) in an interacting particle system. Numerical simulations are in excellent agreement with our analytical results.

Stochastic resetting and noise-enhanced stability are two phenomena that can affect the lifetime and relaxation of nonequilibrium states. They can be considered measures of controlling the efficiency of the completion process when a stochastic system has to reach the desired state. Here, we study the interaction of random (Poissonian) resetting and stochastic dynamics in unstable potentials. Unlike noise-induced stability that increases the relaxation time, the stochastic resetting may eliminate winding trajectories contributing to the lifetime and accelerate the escape kinetics from unstable states. In this paper, we present a framework to analyze compromises between the two contrasting phenomena in noise-driven kinetics subject to random restarts.

The voter model is a toy model of consensus formation based on nearest-neighbor interactions. A voter sits at each vertex in a hypercubic lattice (of dimension d) and is in one of two possible opinion states. The opinion state of each voter flips randomly, at a rate proportional to the fraction of the nearest neighbors that disagree with the voter. If the voters are initially independent and undecided, the model is known to lead to a consensus if and only if d⩽2 . In this paper the model is subjected to stochastic resetting: the voters revert independently to their initial opinion according to a Poisson process of fixed intensity (the resetting rate). This resetting prescription induces kinetic equations for the average opinion state and for the two-point function of the model. For initial conditions consisting of undecided voters except for one decided voter at the origin, the one-point function evolves as the probability of presence of a diffusive random walker on the lattice, whose position is stochastically reset to the origin. The resetting prescription leads to a non-equilibrium steady state. For an initial state consisting of independent undecided voters, the density of domain walls in the steady state is expressed in closed form as a function of the resetting rate. This function is differentiable at zero if and only if d⩾5 .

We investigate the dynamics of a non-interacting spin system, undergoing coherent Rabi oscillations, in the presence of stochastic resetting. We show that resetting generally induces long-range quantum and classical correlations both in the emergent dissipative dynamics and in the non-equilibrium stationary state. Moreover, for the case of conditional reset protocols -- where the system is reinitialized to a state dependent on the outcome of a preceding measurement -- we show that, in the thermodynamic limit, the spin system can feature collective behavior which results in a phenomenology reminiscent of that occurring in non-equilibrium phase transitions. The discussed reset protocols can be implemented on quantum simulators and quantum devices that permit fast measurement and readout of macroscopic observables, such as the magnetisation. Our approach does not require the control of coherent interactions and may therefore highlight a route towards a simple and robust creation of quantum correlations and collective non-equilibrium states, with potential applications in quantum enhanced metrology and sensing.