Stochastic Resetting Research Articles

Generalized Itô’s lemma and the stochastic thermodynamics of diffusion with resetting

Abstract Methods from the theory of stochastic processes are increasingly being used to extend classical thermodynamics to mesoscopic non-equilibrium systems. One characteristic feature of these systems is that averaging the stochastic entropy with respect to an ensemble of stochastic trajectories leads to a second law of thermodynamics that quantifies the degree of departure from thermodynamic equilibrium. A well known mechanism for maintaining a diffusing particle out of thermodynamic equilibrium is stochastic resetting. In its simplest form, the position of the particle instantaneously resets to a fixed position x 0 at a sequence of times generated from a Poisson process of constant rate r. Within the context of stochastic thermodynamics, instantaneous resetting to a single point is a unidirectional process that has no time-reversed equivalent. Hence, the average rate of entropy production calculated using the Gibbs–Shannon entropy cannot be related to the degree of time-reversal symmetry breaking. The problem of unidirectionality can be avoided by considering resetting to a random position or diffusion in an intermittent confining potential. In this paper we show how stochastic entropy production along sample paths of diffusion processes with resetting can be analyzed in terms of extensions of Itô’s formula for stochastic differential equations (SDEs) that include both continuous and discrete processes. First, we use the stochastic calculus of jump-diffusion processes to calculate the rate of stochastic entropy production for instantaneous resetting, and show how previous results are recovered upon averaging over sample trajectories. Second, we formulate single-particle diffusion in a switching potential as a hybrid SDE and develop a hybrid extension of Itô’s stochastic calculus to derive a general expression for the rate of stochastic entropy production. We illustrate the theory by considering overdamped Brownian motion in an intermittent harmonic potential. Finally, we calculate the average rate of entropy production for a population of non-interacting Brownian particles moving in a common switching potential. In particular, we show that the latter induces statistical correlations between the particles, which means that the total entropy is not given by the sum of the 1-particle entropies.

Lattice random walk dynamics with stochastic resetting in heterogeneous space

Abstract We examine the diffusive dynamics of a lattice random walk subject to resetting in a one-dimensional spatially heterogeneous environment composed of two media separated by an interface. At random times the walker may reset its position to the interface, but only when in the left medium. In addition the spatial heterogeneity results from having unequal diffusivities and biases in the two media. We construct the Master equation for the dynamics of the walker occupation probability in unbounded space, solve it exactly in terms of generating functions, and analyse the dynamics of the first and second moment. Making use of the closed form solution in the unbounded case, we build the analytic solution of the Master equation in finite and semi-infinite domains. By bounding the space on the right with a reflecting boundary we study the first-passage dynamics to a single fully absorbing target placed in the left medium away from the interface. As reset strongly increases the time to reach the target, we find that the first-passage dynamics enter the motion-limited regime even for relative small resetting probability. We also identify a surprising non-monotonic dependence of the first-passage probability mode as a function of the bias. By deriving an analytic expression for the mean first-passage time, we show when its value is independent of the diffusivity and bias in the left medium, uncovering another example of the so-called mean disorder indifference phenomenon.

Thermodynamic cost of finite-time stochastic resetting

Recent experiments have implemented resetting by means of an external trap, whereby a system relaxes to the minimum of the trap and is reset in a finite time. In this work, we set up and analyze the thermodynamics of such a protocol. We present a general framework, valid even for non-Poissonian resetting, that captures the thermodynamic work required to maintain a resetting process up to a given observation time, and exactly calculate the moment generating function of this work. Our framework is valid for a wide range of systems, the only assumption being relaxation to equilibrium in the resetting trap. Examples and extensions are considered. In the case of Brownian motion, we investigate optimal resetting schemes that minimize work and its fluctuations, the mean work for arbitrary switching protocols, and comparisons to previously studied resetting schemes. Numerical simulations are performed to validate our findings. Published by the American Physical Society 2024

Emergence of vertical diversity under disturbance.

We propose a statistical physics model of a neutral community, where each agent can represent identical plant species growing in the vertical direction with sunlight in the form of rich-get-richer competition. Disturbance added to this ecosystem, which makes an agent restart from the lowest growth level, is realized as a stochastic resetting. We show that in this model for sufficiently strong competition, vertical diversity characterized by a family of Hill numbers robustly emerges as a local maximum at intermediate disturbance.

Siegmund duality for physicists: a bridge between spatial and first-passage properties of continuous- and discrete-time stochastic processes

Abstract We consider a generic one-dimensional stochastic process x(t), or a random walk Xn , which describes the position of a particle evolving inside an interval [ a , b ] , with absorbing walls located at a and b. In continuous time, x(t) is driven by some equilibrium process θ ( t ) , while in discrete time, the jumps of Xn follow a stationary process that obeys a time-reversal property. An important observable to characterize its behavior is the exit probability E b ( x , t ) , which is the probability for the particle to be absorbed first at the wall b, before or at time t, given its initial position x. In this paper we show that the derivation of this quantity can be tackled by studying a dual process y(t) very similar to x(t) but with hard walls at a and b. More precisely, we show that the quantity E b ( x , t ) for the process x(t) is equal to the probability Φ ~ ( x , t | b ) of finding the dual process inside the interval [ a , x ] at time t, with y ( 0 ) = b . This is known as Siegmund duality in mathematics. Here we show that this duality applies to various processes that are of interest in physics, including models of active particles, diffusing diffusivity models, a large class of discrete- and continuous-time random walks, and even processes subjected to stochastic resetting. For all these cases, we provide an explicit construction of the dual process. We also give simple derivations of this identity both in the continuous and in the discrete time setting, as well as numerical tests for a large number of models of interest. Finally, we use simulations to show that the duality is also likely to hold for more complex processes, such as fractional Brownian motion.

Crossover from anomalous to normal diffusion: Ising model with stochastic resetting

In this paper, we investigate the dynamics of the two-dimensional Ising model with stochastic resetting, utilizing a constant resetting rate procedure with zero-strength initial magnetization. Our results reveal the presence of a characteristic rate rc∼L−z, where L represents the system size and z denotes the dynamical exponent. Below rc, both the equilibrium and dynamical properties remain unchanged. At the same time, for r>rc, the resetting process induces a transition in the probability distribution of the magnetization from a double-peak distribution to a three-peak distribution, ultimately culminating in a single-peak exponential decay. Furthermore, we also find that at the critical points, as r increases, the diffusion of the magnetization changes from anomalous to normal, and the correlation time shifts from being dependent on L to being r-dependent only. Published by the American Physical Society 2024

Optimizing leapover lengths of Lévy flights with resetting.

We consider a one-dimensional search process under stochastic resetting conditions. A target is located at b≥0 and a searcher, starting from the origin, performs a discrete-time random walk with independent jumps drawn from a heavy-tailed distribution. Before each jump, there is a given probability r of restarting the walk from the initial position. The efficiency of a "myopic search"-in which the search stops upon crossing the target for the first time-is usually characterized in terms of the first-passage time τ. On the other hand, great relevance is encapsulated by the leapover length l=x_{τ}-b, which measures how far from the target the search ends. For symmetric heavy-tailed jump distributions, in the absence of resetting the average leapover is always infinite. Here we show instead that resetting induces a finite average leapover ℓ_{b}(r) if the mean jump length is finite. We compute exactly ℓ_{b}(r) and determine the condition under which resetting allows for nontrivial optimization, i.e., for the existence of r^{*} such that ℓ_{b}(r^{*}) is minimal and smaller than the average leapover of the single jump.

Ratcheting by Stochastic Resetting With Fat-Tailed Time Distributions.

We investigated both numerically and analytically the drift of a Brownian particle in a ratchet potential under stochastic resetting with fat-tailed distributions. As a study case we chose a Pareto time distribution with tail index β. We observed that for rectification occurs even if for the mean resetting time is infinite. However, for rectification is completely suppressed. For low noise levels, the drift speed attains a maximum for β immediately above 1, that is for finite but large mean resetting times. In correspondence with such an optimal drift the particle diffusion over the ratchet potential turns from normal to superdiffusive, a property also related to the fat tails of the resetting time distribution.

Fractional Telegrapher's Equation under Resetting: Non-Equilibrium Stationary States and First-Passage Times.

We consider two different time fractional telegrapher's equations under stochastic resetting. Using the integral decomposition method, we found the probability density functions and the mean squared displacements. In the long-time limit, the system approaches non-equilibrium stationary states, while the mean squared displacement saturates due to the resetting mechanism. We also obtain the fractional telegraph process as a subordinated telegraph process by introducing operational time such that the physical time is considered as a Lévy stable process whose characteristic function is the Lévy stable distribution. We also analyzed the survival probability for the first-passage time problem and found the optimal resetting rate for which the corresponding mean first-passage time is minimal.

Stochastic resetting can optimize the intermittent search strategy in a two-dimensional confined topography

Based on the intermittent two-state random walk model, by calculating and discussing the mean first capture time, the search efficiency of the intermittent search strategy in a two-dimensional confined topography in the absence and presence of stochastic resetting is studied. Results reveal that for the close target, the regular Brownian motion is the efficient search strategy, whereas, for the target far from the starting point, the typical intermittent search strategy with relocation times being power-law distributed turns out to be the advantageous one, and the search efficiency impressively enhances with the distance of the target from the starting point increasing. We demonstrate that though in the absence of stochastic resetting there does not exist a universal optimal search strategy in the confined topography, the introduction of stochastic resetting makes the Brownian motion surprisingly to be the universal optimal search process, and distinctly outperforms the ones without stochastic resetting.

Fractional heterogeneous telegraph processes: Interplay between heterogeneity, memory, and stochastic resetting.

Fractional heterogeneous telegraph processes are considered in the framework of telegrapher's equationsaccompanied by memory effects. The integral decomposition method is developed for the rigorous treating of the problem. Exact solutions for the probability density functions and the mean squared displacements are obtained. A relation between the fractional heterogeneous telegrapher's equationand the corresponding Langevin equationhas been established in the framework of the developed subordination approach. The telegraph process in the presence of stochastic resetting has been studied, as well. An exact expression for both the nonequilibrium stationary distributions/states and the mean squared displacements are obtained.

Hitting probabilities for fast stochastic search *

Many physical phenomena are modeled as stochastic searchers looking for targets. In these models, the probability that a searcher finds a particular target, its so-called hitting probability, is often of considerable interest. In this work we determine hitting probabilities for stochastic search processes conditioned on being faster than a random short time. Such times have been used to model stochastic resetting or stochastic inactivation. These results apply to any search process, diffusive or otherwise, whose unconditional short-time behavior can be adequately approximated, which we characterize for broad classes of stochastic search. We illustrate these results in several examples and show that the conditional hitting probabilities depend predominantly on the relative geodesic lengths between the initial position of the searcher and the targets. Finally, we apply these results to a canonical evidence accumulation model for decision making.

Effect of stochastic resettings on the counting of level crossings for inertial random processes.

We study the counting of level crossings for inertial random processes exposed to stochastic resetting events. We develop the general approach of stochastic resetting for inertial processes with sudden changes in the state characterized by position and velocity. We obtain the level-crossing intensity in terms of that of underlying reset-free process for resetting events with Poissonian statistics. We apply this result to the random acceleration process and the inertial Brownian motion. In both cases, we show that there is an optimal resetting rate that maximizes the crossing intensity, and we obtain the asymptotic behavior of the crossing intensity for large and small resetting rates. Finally, we discuss the stationary distribution and the mean first-arrival time in the presence of resettings.

Unbiased density computation for stochastic resetting *

We establish a practical means for unbiased computation of the marginal probability density function of the dynamics under stochastic resetting. In contrast to conventional dynamics devoid of resetting, the marginal probability density function under resetting may exhibit cusps and, in multi-dimensions, explosions at reset positions, arising from the compelled displacement of mass. Standard numerical techniques, such as kernel density estimation, fall short in accurately reproducing those characteristics due to their inherent smoothing effects. The proposed unbiased estimation formulas are derived using advanced stochastic calculus in their general formulations, yet their implementation in specific problem settings involves only elementary numerical operations, requiring minimal mathematical expertise and marking the very first instance of a numerical method free from bias in this context. We present numerical results throughout to validate the derived estimation formulas and, more broadly, to demonstrate the effectiveness of our approach in accurately capturing the irregularities of the marginal probability density function.

Active Brownian particle under stochastic orientational resetting

We employ renewal processes to characterize the spatiotemporal dynamics of an active Brownian particle under stochastic orientational resetting. By computing the experimentally accessible intermediate scattering function (ISF) and reconstructing the full time-dependent distribution of the displacements, we study the interplay of rotational diffusion and resetting. The resetting process introduces a new spatiotemporal regime reflecting the directed motion of agents along the resetting direction at large length scales, which becomes apparent in an imaginary part of the ISF. We further derive analytical expressions for the low-order moments of the displacements and find that the variance displays an effective diffusive regime at long times, which decreases for increasing resetting rates. At intermediate times the dynamics are characterized by a negative skewness as well as a non-zero non-Gaussian parameter.

Random walks with spatial and temporal resets can explain individual and colony-level searching patterns in ants.

Central place foragers, such as many ants, exploit the environment around their nest. The extent of their foraging range is a function of individual movement, but how the movement patterns of large numbers of foragers result in an emergent colony foraging range remains unclear. Here, we introduce a random walk model with stochastic resetting to depict the movements of searching ants. Stochastic resetting refers to spatially resetting at random times the position of agents to a given location, here the nest of searching ants. We investigate the effect of a range of resetting mechanisms and compare the macroscopic predictions of our model to laboratory and field data. We find that all returning mechanisms very robustly ensure that scouts exploring the surroundings of a nest will be exponentially distributed with distance from the nest. We also find that a decreasing probability for searching ants to return to their nest is compatible with empirical data, resulting in scouts going further away from the nest as the number of foraging trips increases. Our findings highlight the importance of resetting random walk models for depicting the movements of central place foragers and nurture novel questions regarding the searching behaviour of ants.

Kuramoto model subject to subsystem resetting: How resetting a part of the system may synchronize the whole of it.

We introduce and investigate the effects of a new class of stochastic resetting protocol called subsystem resetting, whereby a subset of the system constituents in a many-body interacting system undergoes bare evolution interspersed with simultaneous resets at random times, while the remaining constituents evolve solely under the bare dynamics. Here, by reset is meant a reinitialization of the dynamics from a given state. We pursue our investigation within the ambit of the well-known Kuramoto model of coupled phase-only oscillators of distributed natural frequencies. Here, the reset protocol corresponds to a chosen set of oscillators being reset to a synchronized state at random times. We find that the mean ω_{0} of the natural frequencies plays a defining role in determining the long-time state of the system. For ω_{0}=0, the system reaches a synchronized stationary state at long times, characterized by a time-independent nonzero value of the synchronization order parameter that quantifies macroscopic order in the system. Moreover, we find that resetting even an infinitesimal fraction of the total number of oscillators, in the extreme limit of infinite resetting rate, has the drastic effect of synchronizing the entire system, even in parameter regimes in which the bare evolution does not support global synchrony. By contrast, for ω_{0}≠0, the dynamics allows at long times either a synchronized stationary state or an oscillatory synchronized state, with the latter characterized by an oscillatory behavior as a function of time of the order parameter, with a nonzero time-independent time average. Our results thus imply that the nonreset subsystem always gets synchronized at long times through the act of resetting of the reset subsystem. Our results, analytical using the Ott-Antonsen ansatz as well as those based on numerical simulations, are obtained for two representative oscillator frequency distributions, namely, a Lorentzian and a Gaussian. Given that it is easier to reset a fraction of the system constituents than the entire system, we discuss how subsystem resetting may be employed as an efficient mechanism to control attainment of global synchrony in the Kuramoto system.

Full-record statistics of one-dimensional random walks.

We develop a comprehensive framework for analyzing full-record statistics, covering record counts M(t_{1}),M(t_{2}),..., their corresponding attainment times T_{M(t_{1})},T_{M(t_{2})},..., and the intervals until the next record. From this multiple-time distribution, we derive general expressions for various observables related to record dynamics, including the conditional number of records given the number observed at a previous time and the conditional time required to reach the current record given the occurrence time of the previous one. Our formalism is exemplified by a variety of stochastic processes, including biased nearest-neighbor random walks, asymmetric run-and-tumble dynamics, and random walks with stochastic resetting.

Thermodynamics of Computations with Absolute Irreversibility, Unidirectional Transitions, and Stochastic Computation Times

Developing a thermodynamic theory of computation is a challenging task at the interface of nonequilibrium thermodynamics and computer science. In particular, this task requires dealing with difficulties such as stochastic halting times, unidirectional (possibly deterministic) transitions, and restricted initial conditions, features common in real-world computers. Here, we present a framework which tackles all such difficulties by extending the martingale theory of nonequilibrium thermodynamics to generic nonstationary Markovian processes, including those with broken detailed balance and/or absolute irreversibility. We derive several universal fluctuation relations and second-law-like inequalities that provide both lower and upper bounds on the intrinsic dissipation (mismatch cost) associated with any periodic process—in particular, the periodic processes underlying all current digital computation. Crucially, these bounds apply even if the process has stochastic stopping times, as it does in many computational machines. We illustrate our results with exhaustive numerical simulations of deterministic finite automata processing bit strings, one of the fundamental models of computation from theoretical computer science. We also provide universal equalities and inequalities for the acceptance probability of words of a given length by a deterministic finite automaton in terms of thermodynamic quantities, and outline connections between computer science and stochastic resetting. Our results, while motivated from the computational context, are applicable far more broadly. Published by the American Physical Society 2024

Lévy flights and Lévy walks under stochastic resetting.

Stochastic resetting is a protocol of starting anew, which can be used to facilitate the escape kinetics. We demonstrate that restarting can accelerate the escape kinetics from a finite interval restricted by two absorbing boundaries also in the presence of heavy-tailed, Lévy-type, α-stable noise. However, the width of the domain where resetting is beneficial depends on the value of the stability index α determining the power-law decay of the jump length distribution. For heavier (smaller α) distributions, the domain becomes narrower in comparison to lighter tails. Additionally, we explore connections between Lévy flights (LFs) and Lévy walks (LWs) in the presence of stochastic resetting. First of all, we show that for Lévy walks, the stochastic resetting can also be beneficial in the domain where the coefficient of variation is smaller than 1. Moreover, we demonstrate that in the domain where LWs are characterized by a finite mean jump duration (length), with the increasing width of the interval, the LWs start to share similarities with LFs under stochastic resetting.