Resetting transition is governed by an interplay between thermal and potential energy

Combining stochastic resetting with Metadynamics to speed-up molecular dynamics simulations

Metadynamics is a powerful method to accelerate molecular dynamics simulations, but its efficiency critically depends on the identification of collective variables that capture the slow modes of the process. Unfortunately, collective variables are usually not known a priori and finding them can be very challenging. We recently presented a collective variables-free approach to enhanced sampling using stochastic resetting. Here, we combine the two methods, showing that it can lead to greater acceleration than either of them separately. We also demonstrate that resetting Metadynamics simulations performed with suboptimal collective variables can lead to speedups comparable with those obtained with optimal collective variables. Therefore, applying stochastic resetting can be an alternative to the challenging task of improving suboptimal collective variables, at almost no additional computational cost. Finally, we propose a method to extract unbiased mean first-passage times from Metadynamics simulations with resetting, resulting in an improved tradeoff between speedup and accuracy. This work enables combining stochastic resetting with other enhanced sampling methods to accelerate a broad range of molecular simulations.

Results for Nonlinear Diffusion Equations with Stochastic Resetting

In this study, we investigate a nonlinear diffusion process in which particles stochastically reset to their initial positions at a constant rate. The nonlinear diffusion process is modeled using the porous media equation and its extensions, which are nonlinear diffusion equations. We use analytical and numerical calculations to obtain and interpret the probability distribution of the position of the particles and the mean square displacement. These results are further compared and shown to agree with the results of numerical simulations. Our findings show that a system of this kind exhibits non-Gaussian distributions, transient anomalous diffusion (subdiffusion and superdiffusion), and stationary states that simultaneously depend on the nonlinearity and resetting rate.

Optimization of escape kinetics by reflecting and resetting

Stochastic restarting is a strategy of starting anew. Incorporation of the resetting to the random walks can result in a decrease in the mean first passage time due to the ability to limit unfavorably meandering, sub-optimal trajectories. In this paper, we examine how stochastic resetting influences escape dynamics from the (-∞,1) interval in the presence of the single-well power-law |x|κ potentials with κ>0. Examination of the mean first passage time is complemented by the analysis of the coefficient of variation, which provides a robust and reliable indicator assessing the efficiency of stochastic resetting. The restrictive nature of resetting is compared to placing a reflective boundary in the system at hand. In particular, for each potential, the position of the reflecting barrier giving the same mean first passage time as the optimal resetting rate is determined. Finally, in addition to reflecting, we compare the effectiveness of other resetting strategies with respect to optimization of the mean first passage time.

Effects of mortality on stochastic search processes with resetting

We study the first-passage time to the origin of a mortal Brownian particle, with mortality rate μ, diffusing in one dimension. The particle starts its motion from x>0 and it is subject to stochastic resetting with constant rate r. We first unveil the relation between the probability of reaching the target and the mean first-passage time of the corresponding problem in absence of mortality, which allows us to deduce under which conditions the former can be increased by adjusting the restart rate. We then consider the first-passage time conditioned on the event that the particle reaches the target before dying, and provide exact expressions for the mean and the variance as functions of r, corroborated by numerical simulations. By studying the impact of resetting for different mortality regimes, we also show that, if the average lifetime τ_{μ}=1/μ is long enough with respect to the diffusive time scale τ_{D}=x^{2}/(4D), there exist both a resetting rate r_{μ}^{*} that maximizes the probability and a rate r_{m} that minimizes the mean first-passage time. However, the two never coincide for positive μ, making the optimization problem highly nontrivial.

First-passage time of run-and-tumble particles with noninstantaneous resetting

We study the statistics of the first-passage time of a single run-and-tumble particle (RTP) in one spatial dimension, with or without resetting, to a fixed target located at L>0. First, we compute the first-passage time distribution of a free RTP, without resetting or in a confining potential, but averaged over the initial position drawn from an arbitrary distribution p(x). Recent experiments used a noninstantaneous resetting protocol that motivated us to study in particular the case where p(x) corresponds to the stationary non-Boltzmann distribution of an RTP in the presence of a harmonic trap. This distribution p(x) is characterized by a parameter ν>0, which depends on the microscopic parameters of the RTP dynamics. We show that the first-passage time distribution of the free RTP, drawn from this initial distribution, develops interesting singular behaviors, depending on the value of ν. We then switch on resetting, mimicked by relaxation of the RTP in the presence of a harmonic trap. Resetting leads to a finite mean first-passage time and we study this as a function of the resetting rate for different values of the parameters ν and b=L/c, where c is the position of the right edge of the initial distribution p(x). In the diffusive limit of the RTP dynamics, we find a rich phase diagram in the (b,ν) plane, with an interesting reentrance phase transition. Away from the diffusive limit, qualitatively similar rich behaviors emerge for the full RTP dynamics.

Expediting Feller process with stochastic resetting

We explore the effect of stochastic resetting on the first-passage properties of the Feller process. The Feller process can be envisioned as space-dependent diffusion, with diffusion coefficient D(x)=x, in a potential U(x)=x(x/2-θ) that owns a minimum at θ. This restricts the process to the positive side of the origin and therefore, Feller diffusion can successfully model a vast array of phenomena in biological and social sciences, where realization of negative values is forbidden. In our analytically tractable model system, a particle that undergoes Feller diffusion is subject to Poissonian resetting, i.e., taken back to its initial position at a constant rate r, after random time epochs. We addressed the two distinct cases that arise when the relative position of the absorbing boundary (x_{a}) with respect to the initial position of the particle (x_{0}) differ, i.e., for (a) x_{0}<x_{a} and (b) x_{a}<x_{0}. Utilizing the Fokker-Planck description of the system, we obtained closed-form expressions for the Laplace transform of the survival probability and hence derived the exact expressions of the mean first-passage time 〈T_{r}〉. Performing a comprehensive analysis on the optimal resetting rate (r^{★}) that minimize 〈T_{r}〉 and the maximal speedup that r^{★} renders, we identify the phase space where Poissonian resetting facilitates first-passage for Feller diffusion. We observe that for x_{0}<x_{a}, resetting accelerates first-passage when θ<θ_{c}, where θ_{c} is a critical value of θ that decreases when x_{a} is moved away from the origin. In stark contrast, for x_{a}<x_{0}, resetting accelerates first-passage when θ>θ_{c}, where θ_{c} is a critical value of θ that increases when x_{0} is moved away from the origin. Our study opens up the possibility of a series of subsequent works with more case-specific models of Feller diffusion with resetting.

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

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.

Interplay of noise induced stability and stochastic resetting

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.

First passage in the presence of stochastic resetting and a potential barrier

Diffusion and first passage in the presence of stochastic resetting and potential bias have been of recent interest. We study a few models, systematically progressing in their complexity, to understand the usefulness of resetting. In the parameter space of the models, there are multiple continuous and discontinuous transitions where the advantage of resetting vanishes. We show these results analytically exactly for a tent potential, and numerically accurately for a quartic potential relevant to a magnetic system at low temperatures. We find that the spatial asymmetry of the potential across the barrier, and the number of absorbing boundaries, play a crucial role in determining the type of transition.

First passage of a diffusing particle under stochastic resetting in bounded domains with spherical symmetry

We investigate the first passage properties of a Brownian particle diffusing freely inside a d-dimensional sphere with absorbing spherical surface subject to stochastic resetting. We derive the mean time to absorption (MTA) as functions of resetting rate γ and initial distance r of the particle to the center of the sphere. We find that when r>r_{c} there exists a nonzero optimal resetting rate γ_{opt} at which the MTA is a minimum, where r_{c}=sqrt[d/(d+4)]R and R is the radius of the sphere. As r increases, γ_{opt} exhibits a continuous transition from zero to nonzero at r=r_{c}. Furthermore, we consider that the particle lies between two two-dimensional or three-dimensional concentric spheres with absorbing boundaries, and obtain the domain in which resetting expedites the MTA, which is (R_{1},r_{c_{1}})∪(r_{c_{2}},R_{2}), with R_{1} and R_{2} being the radii of inner and outer spheres, respectively. Interestingly, when R_{1}/R_{2} is less than a critical value, γ_{opt} exhibits a discontinuous transition at r=r_{c_{1}}; otherwise, such a transition is continuous. However, at r=r_{c_{2}} the transition is always continuous.