Inspired by many examples in nature, stochastic resetting of random processes has been studied extensively in the past decade. In particular, various models of stochastic particle motion were considered where upon resetting the particle is returned to its initial position. Here we generalize the model of diffusion with resetting to account for situations where a particle is returned only a fraction of its distance to the origin, e.g., half way. We show that this model always attains a steady-state distribution which can be written as an infinite sum of independent, but not identical, Laplace random variables. As a result, we find that the steady-state transitions from the known Laplace form which is obtained in the limit of full resetting to a Gaussian form which is obtained close to the limit of no resetting. A similar transition is shown to be displayed by drift-diffusion whose steady-state can also be expressed as an infinite sum of independent random variables. Finally, we extend our analysis to capture the temporal evolution of drift-diffusion with partial resetting, providing a bottom-up probabilistic construction that yields a closed form solution for the time dependent distribution of this process in Fourier-Laplace space. Possible extensions and applications of diffusion with partial resetting are discussed.
Stochastic resetting models diverse phenomena across numerous scientific disciplines. Current understanding stems from the renewal framework, which relates systems subject to global resetting to their non-resetting counterparts. Yet, in interacting many-body systems, even the simplest scenarios involving resetting give rise to the notion of local resetting, whose analysis falls outside the scope of the renewal approach. A prime example is that of diffusing particles with excluded volume interactions that independently attempt to reset their position to the origin of a 1D lattice. With renewal …
We consider properties of one-dimensional diffusive dichotomous flow and discuss effects of resonant activation in the presence of statistically independent random resetting mechanism. Resonant activation and stochastic resetting are two similar effects, as both of them can optimize the noise induced escape. Our studies show completely different origins of optimization in adapted setups. Efficiency of stochastic resetting relies on elimination of suboptimal trajectories while resonant activation is associated with matching of time scales in the dynamic environment. Consequently, both effects c…
We study simple diffusion where a particle stochastically resets to its initial position at a constant rate r. A finite resetting rate leads to a nonequilibrium stationary state with non-Gaussian fluctuations for the particle position. We also show that the mean time to find a stationary target by a diffusive searcher is finite and has a minimum value at an optimal resetting rate r^*. Resetting also alters fundamentally the late time decay of the survival probability of a stationary target when there are multiple searchers: while the typical survival probability decays exponentially with time,…
We consider the problem of diffusion with stochastic resetting in a population of random walks where the diffusion coefficient is not constant, but behaves as a power-law of the average resetting rate of the population. Resetting occurs only beyond a threshold distance from the origin. This problem is motivated by physical realizations like soft matter under shear, where diffusion of a walk is induced by resetting events of other walks. We first reformulate in the broader context of diffusion with stochastic resetting the so-called Hébraud-Lequeux model for plasticity in dense soft matter, in …
The canonical Evans--Majumdar model for diffusion with stochastic resetting to the origin assumes that resetting takes zero time: upon resetting the diffusing particle is teleported back to the origin to start its motion anew. However, in reality, getting from one place to another takes a finite amount of time which must be accounted for as diffusion with resetting already serves as a model for a myriad of processes in physics and beyond. Here we consider a situation where upon resetting the diffusing particle returns to the origin at a finite (rather than infinite) speed. This creates a coupl…
Diffusion with stochastic resetting is a paradigm of resetting processes. Standard renewal or master equation approach are typically used to study steady state and other transport properties such as average, mean squared displacement etc. What remains less explored is the two time point correlation functions whose evaluation is often daunting since it requires the implementation of the exact time dependent probability density functions of the resetting processes which are unknown for most of the problems. We adopt a different approach that allows us to write a stochastic solution in the level …
We study the Brownian motion of a particle in a bounded circular 2-dimensional domain, in search for a stationary target on the boundary of the domain. The process switches between two modes: one where it performs a two-dimensional diffusion inside the circle and one where it travels along the one-dimensional boundary. During the diffusion, the Brownian particle resets to its initial position with a constant rate $r$. The Fokker-Planck formalism allows us to calculate the mean time to absorption (MTA) as well as the optimal resetting rate for which the MTA is minimized. From the derived analyt…
We consider a continuous-space and continuous-time diffusion process under resetting with memory. A particle resets to a position chosen from its trajectory in the past according to a memory kernel. Depending on the form of the memory kernel, we show analytically how different asymptotic behaviours of the variance of the particle position emerge at long times. These range from standard diffusive ($σ^2 \sim t$) all the way to anomalous ultraslow growth $σ^2 \sim \ln \ln t$.
Mirror transitions in diffusion with stochastic resetting confined on a ring
Universidad Autónoma Metropolitana（墨西哥自治 metropolitan 大学）；Universidad Autónoma de Chiapas（恰帕斯自治大学）；Instituto de Física, Universidad Nacional Autónoma de México（墨西哥国立自治大学物理研究所）
We explore the effect of stochastic resetting on the first-passage properties of space-dependent diffusion in presence of a constant bias. In our analytically tractable model system, a particle diffusing in a linear potential $U(x)\proptoμ|x|$ with a spatially varying diffusion coefficient $D(x)=D_0|x|$ undergoes stochastic resetting, i.e., returns to its initial position $x_0$ at random intervals of time, with a constant rate $r$. Considering an absorbing boundary placed at $x_a<x_0$, we first derive an exact expression of the survival probability of the diffusing particle in the Laplace spac…
Heterogeneous planar diffusion with axisymmetric power-law decaying diffusion coefficient under stochastic resetting
Research Center for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts（马其顿科学院与艺术院计算机科学与信息技术研究中心）；Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss. Cyril and Methodius University（圣西里尔和梅托迪乌斯大学自然科学与数学学院物理研究所）；Department of Physics, Korea University（高丽大学物理系）；Laboratoire de Physique et Chimie Théoriques (CNRS UMR 7019), Université de Lorraine Nancy（洛林大学南锡分校理论与物理化学实验室）；Centro de Física Teórica e Computacional, Universidade de Lisboa（里斯本大学理论与计算物理中心）；Departamento de Física, Universidade Estadual de Maringá（马里纳州立大学物理系）
This paper presents a diffusion process with a novel resetting mechanism in which the amplitude of the process is instantaneously converted to a proportion of its value at random times. This model is described by a Langevin equation with both additive Gaussian white noise and multiplicative Poisson shot noise terms. The distribution function obeys a pantograph equation, a functional partial differential equation evaluated at two amplitudes simultaneously. From this equation the exact statistical moments and steady-state distribution of the process are calculated. The distribution interpolates …
We analyze diffusion processes with finite propagation speed in a non-homogeneous medium in terms of the heterogeneous telegrapher's equation. In the diffusion limit of infinite-velocity propagation we recover the results for the heterogeneous diffusion process. The heterogeneous telegrapher's process exhibits a rich variety of diffusion regimes including hyperdiffusion, ballistic motion, superdiffusion, normal diffusion and subdiffusion, and different crossover dynamics characteristic for complex systems in which anomalous diffusion is observed. The anomalous diffusion exponent in the short t…
We study the diffusive motion of a test particle in a two-dimensional comb structure consisting of a main backbone channel with continuously distributed side branches, in the presence of stochastic Markovian resetting to the initial position of the particle. We assume that the motion along the infinitely long branches is biased by a confining potential. The crossover to the steady state is quantified in terms of a large deviation function, which is derived for the first time for comb structures in present paper. We show that the relaxation region is demarcated by a nonlinear "light-cone" beyon…
We consider diffusion in arbitrary spatial dimension d with the addition of a resetting process wherein the diffusive particle stochastically resets to a fixed position at a constant rate $r$. We compute the non-equilibrium stationary state which exhibits non-Gaussian behaviour. We then consider the presence of an absorbing target centred at the origin and compute the survival probability and mean time to absorption of the diffusive particle by the target. The mean absorption time is finite and has a minimum value at an optimal resetting rate $r^*$ which depends on dimension. Finally we consid…
We investigate an intermittent stochastic process in which the diffusive motion with time-dependent diffusion coefficient $D(t) \sim t^{α-1}$ with $α> 0$ (scaled Brownian motion) is stochastically reset to its initial position, and starts anew. \color{black} In the present work we discuss the situation, in which the memory on the value of the diffusion coefficient at a resetting time is erased, so that the whole process is a fully renewal one. The situation when the resetting of coordinate does not affect the diffusion coefficient's time dependence is considered in the other work of this serie…
Generalized diffusion process with nonlocal interactions: Continuous time random walk model and stochastic resetting
Research Center for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts（马其顿科学院计算机科学与信息技术研究中心）；Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss. Cyril and Methodius University（圣西里尔和麦托迪乌斯大学自然科学与数学学院物理研究所）；Institute of Nuclear Physics, Polish Academy of Sciences（波兰科学院核物理研究所）；Faculty of Computer Science and Engineering, Ss. Cyril and Methodius University（圣西里尔和麦托迪乌斯大学计算机科学与工程学部）；Department of Physics, Korea University（韩国大学物理系）
Diffusion with stochastic resetting has recently emerged as a powerful modeling tool with a myriad of potential applications. Here, we study local time in this model, covering situations of free and biased diffusion with, and without, the presence of an absorbing boundary. Given a Brownian trajectory that evolved for $t$ units of time, the local time is simply defined as the total time the trajectory spent in a small vicinity of its initial position. However, as Brownian trajectories are stochastic --- the local time itself is a random variable which fluctuates round and about its mean value. …
Resetting or restart, when applied to a stochastic process, usually brings its dynamics to a time-independent stationary state. In turn, the optimal resetting rate makes the mean time to reach a target to be the shortest one. These and other intriguing problems have been intensively studied in the case of ordinary diffusive processes over the last decade. In this paper we consider the influence of stochastic resetting on a diffusive motion modeled in terms of the non-linear differential equation. The reason for its non-linearity is the power-law dependence of the diffusion coefficient on the p…
The zero-temperature limit of the backgammon model under resetting is studied. The model is a balls-in-boxes model whose relaxation dynamics is governed by the density of boxes containing just one particle. As these boxes become rare at large times, the model presents an entropy barrier. As a preliminary step, a related model with faster relaxation, known to be mapped to a symmetric random walk, is studied by mapping recent results on diffusion with resetting onto the balls-in-boxes problem. Diffusion with an absorbing target at the origin (and diffusion constant equal to one), stochastically …
In this paper, we investigate the effects of stochastic resetting on diffusion in $\R^d\backslash \calU$, where $\calU$ is a bounded obstacle with a partially absorbing surface $\partial \calU$. We begin by considering a Robin boundary condition with a constant reactivity $κ_0$, and show how previous results are recovered in the limits $κ_0\rightarrow 0,\infty$. We then generalize the Robin boundary condition to a more general probabilistic model of diffusion-mediated surface reactions using an encounter-based approach. The latter considers the joint probability density or propagator $P(\x,\el…
We address the effect of stochastic resetting on diffusion and subdiffusion process. For diffusion we find that MSD relaxes to a constant only when the distribution of reset times possess finite mean and variance. In this case, the leading order contribution to the PDF of a Gaussian propagator under resetting exhibits a cusp independent of the specific details of the reset time distribution. For subdiffusion we derive the PDF in Laplace space for arbitrary resetting protocol. Resetting at constant rate allows evaluation of the PDF in terms of H-function. We analyze the steady state and derive …
We explore the effect of stochastic resetting on the first-passage properties of Feller process. The Feller process can be envisioned as space-dependent diffusion, with diffusion coefficient $D(x)=x$, in a potential $U(x)=x\left(\frac{x}{2}-θ\right)$ 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 su…
We investigate an intermittent stochastic process, in which the diffusive motion with time-dependent diffusion coefficient $D(t)\sim t^{α-1}$, $α>0$ (scaled Brownian motion), is stochastically reset to its initial position and starts anew. The resetting follows a renewal process with either exponential or power-law distribution of the waiting times between successive renewals. The resetting events, however, do not affect the time dependence of the diffusion coefficient, so that the whole process appears to be a non-renewal one. We discuss the mean squared displacement of a particle and probabi…
We study the dynamics of a Brownian motion with a diffusion coefficient which evolves stochastically. We first study this process in arbitrary dimensions and find the scaling form and the corresponding scaling function of the position distribution. We find that the tails of the distribution have exponential tails with a ballistic scaling. We then introduce the resetting dynamics where, at a constant rate, both the position and the diffusion coefficient are reset to zero. This eventually leads to a nonequilibrium stationary state, which we study in arbitrary dimensions. In stark contrast to ord…
We consider the motion of a randomly accelerated particle in one dimension under stochastic resetting mechanism. Denoting the position and velocity by $x$ and $v$ respectively, we consider two different resetting protocols - (i) complete resetting: here both $x$ and $v$ reset to their initial values $x_0$ and $v_0$ at a constant rate $r$, (ii) partial resetting: here only $x$ resets to $x_0$ while $v$ evolves without interruption. For complete resetting, we find that the particle attains stationary state in both $x$ and $v$. We compute the non-equilibrium joint stationary state of $x$ and $v$ …
We consider the dynamical evolution of a Brownian particle undergoing stochastic resetting, meaning that after random periods of time it is forced to return to the starting position. The intervals after which the random motion is stopped are drawn from a Gamma distribution of shape parameter $α$ and scale parameter $r$, while the return motion is performed at constant velocity $v$, so that the time cost for a reset is correlated to the last position occupied during the stochastic phase. We show that for any value of $α$ the process reaches a non-equilibrium steady state and unveil the dependen…
Non-equilibrium dynamics of drift-diffusion process under threshold resetting
The Institute of Mathematical Sciences（马哈拉施特拉数学科学研究所）；Homi Bhabha National Institute（霍米·巴巴国立研究所）；Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), Université de Paris-Sud（巴黎南大学理论与统计物理实验室）
Stochastic resetting is prevalent in natural and man-made systems giving rise to a long series of non-equilibrium phenomena. Diffusion with stochastic resetting serves as a paradigmatic model to study these phenomena, but the lack of a well-controlled platform by which this process can be studied experimentally has been a major impediment to research in the field. Here, we report the experimental realization of colloidal particle diffusion and resetting via holographic optical tweezers. We provide the first experimental corroboration of central theoretical results, and go on to measure the ene…
The steady state of a Brownian particle diffusing in an arbitrary potential under the stochastic resetting mechanism has been studied. We show that there are different classes of nonequilibrium steady states depending on the nature of the potential. In the stable potential landscape, the system attains a well defined steady state however existence of the steady state for the unstable landscape is constrained. We have also investigated the transient properties of the propagator towards the steady state under the stochastic resetting mechanism. Finally, we have done numerical simulations to veri…
The effects of Poissonian resetting at a constant rate $r$ on the reaction time between a Brownian particle and a stochastically gated target are studied. The target switches between a reactive state and a non-reactive one. We calculate the mean time at which the particle subject to resetting hits the target for the first time, while the latter is in the reactive state. The search time is minimum at a resetting rate that depends on the target transition rates. When the target relaxation rate is much larger than both the resetting rate and the inverse diffusion time, the system becomes equivale…
We extend the theoretical framework used to study search processes with stochastic resetting to the case of partially absorbing targets. Instead of an absorption event occurring when the search particle reaches the boundary of a target, the particle can diffuse freely in and out of the target region and is absorbed at a rate $κ$ when inside the target. In the context of cell biology, the target could represent a chemically reactive substrate within a cell or a region where a particle can be offloaded onto a nearby compartment. We apply this framework to a partially absorbing interval and to sp…
The escape of the randomly accelerated undamped particle from the finite interval under action of stochastic resetting is studied. The motion of such a particle is described by the full Langevin equation and the particle is characterized by the position and velocity. We compare three resetting protocols, which restarts velocity or position (partial resetting) and the whole motion (position and velocity -- full resetting). Using the mean first passage time we assess efficiency of restarting protocols in facilitating or suppressing the escape kinetics. There are fundamental differences between p…
What happens when a continuously evolving stochastic process is interrupted with large changes at random intervals $τ$ distributed as a power-law $\sim τ^{-(1+α)};α>0$? Modeling the stochastic process by diffusion and the large changes as abrupt resets to the initial condition, we obtain {\em exact} closed-form expressions for both static and dynamic quantities, while accounting for strong correlations implied by a power-law. Our results show that the resulting dynamics exhibits a spectrum of rich long-time behavior, from an ever-spreading spatial distribution for $α< 1$, to one that is time i…
We consider a walker moving in a one-dimensional interval with absorbing boundaries under the effect of Markovian resettings to the initial position. The walker's motion follows a random walk characterized by a general waiting time distribution between consecutive short jumps. We investigate the existence of an optimal reset rate, which minimizes the mean exit passage time, in terms of the statistical properties of the waiting time probability. Generalizing previous results restricted to Markovian random walks, we here find that, depending on the value of the relative standard deviation of the…
We consider the mean time to absorption by an absorbing target of a diffusive particle with the addition of a process whereby the particle is reset to its initial position with rate $r$. We consider several generalisations of the model of M. R. Evans and S. N. Majumdar (2011), Diffusion with stochastic resetting, Phys. Rev. Lett. 106, 160601: (i) a space dependent resetting rate $r(x)$ ii) resetting to a random position $z$ drawn from a resetting distribution ${\cal P}(z)$ iii) a spatial distribution for the absorbing target $P_T(x)$. As an example of (i) we show that the introduction of a non…
We investigate a diffusion process in heterogeneous media where particles stochastically reset to their initial positions at a constant rate. The heterogeneous media is modeled using a spatial-dependent diffusion coefficient with a power-law dependence on particles' positions. We use the Green function approach to obtain exact solutions for the probability distribution of particles' positions and the mean square displacement. These results are further compared and agree with numerical simulations of a Langevin equation. We also study the first-passage time problem associated with this diffusio…
The effect of partial absorption on a diffusive particle which stochastically resets its position with a finite rate $r$ is considered. The particle is absorbed by a target at the origin with absorption `velocity' $a$; as the velocity $a$ approaches $\infty$ the absorption property of the target approaches that of a perfectly-absorbing target. The effect of partial absorption on first-passage time problems is studied, in particular, it is shown that the mean time to absorption (MTA) is increased by an additive term proportional to $1/a$. The results are extended to multiparticle systems where …
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 th…
A dynamical process that takes a random time to complete, e.g., a chemical reaction, may either be accelerated or hindered due to resetting. Tuning system parameters such as temperature, viscosity or concentration, can invert the effect of resetting on the mean completion time of the process, which leads to a resetting transition. Though the resetting transition was recently studied for diffusion in a handful of model potentials, it is yet unknown whether the results follow any universality in terms of well-defined physical parameters. To bridge this gap, we propose a general framework which r…
Markov processes restarted or reset at random times to a fixed state or region in space have been actively studied recently in connection with random searches, foraging, and population dynamics. Here we study the large deviations of time-additive functions or observables of Markov processes with resetting. By deriving a renewal formula linking generating functions with and without resetting we are able to obtain the rate function of such observables, characterizing the likelihood of their fluctuations in the long-time limit. We consider as an illustration the large deviations of the area of th…
Starting from a frequency diffusion process for a tagged photon which simulates relaxation to the Planck law, we introduce a resetting where photons lower their frequency at random times. We consider two versions, one where the resetting to low frequency is independent of the existing frequency and a second case where the reduction in frequency scales with the original frequency. The result is a nonlinear Markov process where the stationary distribution modifies the Planck law by abundance of low-frequency occupation. The physical relevance of such photon resetting processes can be found in ex…
