We study the effect of resetting on diffusion in a logarithmic potential. In this model, a particle diffusing in a potential $U(x) = U_0\log|x|$ is reset, i.e., taken back to its initial position, with a constant rate $r$. We show that this analytically tractable model system exhibits a series of phase transitions as a function of a single parameter, $βU_0$, the ratio of the strength of the potential to the thermal energy. For $βU_0<-1$ the potential is strongly repulsive, preventing the particle from reaching the origin. Resetting then generates a non-equilibrium steady state which is characterized exactly and thoroughly analyzed. In contrast, for $βU_0>-1$ the potential is either weakly repulsive or attractive and the diffusing particle eventually reaches the origin. In this case, we provide a closed form expression for the subsequent first-passage time distribution and show that a resetting transition occurs at $βU_0=5$. Namely, we find that resetting can expedite arrival to the origin when $-1<βU_0<5$, but not when $βU_0>5$. The results presented herein generalize results for simple diffusion with resetting -- a widely applicable model that is obtained from ours by setting $U_0=0$. Extending to general potential strengths, our work opens the door to theoretical and experimental investigation of a plethora of problems that bring together resetting and diffusion in logarithmic potential.
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…
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…
Stochastic resetting can be naturally understood as a renewal process governing the evolution of an underlying stochastic process. In this work, we formally derive well-known results of diffusion with resets from a renewal theory perspective. Parallel to the concepts from renewal theory, we introduce the conditioned backward and forward times for stochastic processes with resetting to be the times since the last and until the next reset, given that the current state of the system is known. We focus on studying diffusion under Markovian and non-Markovian resetting. For these cases, we find the …
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…
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 …
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 r…
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 study a Brownian particle diffusing under a time-modulated stochastic resetting mechanism to a fixed position. The rate of resetting r(t) is a function of the time t since the last reset event. We derive a sufficient condition on r(t) for a steady-state probability distribution of the position of the particle to exist. We derive the form of the steady-state distributions under some particular choices of r(t) and also consider the late time relaxation behavior of the probability distribution. Finally we consider first passage time properties for the Brownian particle to reach the origin and …
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 $σ^2 \sim \ln \ln t$.
In this paper we consider diffusion in a domain $Ω$ containing a partially absorbing target $\calM$ with position and occupation time resetting. The occupation time $A_t$ is a Brownian functional that determines the amount of time that the particle spends in $ \calM$ over the time interval $[0,t]$. We assume that there exists some internal state $\calU_t$ of the particle at time $t$ which is modified whenever the particle is diffusing within $\calM $. The state $\calU_t$ is taken to be a monotonically increasing function of $A_t$, and absorption occurs as soon as $\calU_t$ crosses some fixed 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…
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…
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 …
Recent studies in one dimension have revealed that the temporal advantage rendered by stochastic resetting to diffusing particles in attaining first passage, may be annulled by a sufficiently strong attractive potential. We extend the results to higher dimensions. For a diffusing particle in an attractive potential $V({R})=k {R}^n$, in general $d$ dimensions, we study the critical strength $k = k_c$ above which resetting becomes disadvantageous. The point of continuous transition may be exactly found even in cases where the problem with resetting is not solvable, provided the first two moments…
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 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…
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…
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 study the effect of stochastic resetting on a run and tumble particle (RTP) in two spatial dimensions. We consider a resetting protocol which affects both the position and orientation of the RTP: with a constant rate the particle undergoes a positional resetting to a fixed point in space and orientation randomization. We compute the radial and $x$-marginal stationary state distributions and show that while the former approaches a constant value as $r \to 0$, the latter diverges logarithmically as $x \to 0$. On the other hand, both the marginal distributions decay exponentially with the same…
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 the Brownian diffusion of particles in one spatial dimension and in the presence of finite regions within which particles can either evaporate or be reset to a given location. For open boundary conditions, we highlight the appearance of a Brownian yet non-Gaussian diffusion: at long times, the particle distribution is non-Gaussian but its variance grows linearly in time. Moreover, we show that the effective diffusion coefficient of the particles in such systems is bounded from below by $(1-2/π)$ times their bare diffusion coefficient. For periodic boundary conditions, i.e., for …
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…
In this paper we consider the diffusive search for a bounded target $Ω\in \R^d$ with its boundary $\partial Ω$ totally absorbing. We assume that the target is surrounded by a semipermeable interface given by the closed surface $\partial \calM$ with $Ω\subset \calM\subset \R^d$. That is, the interface totally surrounds the target and thus partially screens the diffusive search process. We also assume that the position of the diffusing particle (searcher) randomly resets to its initial position $\x_0$ according to a Poisson process with a resetting rate $r$. The location $\x_0$ is taken to be ou…
In this paper we develop the theory of drift-diffusion on a semi-infinite Cayley tree with stochastic resetting. In the case of a homogeneous tree with a closed terminal node and no resetting, it is known that the system undergoes a classical localization-delocalization (LD) transition at a critical mean velocity $v_c= -(D/L)\ln (z-1)$ where $D$ is the diffusivity, $L$ is the branch length and $z$ is the coordination number of the tree. If $v <v_c$ then the steady state concentration at the terminal node is non-zero (drift-dominated localized state), whereas it is zero when $v >v_c $ (diffusio…
Analytic and Monte Carlo Approximations to the Distribution of the First Passage Time of the Drifted Diffusion with Stochastic Resetting and Mixed Boundary Conditions
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）
