4 pages, 3 figures, revtex 4-1 format. Version 1 contains changes in response to referee comments. Version 2: A missing factor of 2 in an inline formula has been corrected
We investigate classic diffusion with the added feature that a diffusing particle is reset to its starting point each time the particle reaches a specified threshold. In an infinite domain, this process is non-stationary and its probability distribution exhibits rich features. In a finite domain, we define a non-trivial optimization in which a cost is incurred whenever the particle is reset and a reward is obtained while the particle stays near the reset point. We derive the condition to optimize the net gain in this system, namely, the reward minus the cost.
We determine the full distribution and moments of the first passage time for a wide class of stochastic search processes in the limit of frequent stochastic resetting. Our results apply to any system whose short-time behavior of the search process without resetting can be estimated. In addition to the typical case of exponentially distributed resetting times, we prove our results for a wide array of resetting time distributions. We illustrate our results in several examples and show that the errors of our approximations vanish exponentially fast in typical scenarios of diffusive search.
We revisit the simple lattice random walk (Pólya walk) and the Sisyphus random walk in $\mathbb{Z}$, in the presence of random restarts. We use a relatively direct approach namely First passage under restart for discrete space and time which was recently developed by us in PRE 103, 052129 (2021) and rederive the first passage properties of these walks under the memoryless geometric restart mechanism. Subsequently, we show how our method could be generalized to arbitrary first passage process subject to more complex restart mechanisms such as sharp, Poisson and Zeta distribution where the latte…
We study experimentally and theoretically the optimal mean time needed by a free diffusing Brownian particle to reach a target at a distance L from an initial position in the presence of resetting. Both the initial position and the resetting position are Gaussian distributed with width $σ$. We derived and tested two resetting protocols, one with a periodic and one with random (Poissonian) resetting times. We computed and measured the full first-passage probability distribution that displays spectacular spikes immediately after each resetting time for close targets. We study the optimal mean fi…
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…
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 study discrete-time random walks on arbitrary networks with first-passage resetting processes. To the end, a set of nodes are chosen as observable nodes, and the walker is reset instantaneously to a given resetting node whenever it hits either of observable nodes. We derive exact expressions of the stationary occupation probability, the average number of resets in the long time, and the mean first-passage time between arbitrary two non-observable nodes. We show that all the quantities can be expressed in terms of the fundamental matrix $\textbf{Z}=(\textbf{I}-\textbf{Q})^{-1}$, where $\text…
We study experimentally, numerically and theoretically the optimal mean time needed by a Brownian particle, freely diffusing either in one or two dimensions, to reach, within a tolerance radius $R_{\text tol}$, a target at a distance $L$ from an initial position in the presence of resetting. The reset position is Gaussian distributed with width $σ$. We derived and tested two resetting protocols, one with a periodic and one with random (Poissonian) resetting times. We computed and measured the full first-passage probability distribution that displays spectacular spikes immediately after each re…
We combine the processes of resetting and first-passage to define \emph{first-passage resetting}, where the resetting of a random walk to a fixed position is triggered by a first-passage event of the walk itself. In an infinite domain, first-passage resetting of isotropic diffusion is non-stationary, with the number of resetting events growing with time as $\sqrt{t}$. We calculate the resulting spatial probability distribution of the particle analytically, and also obtain this distribution by a geometric path decomposition. In a finite interval, we define an optimization problem that is contro…
We consider the dynamics of lattice random walks with resetting. The walker moving randomly on a lattice of arbitrary dimensions resets at every time step to a given site with a constant probability $r$. We construct a discrete renewal equation and present closed-form expressions for different quantities of the resetting dynamics in terms of the underlying reset-free propagator or Green's function. We apply our formalism to the biased random walk dynamics in one-dimensional unbounded space and show how one recovers in the continuous limits results for diffusion with resetting. The resetting dy…
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 nor in a confining potential, but averaged over the initial position drawn from an arbitrary distribution $p(x)$. Recent experiments used a non-instantaneous 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 harm…
First passage of stochastic processes under resetting has recently been an active research topic in the field of statistical physics. However, most of previous studies mainly focused on the systems with continuous time and space. In this paper, we study the effect of stochastic resetting on first passage properties of discrete-time absorbing Markov chains, described by a transition matrix $\brm{Q}$ between transient states and a transition matrix $\brm{R}$ from transient states to absorbing states. Using a renewal approach, we exactly derive the unconditional mean first passage time (MFPT) to …
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…
Non-equilibrium and first-passage properties of generalized Ornstein-Uhlenbeck processes under stochastic resetting
Moscow Institute of Electronics and Mathematics, National Research University Higher School of Economics（莫斯科电子与数学研究所，国立高等经济大学）；Chemistry Department, M.V.Lomonosov Moscow State University（莫斯科大学化学系）
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 study the statistical properties of first-passage time functionals of a one dimensional Brownian motion in the presence of stochastic resetting. A first-passage functional is defined as $V=\int_0^{t_f} Z[x(τ)]$ where $t_f$ is the first-passage time of a reset Brownian process $x(τ)$, i.e., the first time the process crosses zero. In here, the particle is reset to $x_R>0$ at a constant rate $r$ starting from $x_0>0$ and we focus on the following functionals: (i) local time $T_{loc} = \int _0^{t_f}d τ~ δ(x-x_R)$, (ii) residence time $T_{res} = \int _0^{t_f} d τ~θ(x-x_R)$, and (iii) functional…
We consider a Brownian particle diffusing in a one dimensional interval with absorbing end points. We study the ramifications when such motion is interrupted and restarted from the same initial configuration. We provide a comprehensive study of the first passage properties of this trapping phenomena. We compute the mean first passage time and derive the criterion upon which restart always expedites the underlying completion. We show how this set-up is a manifestation of a success-failure problem. We obtain the success and failure rates and relate them with the splitting probabilities, namely, …
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 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 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 $γ_{\rm opt}$ at which the MTA is a minimum, where $r_c=\sqrt {d/\left( {d + 4} \right)} R$ and $R$ is the radius of sphere. As $r$ increases, $γ_{\rm opt}$ exhibits a continuous transition from ze…
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）
Stochastic synthesis-degradation processes: first-passage properties and connections with resetting
Pritzker School of Molecular Engineering, University of Chicago（芝加哥大学普里茨克分子工程学院）；Instituto de Física, Universidad Nacional Autónoma de México（墨西哥国立自治大学物理研究所）
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 reac…
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…
The spectral theory of random walks on networks of arbitrary topology can be readily extended to study random walks and Lévy flights subject to resetting on these structures. When a discrete-time process is stochastically brought back from time to time to its starting node, the mean search time needed to reach another node of the network may be significantly decreased. In other cases, however, resetting is detrimental to search. Using the eigenvalues and eigenvectors of the transition matrix defining the process without resetting, we derive a general criterion for finite networks that establis…
During a random search, resetting the searcher's position from time to time to the starting point often reduces the mean completion time of the process. Although many different resetting models have been studied over the past ten years, only a few can be physically implemented. Here we study theoretically a protocol that can be realised experimentally and which exhibits unusual optimization properties. A Brownian particle is subject to an arbitrary confining potential $v(x)$ which is switched on and off intermittently at fixed rates. Motion is constrained between an absorbing wall located at t…
University of Chicago（芝加哥大学）；LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay（巴黎第十一大学、法国国家科学研究中心理论物理实验室）；Sorbonne Université, Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589（索邦大学、CNRS高能理论物理联合研究单位）
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 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 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…
Detailed knowledge of individual income dynamics is crucial for investigating the existence of the American dream: Are we able to improve our income status during our working life? This key question simply boils down to observing individual status and how it moves between two thresholds: the current income and the desired income. Yet, our knowledge of these temporal properties of income remains limited since we rely on estimates coming from transition matrices which simplify income dynamics by aggregating the individual changes into quantiles and thus overlooking significant microscopic variat…
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…
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）
