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 first-passage time as a function of the resetting period/rate for different values of the ratio b = L/$σ$ and find an interesting phase transtion at a critical value b = bc. For bc < b < $\infty$, there is a metastable optimum time which disappears for b < bc. The intrinsic diffculties in implementing these protocols in experiments are also discussed.
We investigate a stochastic search process in one, two, and three dimensions in which $N$ diffusing searchers that all start at $x_0$ seek a target at the origin. Each of the searchers is also reset to its starting point, either with rate $r$, or deterministically, with a reset time $T$. In one dimension and for a small number of searchers, the search time and the search cost are minimized at a non-zero optimal reset rate (or time), while for sufficiently large $N$, resetting always hinders the search. In general, a single searcher leads to the minimum search cost in one, two, and three dimens…
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 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…
Resetting mediated navigation of active Brownian searcher in a homogeneous topography
Indian Statistical Institute（印度统计研究所）；International Institute of Information Technology（国际信息技术学院）；The Institute of Mathematical Sciences（数学科学研究所）；Homi Bhabha National Institute（霍米·巴哈国家研究所）
Proxitaxis: an adaptive search strategy based on proximity and stochastic resetting
LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay（巴黎第十一大学）；International Centre for Theoretical Sciences, Tata Institute of Fundamental Research（塔塔基础研究院理论科学中心）；Raman Research Institute（拉曼研究所）
We study analytically an intermittent search process in one dimension. There is an immobile target at the origin and a searcher undergoes a discrete time jump process starting at $x_0\geq0$, where successive jumps are drawn independently from an arbitrary jump distribution $f(η)$. In addition, with a probability $0\leq r \leq1$ the position of the searcher is reset to its initial position $x_0$. The efficiency of the search strategy is characterized by the mean time to find the target, i.e., the mean first passage time (MFPT) to the origin. For arbitrary jump distribution $f(η)$, initial posit…
We study first-passage time problems for a diffusive particle with stochastic resetting with a finite rate $r$. The optimal search time is compared quantitatively with that of an effective equilibrium Langevin process with the same stationary distribution. It is shown that the intermittent, nonequilibrium strategy with non-vanishing resetting rate is more efficient than the equilibrium dynamics. Our results are extended to multiparticle systems where a team of independent searchers, initially uniformly distributed with a given density, looks for a single immobile target. Both the average and t…
Rank-dependent optimal resetting in multiparticle search
The Raymond and Beverley School of Chemistry, Tel Aviv University（特拉维夫大学雷蒙德和贝弗利化学学院）；The Raymond and Beverley School of Physics & Astronomy, Tel Aviv University（特拉维夫大学雷蒙德和贝弗利物理与天文学院）
In this paper we use asymptotic analysis to determine the steady-state mean number of resources in each of $N$ small interior targets within a three-dimensional bounded domain. The accumulation of resources is based on multiple rounds of search-and-capture events; whenever a searcher finds a target it delivers a resource packet to the target, after which it escapes and returns to its initial position (resetting after capture). The searcher is then resupplied with cargo and a new search process is initiated after a random delay. Assuming that the accumulation of resources is counterbalanced by …
In this paper, we analyze the mean first passage time (MFPT) for a single Brownian particle to find a stochastically-gated target under the additional condition that the position of the particle is reset to a fixed position $\x_r$ at a rate $r$. The gate switches between an open and closed state according to a two-state Markov chain and can only be detected by the searcher in the open state. One possible example of such a target is a protein switching between different conformational states. As expected, the MFPT with or without resetting is an increasing function of the fraction of time $ρ_0$…
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 …
We study the dynamics of an overdamped Brownian particle subjected to Poissonian stochastic resetting in a nonthermal bath, characterized by a Poisson white noise and a Gaussian noise. Applying the renewal theory we find an exact analytical expression for the spatial distribution at the steady state. Unlike the single exponential distribution as observed in the case of a purely thermal bath, the distribution is double exponential. Relaxation of the transient spatial distributions to the stationary one, for the limiting cases of Poissonian rate, is investigated carefully. In addition, we study …
We introduce a resetting Brownian bridge as a simple model to study search processes where the total search time $t_f$ is finite and the searcher returns to its starting point at $t_f$. This is simply a Brownian motion with a Poissonian resetting rate $r$ to the origin which is constrained to start and end at the origin at time $t_f$. We first provide a rejection-free algorithm to generate such resetting bridges in all dimensions by deriving an effective Langevin equation with an explicit space-time dependent drift $\tilde μ({\bf x},t)$ and resetting rate $\tilde r({\bf x}, t)$. We also study …
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…
We consider a random walker on a ring, subjected to resetting at Poisson-distributed times to the initial position (the walker takes the shortest path along the ring to the initial position at resetting times). In the case of a Brownian random walker the mean first-completion time of a turn is expressed in closed form as a function of the resetting rate. The value is shorter than in the ordinary process if the resetting rate is low enough. Moreover, the mean first-completion time of a turn can be minimised in the resetting rate. At large time the distribution of winding numbers does not reach …
Optimal threshold resetting in collective diffusive search
The Institute of Mathematical Sciences（马修研究所）；Homi Bhabha National Institute（霍米·巴巴国立学院）；School of Engineering and Applied Sciences, Harvard University（哈佛大学工程与应用科学学院）；Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), Université de Paris-Sud, Université Paris-Saclay（巴黎第十一大学/萨克雷大学理论与统计物理实验室）
The Institute of Mathematical Sciences（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), CNRS, Univ. Paris-Sud, Universite Paris-Saclay（理论与统计物理实验室（LPTMS），法国国家科学研究中心，巴黎第十一大学，巴黎萨克雷大学）
We study a non-interacting quantum particle, moving on a one-dimensional lattice, which is subjected to repetitive measurements. We investigate the consequence when such motion is interrupted and restarted from the same initial configuration, known as the quantum resetting problem. We show that such systems can be described by the time evolution under certain time-dependent non-Hermitian Hamiltonians. We construct two such Hamiltonians and compare the results with the exact dynamics. Using this effective non-Hermitian description we evaluate the timescale of the survival probability as well as…
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…
We consider one dimensional diffusive search strategies subjected to external potentials. The location of a single target is drawn from a given probability density function (PDF) $f_G(x)$ and is fixed for each stochastic realization of the process. We optimize the quality of the search strategy as measured by the mean first passage time (MFPT) to the position of the target. For a symmetric but otherwise arbitrary distribution $f_G(x)$ we find the optimal potential that minimizes the MFPT. The minimal MFPT is given by a nonstandard measure of the dispersion, which can be related to the cumulati…
