1
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Chatterjee P, Aravinda S, Modak R. Quest for optimal quantum resetting: Protocols for a particle on a chain. Phys Rev E 2024; 110:034132. [PMID: 39425336 DOI: 10.1103/physreve.110.034132] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/23/2023] [Accepted: 09/04/2024] [Indexed: 10/21/2024]
Abstract
In the classical context, it is well known that, sometimes, if a search does not find its target, it is better to start the process anew. This is known as resetting. The quantum counterpart of resetting also indicates speeding up the detection process by eliminating the dark states, i.e., situations in which the particle avoids detection. In this work, we introduce the most probable position resetting (MPR) protocol, in which, at a given resetting step, resets are done with certain probabilities to the set of possible peak positions (where the probability of finding the particle is maximum) that could occur because of the previous resets and followed by uninterrupted unitary evolution, irrespective of which path was taken by the particle in previous steps. In a tight-binding lattice model, there exists a twofold degeneracy (left and right) of the positions of maximum probability. The survival probability with optimal restart rate approaches 0 (detection probability approaches 1) when the particle is reset with equal probability on both sides path independently. This protocol significantly reduces the optimal mean first-detected-passage time (FDT), and it performs better even if the detector is far apart compared to the usual resetting protocols in which the particle is brought back to the initial position. We propose a modified protocol, an adaptive two-stage MPR, by making the associated probabilities of going to the right and left a function of steps. In this protocol, we see a further reduction of the optimal mean FDT and improvement in the search process when the detector is far apart.
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2
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Montero M, Palassini M, Masoliver J. Effect of stochastic resettings on the counting of level crossings for inertial random processes. Phys Rev E 2024; 110:014116. [PMID: 39160907 DOI: 10.1103/physreve.110.014116] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/23/2023] [Accepted: 06/17/2024] [Indexed: 08/21/2024]
Abstract
We study the counting of level crossings for inertial random processes exposed to stochastic resetting events. We develop the general approach of stochastic resetting for inertial processes with sudden changes in the state characterized by position and velocity. We obtain the level-crossing intensity in terms of that of underlying reset-free process for resetting events with Poissonian statistics. We apply this result to the random acceleration process and the inertial Brownian motion. In both cases, we show that there is an optimal resetting rate that maximizes the crossing intensity, and we obtain the asymptotic behavior of the crossing intensity for large and small resetting rates. Finally, we discuss the stationary distribution and the mean first-arrival time in the presence of resettings.
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3
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Yin R, Wang Q, Barkai E. Instability in the quantum restart problem. Phys Rev E 2024; 109:064150. [PMID: 39020895 DOI: 10.1103/physreve.109.064150] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/24/2024] [Accepted: 06/06/2024] [Indexed: 07/20/2024]
Abstract
Repeatedly monitored quantum walks with a rate 1/τ yield discrete-time trajectories which are inherently random. With these paths the first-hitting time with sharp restart is studied. We find an instability in the optimal mean hitting time, which is not found in the corresponding classical random-walk process. This instability implies that a small change in parameters can lead to a rather large change of the optimal restart time. We show that the optimal restart time versus τ, as a control parameter, exhibits sets of staircases and plunges. The plunges, are due to the mentioned instability, which in turn is related to the quantum oscillations of the first-hitting time probability, in the absence of restarts. Furthermore, we prove that there are only two patterns of staircase structures, dependent on the parity of the distance between the target and the source in units of lattice constant. The global minimum of the hitting time is controlled not only by the restart time, as in classical problems, but also by the sampling time τ. We provide numerical evidence that this global minimum occurs for the τ minimizing the mean hitting time, given restarts taking place after each measurement. Last, we numerically show that the instability found in this work is relatively robust against stochastic perturbations in the sampling time τ.
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4
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Majumder R, Chattopadhyay R, Gupta S. Kuramoto model subject to subsystem resetting: How resetting a part of the system may synchronize the whole of it. Phys Rev E 2024; 109:064137. [PMID: 39020942 DOI: 10.1103/physreve.109.064137] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/25/2024] [Accepted: 05/23/2024] [Indexed: 07/20/2024]
Abstract
We introduce and investigate the effects of a new class of stochastic resetting protocol called subsystem resetting, whereby a subset of the system constituents in a many-body interacting system undergoes bare evolution interspersed with simultaneous resets at random times, while the remaining constituents evolve solely under the bare dynamics. Here, by reset is meant a reinitialization of the dynamics from a given state. We pursue our investigation within the ambit of the well-known Kuramoto model of coupled phase-only oscillators of distributed natural frequencies. Here, the reset protocol corresponds to a chosen set of oscillators being reset to a synchronized state at random times. We find that the mean ω_{0} of the natural frequencies plays a defining role in determining the long-time state of the system. For ω_{0}=0, the system reaches a synchronized stationary state at long times, characterized by a time-independent nonzero value of the synchronization order parameter that quantifies macroscopic order in the system. Moreover, we find that resetting even an infinitesimal fraction of the total number of oscillators, in the extreme limit of infinite resetting rate, has the drastic effect of synchronizing the entire system, even in parameter regimes in which the bare evolution does not support global synchrony. By contrast, for ω_{0}≠0, the dynamics allows at long times either a synchronized stationary state or an oscillatory synchronized state, with the latter characterized by an oscillatory behavior as a function of time of the order parameter, with a nonzero time-independent time average. Our results thus imply that the nonreset subsystem always gets synchronized at long times through the act of resetting of the reset subsystem. Our results, analytical using the Ott-Antonsen ansatz as well as those based on numerical simulations, are obtained for two representative oscillator frequency distributions, namely, a Lorentzian and a Gaussian. Given that it is easier to reset a fraction of the system constituents than the entire system, we discuss how subsystem resetting may be employed as an efficient mechanism to control attainment of global synchrony in the Kuramoto system.
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5
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Nikitin I, Belan S. Constructing efficient strategies for the process optimization by restart. Phys Rev E 2024; 109:054117. [PMID: 38907416 DOI: 10.1103/physreve.109.054117] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/26/2023] [Accepted: 04/19/2024] [Indexed: 06/24/2024]
Abstract
Optimization of the mean completion time of random processes by restart is a subject of active theoretical research in statistical physics and has long found practical application in computer science. Meanwhile, one of the key issues remains largely unsolved: how to construct a restart strategy for a process whose detailed statistics are unknown to ensure that the expected completion time will reduce? Addressing this query here we propose several constructive criteria for the effectiveness of various protocols of noninstantaneous restart in the mean completion time problem and in the success probability problem. Being expressed in terms of a small number of easily estimated statistical characteristics of the original process (MAD, median completion time, low-order statistical moments of completion time), these criteria allow informed restart decision based on partial information.
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6
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Żbik B, Dybiec B. Lévy flights and Lévy walks under stochastic resetting. Phys Rev E 2024; 109:044147. [PMID: 38755837 DOI: 10.1103/physreve.109.044147] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/27/2023] [Accepted: 03/27/2024] [Indexed: 05/18/2024]
Abstract
Stochastic resetting is a protocol of starting anew, which can be used to facilitate the escape kinetics. We demonstrate that restarting can accelerate the escape kinetics from a finite interval restricted by two absorbing boundaries also in the presence of heavy-tailed, Lévy-type, α-stable noise. However, the width of the domain where resetting is beneficial depends on the value of the stability index α determining the power-law decay of the jump length distribution. For heavier (smaller α) distributions, the domain becomes narrower in comparison to lighter tails. Additionally, we explore connections between Lévy flights (LFs) and Lévy walks (LWs) in the presence of stochastic resetting. First of all, we show that for Lévy walks, the stochastic resetting can also be beneficial in the domain where the coefficient of variation is smaller than 1. Moreover, we demonstrate that in the domain where LWs are characterized by a finite mean jump duration (length), with the increasing width of the interval, the LWs start to share similarities with LFs under stochastic resetting.
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Affiliation(s)
- Bartosz Żbik
- Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, Łojasiewicza 11, 30-348 Kraków, Poland
| | - Bartłomiej Dybiec
- Institute of Theoretical Physics and Mark Kac Center for Complex Systems Research, Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, Łojasiewicza 11, 30-348 Kraków, Poland
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7
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Méndez V, Flaquer-Galmés R, Campos D. First-passage time of a Brownian searcher with stochastic resetting to random positions. Phys Rev E 2024; 109:044134. [PMID: 38755900 DOI: 10.1103/physreve.109.044134] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/04/2024] [Accepted: 03/22/2024] [Indexed: 05/18/2024]
Abstract
We study the effect of a resetting point randomly distributed around the origin on the mean first-passage time of a Brownian searcher moving in one dimension. We compare the search efficiency with that corresponding to reset to the origin and find that the mean first-passage time of the latter can be larger or smaller than the distributed case, depending on whether the resetting points are symmetrically or asymmetrically distributed. In particular, we prove the existence of an optimal reset rate that minimizes the mean first-passage time for distributed resetting to a finite interval if the target is located outside this interval. When the target position belongs to the resetting interval or it is infinite then no optimal reset rate exists, but there is an optimal resetting interval width or resetting characteristic scale which minimizes the mean first-passage time. We also show that the first-passage density averaged over the resetting points depends on its first moment only. As a consequence, there is an equivalent point such that the first-passage problem with resetting to that point is statistically equivalent to the case of distributed resetting. We end our study by analyzing the fluctuations of the first-passage times for these cases. All our analytical results are verified through numerical simulations.
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Affiliation(s)
- V Méndez
- Grup de Física Estadística, Departament de Física. Facultat de Ciències, Universitat Autònoma de Barcelona, 08193 Barcelona, Spain
| | - R Flaquer-Galmés
- Grup de Física Estadística, Departament de Física. Facultat de Ciències, Universitat Autònoma de Barcelona, 08193 Barcelona, Spain
| | - D Campos
- Grup de Física Estadística, Departament de Física. Facultat de Ciències, Universitat Autònoma de Barcelona, 08193 Barcelona, Spain
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8
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Biroli M, Kulkarni M, Majumdar SN, Schehr G. Dynamically emergent correlations between particles in a switching harmonic trap. Phys Rev E 2024; 109:L032106. [PMID: 38632730 DOI: 10.1103/physreve.109.l032106] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/07/2023] [Accepted: 02/09/2024] [Indexed: 04/19/2024]
Abstract
We study a one dimensional gas of N noninteracting diffusing particles in a harmonic trap, whose stiffness switches between two values μ_{1} and μ_{2} with constant rates r_{1} and r_{2}, respectively. Despite the absence of direct interaction between the particles, we show that strong correlations between them emerge in the stationary state at long times, induced purely by the dynamics itself. We compute exactly the joint distribution of the positions of the particles in the stationary state, which allows us to compute several physical observables analytically. In particular, we show that the extreme value statistics (EVS), i.e., the distribution of the position of the rightmost particle, has a nontrivial shape in the large N limit. The scaling function characterizing this EVS has a finite support with a tunable shape (by varying the parameters). Remarkably, this scaling function turns out to be universal. First, it also describes the distribution of the position of the kth rightmost particle in a 1d trap. Moreover, the distribution of the position of the particle farthest from the center of the harmonic trap in d dimensions is also described by the same scaling function for all d≥1. Numerical simulations are in excellent agreement with our analytical predictions.
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Affiliation(s)
- Marco Biroli
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Manas Kulkarni
- ICTS, Tata Institute of Fundamental Research, Bengaluru 560089, India
| | - Satya N Majumdar
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Grégory Schehr
- Sorbonne Université, Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589, 4 Place Jussieu, 75252 Paris Cedex 05, France
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9
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Julián-Salgado P, Dagdug L, Boyer D. Diffusion with two resetting points. Phys Rev E 2024; 109:024134. [PMID: 38491676 DOI: 10.1103/physreve.109.024134] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/17/2023] [Accepted: 02/02/2024] [Indexed: 03/18/2024]
Abstract
We study the problem of a target search by a Brownian particle subject to stochastic resetting to a pair of sites. The mean search time is minimized by an optimal resetting rate which does not vary smoothly, in contrast with the well-known single site case, but exhibits a discontinuous transition as the position of one resetting site is varied while keeping the initial position of the particle fixed, or vice versa. The discontinuity vanishes at a "liquid-gas" critical point in position space. This critical point exists provided that the relative weight m of the further site is comprised in the interval [2.9028...,8.5603...]. When the initial position is a random variable that follows the resetting point distribution, a discontinuous transition also exists for the optimal rate as the distance between the resetting points is varied, provided that m exceeds the critical value m_{c}=6.6008.... This setup can be mapped onto an intermittent search problem with switching diffusion coefficients and represents a minimal model for the study of distributed resetting.
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Affiliation(s)
- Pedro Julián-Salgado
- Basic Sciences and Engineering, Universidad Autónoma Metropolitana, Apartado Postal 55-534, Mexico City 09340, Mexico
| | - Leonardo Dagdug
- Basic Sciences and Engineering, Universidad Autónoma Metropolitana, Apartado Postal 55-534, Mexico City 09340, Mexico
| | - Denis Boyer
- Instituto de Física, Universidad Nacional Autónoma de México, Mexico City 04510, Mexico
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10
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Guo W, Yan H, Chen H. Extremal statistics for a resetting Brownian motion before its first-passage time. Phys Rev E 2023; 108:044115. [PMID: 37978585 DOI: 10.1103/physreve.108.044115] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/29/2023] [Accepted: 09/08/2023] [Indexed: 11/19/2023]
Abstract
We study the extreme value statistics of a one-dimensional resetting Brownian motion (RBM) till its first passage through the origin starting from the position x_{0} (>0). By deriving the exit probability of RBM in an interval [0,M] from the origin, we obtain the distribution P_{r}(M|x_{0}) of the maximum displacement M and thus gives the expected value 〈M〉 of M as functions of the resetting rate r and x_{0}. We find that 〈M〉 decreases monotonically as r increases, and tends to 2x_{0} as r→∞. In the opposite limit, 〈M〉 diverges logarithmically as r→0. Moreover, we derive the propagator of RBM in the Laplace domain in the presence of both absorbing ends, and then leads to the joint distribution P_{r}(M,t_{m}|x_{0}) of M and the time t_{m} at which this maximum is achieved in the Laplace domain by using a path decomposition technique, from which the expected value 〈t_{m}〉 of t_{m} is obtained explicitly. Interestingly, 〈t_{m}〉 shows a nonmonotonic dependence on r, and attains its minimum at an optimal r^{*}≈2.71691D/x_{0}^{2}, where D is the diffusion coefficient. Finally, we perform extensive simulations to validate our theoretical results.
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Affiliation(s)
- Wusong Guo
- School of Physics and Optoelectronic Engineering, Anhui University, Hefei 230601, China
| | - Hao Yan
- School of Physics and Optoelectronic Engineering, Anhui University, Hefei 230601, China
| | - Hanshuang Chen
- School of Physics and Optoelectronic Engineering, Anhui University, Hefei 230601, China
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11
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Capała K, Dybiec B. Optimization of escape kinetics by reflecting and resetting. CHAOS (WOODBURY, N.Y.) 2023; 33:103124. [PMID: 37832519 DOI: 10.1063/5.0159475] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/24/2023] [Accepted: 09/27/2023] [Indexed: 10/15/2023]
Abstract
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.
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Affiliation(s)
- Karol Capała
- Personal Health Data Science Group, Sano-Centre for Computational Personalised Medicine, Czarnowiejska 36, 30-054 Kraków, Poland
- Institute of Theoretical Physics and Mark Kac Center for Complex Systems Research, Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, Łojasiewicza 11, 30-348 Kraków, Poland
| | - Bartłomiej Dybiec
- Institute of Theoretical Physics and Mark Kac Center for Complex Systems Research, Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, Łojasiewicza 11, 30-348 Kraków, Poland
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12
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Starkov D, Belan S. Universal performance bounds of restart. Phys Rev E 2023; 107:L062101. [PMID: 37464603 DOI: 10.1103/physreve.107.l062101] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/13/2022] [Accepted: 05/11/2023] [Indexed: 07/20/2023]
Abstract
As has long been known to computer scientists, the performance of probabilistic algorithms characterized by relatively large runtime fluctuations can be improved by applying a restart, i.e., episodic interruption of a randomized computational procedure followed by initialization of its new statistically independent realization. A similar effect of restart-induced process acceleration could potentially be possible in the context of enzymatic reactions, where dissociation of the enzyme-substrate intermediate corresponds to restarting the catalytic step of the reaction. To date, a significant number of analytical results have been obtained in physics and computer science regarding the effect of restart on the completion time statistics in various model problems, however, the fundamental limits of restart efficiency remain unknown. Here we derive a range of universal statistical inequalities that offer constraints on the effect that restart could impose on the completion time of a generic stochastic process. The corresponding bounds are expressed via simple statistical metrics of the original process such as harmonic mean h, median value m, and mode M, and, thus, are remarkably practical. We test our analytical predictions with multiple numerical examples, discuss implications arising from them and important avenues of future work.
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Affiliation(s)
- Dmitry Starkov
- Landau Institute for Theoretical Physics, Russian Academy of Sciences, 1-A Akademika Semenova Av., 142432 Chernogolovka, Russia
- National Research University Higher School of Economics, Faculty of Mathematics, Usacheva 6, 119048 Moscow, Russia
| | - Sergey Belan
- Landau Institute for Theoretical Physics, Russian Academy of Sciences, 1-A Akademika Semenova Av., 142432 Chernogolovka, Russia
- National Research University Higher School of Economics, Faculty of Physics, Myasnitskaya 20, 101000 Moscow, Russia
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13
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Yin R, Barkai E. Restart Expedites Quantum Walk Hitting Times. PHYSICAL REVIEW LETTERS 2023; 130:050802. [PMID: 36800468 DOI: 10.1103/physrevlett.130.050802] [Citation(s) in RCA: 6] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/08/2022] [Accepted: 12/22/2022] [Indexed: 06/18/2023]
Abstract
Classical first-passage times under restart are used in a wide variety of models, yet the quantum version of the problem still misses key concepts. We study the quantum hitting time with restart using a monitored quantum walk. The restart strategy eliminates the problem of dark states, i.e., cases where the particle evades detection, while maintaining the ballistic propagation which is important for a fast search. We find profound effects of quantum oscillations on the restart problem, namely, a type of instability of the mean detection time, and optimal restart times that form staircases, with sudden drops as the rate of sampling is modified. In the absence of restart and in the Zeno limit, the detection of the walker is not possible, and we examine how restart overcomes this well-known problem, showing that the optimal restart time becomes insensitive to the sampling period.
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Affiliation(s)
- R Yin
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 52900, Israel
| | - E Barkai
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 52900, Israel
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14
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Radice M. Effects of mortality on stochastic search processes with resetting. Phys Rev E 2023; 107:024136. [PMID: 36932537 DOI: 10.1103/physreve.107.024136] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/02/2022] [Accepted: 02/10/2023] [Indexed: 06/18/2023]
Abstract
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.
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Affiliation(s)
- Mattia Radice
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, 01187 Dresden, Germany
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15
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Blumer O, Reuveni S, Hirshberg B. Stochastic Resetting for Enhanced Sampling. J Phys Chem Lett 2022; 13:11230-11236. [PMID: 36446130 PMCID: PMC9743203 DOI: 10.1021/acs.jpclett.2c03055] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/07/2022] [Accepted: 11/23/2022] [Indexed: 06/16/2023]
Abstract
We present a method for enhanced sampling of molecular dynamics simulations using stochastic resetting. Various phenomena, ranging from crystal nucleation to protein folding, occur on time scales that are unreachable in standard simulations. They are often characterized by broad transition time distributions, in which extremely slow events have a non-negligible probability. Stochastic resetting, i.e., restarting simulations at random times, was recently shown to significantly expedite processes that follow such distributions. Here, we employ resetting for enhanced sampling of molecular simulations for the first time. We show that it accelerates long time scale processes by up to an order of magnitude in examples ranging from simple models to a molecular system. Most importantly, we recover the mean transition time without resetting, which is typically too long to be sampled directly, from accelerated simulations at a single restart rate. Stochastic resetting can be used as a standalone method or combined with other sampling algorithms to further accelerate simulations.
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Affiliation(s)
- Ofir Blumer
- School
of Chemistry, Tel Aviv University, Tel Aviv6997801, Israel
| | - Shlomi Reuveni
- School
of Chemistry, Tel Aviv University, Tel Aviv6997801, Israel
- The
Center for Computational Molecular and Materials Science, Tel Aviv University, Tel Aviv6997801, Israel
- The
Center for Physics and Chemistry of Living Systems, Tel Aviv University, Tel Aviv6997801, Israel
| | - Barak Hirshberg
- School
of Chemistry, Tel Aviv University, Tel Aviv6997801, Israel
- The
Center for Computational Molecular and Materials Science, Tel Aviv University, Tel Aviv6997801, Israel
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16
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Bressloff PC. A probabilistic model of diffusion through a semi-permeable barrier. Proc Math Phys Eng Sci 2022. [DOI: 10.1098/rspa.2022.0615] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/24/2022] Open
Abstract
Diffusion through semi-permeable structures arises in a wide range of processes in the physical and life sciences. Examples at the microscopic level range from artificial membranes for reverse osmosis to lipid bilayers regulating molecular transport in biological cells to chemical and electrical gap junctions. There are also macroscopic analogues such as animal migration in heterogeneous landscapes. It has recently been shown that one-dimensional diffusion through a barrier with constant permeability
κ
0
is equivalent to snapping out Brownian motion (BM). The latter sews together successive rounds of partially reflecting BMs that are restricted to either the left or the right of the barrier. Each round is killed when its Brownian local time exceeds an exponential random variable parameterized by
κ
0
. A new round is then immediately started in either direction with equal probability. In this article, we use a combination of renewal theory, Laplace transforms and Green’s function methods to show how an extended version of snapping out BM provides a general probabilistic framework for modelling diffusion through a semi-permeable barrier. This includes modifications of the diffusion process away from the barrier (e.g. stochastic resetting) and non-Markovian models of membrane absorption that kill each round of partially reflected BM. The latter leads to time-dependent permeabilities.
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Affiliation(s)
- Paul C. Bressloff
- Department of Mathematics, University of Utah 155 South1400 East, Salt Lake City, UT 84112, USA
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17
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Wang Y, Chen H. Entropy rate of random walks on complex networks under stochastic resetting. Phys Rev E 2022; 106:054137. [PMID: 36559349 DOI: 10.1103/physreve.106.054137] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/15/2022] [Accepted: 10/27/2022] [Indexed: 11/16/2022]
Abstract
Stochastic processes under resetting at random times have attracted a lot of attention in recent years and served as illustrations of nontrivial and interesting static and dynamic features of stochastic dynamics. In this paper, we aim to address how the entropy rate is affected by stochastic resetting in discrete-time Markovian processes, and we explore nontrivial effects of the resetting in the mixing properties of a stochastic process. In particular, we consider resetting random walks (RRWs) with a single resetting node on three different types of networks: degree-regular random networks, a finite-size Cayley tree, and a Barabási-Albert (BA) scale-free network, and we compute the entropy rate as a function of the resetting probability γ. Interestingly, for the first two types of networks, the entropy rate shows a nonmonotonic dependence on γ. There exists an optimal value of γ at which the entropy rate reaches a maximum. Such a maximum is larger than that of maximal-entropy random walks (MREWs) and standard random walks (SRWs) on the same topology, while for the BA network the entropy rate of RRWs either shows a unique maximum or decreases monotonically with γ, depending upon the choice of the resetting node. When the maximum entropy rate of RRWs exists, it can be higher or lower than that of MREWs or SRWs. Our study reveals a nontrivial effect of stochastic resetting on nonequilibrium statistical physics.
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Affiliation(s)
- Yating Wang
- School of Physics and Optoelectronic Engineering, Anhui University, Hefei 230601, China
| | - Hanshuang Chen
- School of Physics and Optoelectronic Engineering, Anhui University, Hefei 230601, China
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18
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Tucci G, Gambassi A, Majumdar SN, Schehr G. First-passage time of run-and-tumble particles with noninstantaneous resetting. Phys Rev E 2022; 106:044127. [PMID: 36397533 DOI: 10.1103/physreve.106.044127] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/04/2022] [Accepted: 09/09/2022] [Indexed: 06/16/2023]
Abstract
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.
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Affiliation(s)
- Gennaro Tucci
- SISSA-International School for Advanced Studies and INFN, via Bonomea 265, I-34136 Trieste, Italy
| | - Andrea Gambassi
- SISSA-International School for Advanced Studies and INFN, via Bonomea 265, I-34136 Trieste, Italy
| | - Satya N Majumdar
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Grégory Schehr
- Sorbonne Université, Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589, 4 Place Jussieu, 75252 Paris Cedex 05, France
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19
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Chen H, Ye Y. Random walks on complex networks under time-dependent stochastic resetting. Phys Rev E 2022; 106:044139. [PMID: 36397577 DOI: 10.1103/physreve.106.044139] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/31/2022] [Accepted: 10/05/2022] [Indexed: 06/16/2023]
Abstract
We study discrete-time random walks on networks subject to a time-dependent stochastic resetting, where the walker either hops randomly between neighboring nodes with a probability 1-ϕ(a) or is reset to a given node with a complementary probability ϕ(a). The resetting probability ϕ(a) depends on the time a since the last reset event (also called a, the age of the walker). Using the renewal approach and spectral decomposition of the transition matrix, we formulate the stationary occupation probability of the walker at each node and the mean first passage time between two arbitrary nodes. Concretely, we consider two different time-dependent resetting protocols that are both exactly solvable. One is that ϕ(a) is a step-shaped function of a and the other one is that ϕ(a) is a rational function of a. We demonstrate the theoretical results on several different networks, also validated by numerical simulations, and find that the time-modulated resetting protocols can be more advantageous than the constant-probability resetting in accelerating the completion of a target search process.
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Affiliation(s)
- Hanshuang Chen
- School of Physics and Optoelectronic Engineering, Anhui University, Hefei 230601, China
| | - Yanfei Ye
- School of Physics and Optoelectronic Engineering, Anhui University, Hefei 230601, China
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20
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Bonomo OL, Pal A, Reuveni S. Mitigating long queues and waiting times with service resetting. PNAS NEXUS 2022; 1:pgac070. [PMID: 36741459 PMCID: PMC9896945 DOI: 10.1093/pnasnexus/pgac070] [Citation(s) in RCA: 5] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 01/24/2022] [Accepted: 05/25/2022] [Indexed: 02/07/2023]
Abstract
What determines the average length of a queue, which stretches in front of a service station? The answer to this question clearly depends on the average rate at which jobs arrive at the queue and on the average rate of service. Somewhat less obvious is the fact that stochastic fluctuations in service and arrival times are also important, and that these are a major source of backlogs and delays. Strategies that could mitigate fluctuations-induced delays are, thus in high demand as queue structures appear in various natural and man-made systems. Here, we demonstrate that a simple service resetting mechanism can reverse the deleterious effects of large fluctuations in service times, thus turning a marked drawback into a favorable advantage. This happens when stochastic fluctuations are intrinsic to the server, and we show that service resetting can then dramatically cut down average queue lengths and waiting times. Remarkably, this strategy is also useful in extreme situations where the variance, and possibly even mean, of the service time diverge-as resetting can then prevent queues from "blowing up." We illustrate these results on the M/G/1 queue in which service times are general and arrivals are assumed to be Markovian. However, the main results and conclusions coming from our analysis are not specific to this particular model system. Thus, the results presented herein can be carried over to other queueing systems: in telecommunications, via computing, and all the way to molecular queues that emerge in enzymatic and metabolic cycles of living organisms.
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Affiliation(s)
- Ofek Lauber Bonomo
- School of Chemistry, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 6997801, Israel,Center for the Physics and Chemistry of Living Systems, Tel Aviv University, Tel Aviv 6997801, Israel,The Sackler Center for Computational Molecular and Materials Science, Tel Aviv University, Tel Aviv 6997801, Israel
| | - Arnab Pal
- School of Chemistry, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 6997801, Israel,Center for the Physics and Chemistry of Living Systems, Tel Aviv University, Tel Aviv 6997801, Israel,The Institute of Mathematical Sciences, IV Cross Road, CIT Campus, Taramani, Chennai 600113, Tamil Nadu, India,Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India
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21
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De Bruyne B, Majumdar SN, Schehr G. Optimal Resetting Brownian Bridges via Enhanced Fluctuations. PHYSICAL REVIEW LETTERS 2022; 128:200603. [PMID: 35657896 DOI: 10.1103/physrevlett.128.200603] [Citation(s) in RCA: 13] [Impact Index Per Article: 6.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/06/2022] [Revised: 04/06/2022] [Accepted: 04/25/2022] [Indexed: 06/15/2023]
Abstract
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 unveil a surprising general mechanism that enhances fluctuations of a Brownian bridge, by introducing a small amount of resetting. This is verified for different observables, such as the mean-square displacement, the hitting probability of a fixed target and the expected maximum. This mechanism, valid for a Brownian bridge in arbitrary dimensions, leads to a finite optimal resetting rate that minimizes the time to search a fixed target. The physical reason behind an optimal resetting rate in this case is entirely different from that of resetting Brownian motions without the bridge constraint. We also derive an exact effective Langevin equation that generates numerically the trajectories of a resetting Brownian bridge in all dimensions via a completely rejection-free algorithm.
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Affiliation(s)
- Benjamin De Bruyne
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Satya N Majumdar
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Grégory Schehr
- Sorbonne Université, Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589, 4 Place Jussieu, 75252 Paris Cedex 05, France
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22
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Lim H, Jung Y. Reaction-path statistical mechanics of enzymatic kinetics. J Chem Phys 2022; 156:134108. [PMID: 35395879 DOI: 10.1063/5.0075831] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We introduce a reaction-path statistical mechanics formalism based on the principle of large deviations to quantify the kinetics of single-molecule enzymatic reaction processes under the Michaelis-Menten mechanism, which exemplifies an out-of-equilibrium process in the living system. Our theoretical approach begins with the principle of equal a priori probabilities and defines the reaction path entropy to construct a new nonequilibrium ensemble as a collection of possible chemical reaction paths. As a result, we evaluate a variety of path-based partition functions and free energies by using the formalism of statistical mechanics. They allow us to calculate the timescales of a given enzymatic reaction, even in the absence of an explicit boundary condition that is necessary for the equilibrium ensemble. We also consider the large deviation theory under a closed-boundary condition of the fixed observation time to quantify the enzyme-substrate unbinding rates. The result demonstrates the presence of a phase-separation-like, bimodal behavior in unbinding events at a finite timescale, and the behavior vanishes as its rate function converges to a single phase in the long-time limit.
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Affiliation(s)
- Hyuntae Lim
- Department of Chemistry, Seoul National University, Seoul 08826, South Korea
| | - YounJoon Jung
- Department of Chemistry, Seoul National University, Seoul 08826, South Korea
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23
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Ahmad S, Rijal K, Das D. First passage in the presence of stochastic resetting and a potential barrier. Phys Rev E 2022; 105:044134. [PMID: 35590648 DOI: 10.1103/physreve.105.044134] [Citation(s) in RCA: 6] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/05/2022] [Accepted: 04/04/2022] [Indexed: 06/15/2023]
Abstract
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.
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Affiliation(s)
- Saeed Ahmad
- Physics Department, Indian Institute of Technology Bombay, Mumbai 400076, India
| | - Krishna Rijal
- Physics Department, Indian Institute of Technology Bombay, Mumbai 400076, India
| | - Dibyendu Das
- Physics Department, Indian Institute of Technology Bombay, Mumbai 400076, India
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24
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Chen H, Huang F. First passage of a diffusing particle under stochastic resetting in bounded domains with spherical symmetry. Phys Rev E 2022; 105:034109. [PMID: 35428076 DOI: 10.1103/physreve.105.034109] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/24/2021] [Accepted: 02/18/2022] [Indexed: 06/14/2023]
Abstract
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.
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Affiliation(s)
- Hanshuang Chen
- School of Physics and Optoelectronics Engineering, Anhui University, Hefei 230601, China
| | - Feng Huang
- Key Laboratory of Advanced Electronic Materials and Devices & School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China
- Key Laboratory of Architectural Acoustic Environment of Anhui Higher Education Institutes, Hefei 230601, China
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25
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Radice M. One-dimensional telegraphic process with noninstantaneous stochastic resetting. Phys Rev E 2021; 104:044126. [PMID: 34781456 DOI: 10.1103/physreve.104.044126] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/01/2021] [Accepted: 10/08/2021] [Indexed: 06/13/2023]
Abstract
In this paper, we consider the one-dimensional dynamical evolution of a particle traveling at constant speed and performing, at a given rate, random reversals of the velocity direction. The particle is subject to stochastic resetting, meaning that at random times it is forced to return to the starting point. Here we consider a return mechanism governed by a deterministic law of motion, so that the time cost required to return is correlated to the position occupied at the time of the reset. We show that in such conditions the process reaches a stationary state which, for some kinds of deterministic return dynamics, is independent of the return phase. Furthermore, we investigate the first-passage properties of the system and provide explicit formulas for the mean first-hitting time. Our findings are supported by numerical simulations.
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Affiliation(s)
- Mattia Radice
- Dipartimento di Scienza e Alta Tecnologia and Center for Nonlinear and Complex Systems, Università degli studi dell'Insubria, Via Valleggio 11, 22100 Como, Italy and I.N.F.N. Sezione di Milano, Via Celoria 16, 20133 Milano, Italy
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26
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Wang W, Cherstvy AG, Kantz H, Metzler R, Sokolov IM. Time averaging and emerging nonergodicity upon resetting of fractional Brownian motion and heterogeneous diffusion processes. Phys Rev E 2021; 104:024105. [PMID: 34525678 DOI: 10.1103/physreve.104.024105] [Citation(s) in RCA: 21] [Impact Index Per Article: 7.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/27/2021] [Accepted: 07/14/2021] [Indexed: 12/12/2022]
Abstract
How different are the results of constant-rate resetting of anomalous-diffusion processes in terms of their ensemble-averaged versus time-averaged mean-squared displacements (MSDs versus TAMSDs) and how does stochastic resetting impact nonergodicity? We examine, both analytically and by simulations, the implications of resetting on the MSD- and TAMSD-based spreading dynamics of particles executing fractional Brownian motion (FBM) with a long-time memory, heterogeneous diffusion processes (HDPs) with a power-law space-dependent diffusivity D(x)=D_{0}|x|^{γ} and their "combined" process of HDP-FBM. We find, inter alia, that the resetting dynamics of originally ergodic FBM for superdiffusive Hurst exponents develops disparities in scaling and magnitudes of the MSDs and mean TAMSDs indicating weak ergodicity breaking. For subdiffusive HDPs we also quantify the nonequivalence of the MSD and TAMSD and observe a new trimodal form of the probability density function. For reset FBM, HDPs and HDP-FBM we compute analytically and verify by simulations the short-time MSD and TAMSD asymptotes and long-time plateaus reminiscent of those for processes under confinement. We show that certain characteristics of these reset processes are functionally similar despite a different stochastic nature of their nonreset variants. Importantly, we discover nonmonotonicity of the ergodicity-breaking parameter EB as a function of the resetting rate r. For all reset processes studied we unveil a pronounced resetting-induced nonergodicity with a maximum of EB at intermediate r and EB∼(1/r)-decay at large r. Alongside the emerging MSD-versus-TAMSD disparity, this r-dependence of EB can be an experimentally testable prediction. We conclude by discussing some implications to experimental systems featuring resetting dynamics.
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Affiliation(s)
- Wei Wang
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, 01187 Dresden, Germany
| | - Andrey G Cherstvy
- Institute for Physics & Astronomy University of Potsdam, Karl-Liebknecht-Straße 24/25, 14476 Potsdam-Golm, Germany.,Institut für Physik, Humboldt-Universität zu Berlin, Newtonstraße 15, 12489 Berlin, Germany
| | - Holger Kantz
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, 01187 Dresden, Germany
| | - Ralf Metzler
- Institute for Physics & Astronomy University of Potsdam, Karl-Liebknecht-Straße 24/25, 14476 Potsdam-Golm, Germany
| | - Igor M Sokolov
- Institut für Physik, Humboldt-Universität zu Berlin, Newtonstraße 15, 12489 Berlin, Germany.,IRIS Adlershof, Zum Großen Windkanal 6, 12489 Berlin, Germany
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27
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Wang S, Chen H, Huang F. Random walks on complex networks with multiple resetting nodes: A renewal approach. CHAOS (WOODBURY, N.Y.) 2021; 31:093135. [PMID: 34598469 DOI: 10.1063/5.0064791] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/27/2021] [Accepted: 09/02/2021] [Indexed: 06/13/2023]
Abstract
Due to wide applications in diverse fields, random walks subject to stochastic resetting have attracted considerable attention in the last decade. In this paper, we study discrete-time random walks on complex networks with multiple resetting nodes. Using a renewal approach, we derive exact expressions of the occupation probability of the walker in each node and mean first-passage time between arbitrary two nodes. All the results can be expressed in terms of the spectral properties of the transition matrix in the absence of resetting. We demonstrate our results on circular networks, stochastic block models, and Barabási-Albert scale-free networks and find the advantage of the resetting processes to multiple resetting nodes in a global search on such networks. Finally, the distribution of resetting probabilities is optimized via a simulated annealing algorithm, so as to minimize the mean first-passage time averaged over arbitrary two distinct nodes.
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Affiliation(s)
- Shuang Wang
- School of Physics Optoelectronics Engineering, Anhui University, Hefei 230601, China
| | - Hanshuang Chen
- School of Physics Optoelectronics Engineering, Anhui University, Hefei 230601, China
| | - Feng Huang
- Key Laboratory of Advanced Electronic Materials and Devices & School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China
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28
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Huang F, Chen H. Random walks on complex networks with first-passage resetting. Phys Rev E 2021; 103:062132. [PMID: 34271762 DOI: 10.1103/physreve.103.062132] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/21/2021] [Accepted: 06/01/2021] [Indexed: 11/07/2022]
Abstract
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 nonobservable nodes. We show that all the quantities can be expressed in terms of the fundamental matrix Z=(I-Q)^{-1}, where I is the identity matrix and Q is the transition matrix between nonobservable nodes. Finally, we use ring networks, two-dimensional square lattices, barbell networks, and Cayley trees to demonstrate the advantage of first-passage resetting in global search on such networks.
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Affiliation(s)
- Feng Huang
- Key Laboratory of Advanced Electronic Materials and Devices & School of Mathematics and Physics, Anhui Jianzhu University, Hefei, 230601, China.,Key Laboratory of Architectural Acoustic Environment of Anhui Higher Education Institutes, Hefei, 230601, China
| | - Hanshuang Chen
- Department of Physics, Anhui University, Hefei, 230601, China
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29
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A Semi-Deterministic Random Walk with Resetting. ENTROPY 2021; 23:e23070825. [PMID: 34203494 PMCID: PMC8303408 DOI: 10.3390/e23070825] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 06/04/2021] [Revised: 06/22/2021] [Accepted: 06/23/2021] [Indexed: 11/17/2022]
Abstract
We consider a discrete-time random walk (xt) which, at random times, is reset to the starting position and performs a deterministic motion between them. We show that the quantity Prxt+1=n+1|xt=n,n→∞ determines if the system is averse, neutral or inclined towards resetting. It also classifies the stationary distribution. Double barrier probabilities, first passage times and the distribution of the escape time from intervals are determined.
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30
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Ray S, Reuveni S. Resetting transition is governed by an interplay between thermal and potential energy. J Chem Phys 2021; 154:171103. [PMID: 34241053 DOI: 10.1063/5.0049642] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
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. Although the resetting transition has been 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 that reveals that the resetting transition is governed by an interplay between the thermal and potential energy. This result is illustrated for different classes of potentials that are used to model a wide variety of stochastic processes with numerous applications.
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Affiliation(s)
- Somrita Ray
- School of Chemistry, The Center for Physics and Chemistry of Living Systems, The Raymond and Beverly Sackler Center for Computational Molecular and Materials Science, and The Ratner Center for Single Molecule Science, Tel Aviv University, Tel Aviv 69978, Israel
| | - Shlomi Reuveni
- School of Chemistry, The Center for Physics and Chemistry of Living Systems, The Raymond and Beverly Sackler Center for Computational Molecular and Materials Science, and The Ratner Center for Single Molecule Science, Tel Aviv University, Tel Aviv 69978, Israel
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31
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Nayak I, Nandi A, Das D. Capture of a diffusive prey by multiple predators in confined space. Phys Rev E 2021; 102:062109. [PMID: 33466016 DOI: 10.1103/physreve.102.062109] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/31/2020] [Accepted: 11/12/2020] [Indexed: 11/07/2022]
Abstract
The first passage search of a diffusing target (prey) by multiple searchers (predators) in confinement is an important problem in the stochastic process literature. While the analogous problem in open space has been studied in some detail, a systematic study in confined space is still lacking. In this paper, we study the first passage times for this problem in one, two, and three dimensions. Due to confinement, the survival probability of the target takes a form ∼e^{-t/τ} at large times t. The characteristic capture timescale τ associated with the rare capture events are rather challenging to measure. We use a computational algorithm that allows us to estimate τ with high accuracy. We study in detail the behavior of τ as a function of the system parameters, namely, the number of searchers N, the relative diffusivity r of the target with respect to the searcher, and the system size. We find that τ deviates from the ∼1/N scaling seen in the case of a static target, and this deviation varies continuously with r and the spatial dimensions.
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Affiliation(s)
- Indrani Nayak
- Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
| | - Amitabha Nandi
- Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
| | - Dibyendu Das
- Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
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32
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Ray S. Space-dependent diffusion with stochastic resetting: A first-passage study. J Chem Phys 2020; 153:234904. [DOI: 10.1063/5.0034432] [Citation(s) in RCA: 24] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Somrita Ray
- School of Chemistry, The Raymond and Beverly Sackler Center for Computational Molecular and Materials Science, The Center for Physics and Chemistry of Living Systems, and The Ratner Center for Single Molecule Science, Tel Aviv University, Tel Aviv 69978, Israel
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33
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Bressloff PC. Target competition for resources under multiple search-and-capture events with stochastic resetting. Proc Math Phys Eng Sci 2020; 476:20200475. [PMID: 33223946 PMCID: PMC7655747 DOI: 10.1098/rspa.2020.0475] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/17/2020] [Accepted: 09/15/2020] [Indexed: 12/19/2022] Open
Abstract
We develop a general framework for analysing the distribution of resources in a population of targets under multiple independent search-and-capture events. Each event involves a single particle executing a stochastic search that resets to a fixed location x r at a random sequence of times. Whenever the particle is captured by a target, it delivers a packet of resources and then returns to x r , where it is reloaded with cargo and a new round of search and capture begins. Using renewal theory, we determine the mean number of resources in each target as a function of the splitting probabilities and unconditional mean first passage times of the corresponding search process without resetting. We then use asymptotic PDE methods to determine the effects of resetting on the distribution of resources generated by diffusive search in a bounded two-dimensional domain with N small interior targets. We show that slow resetting increases the total number of resources M tot across all targets provided that ∑ j = 1 N G ( x r , x j ) < 0 , where G is the Neumann Green's function and x j is the location of the j-th target. This implies that M tot can be optimized by varying r. We also show that the k-th target has a competitive advantage if ∑ j = 1 N G ( x r , x j ) > N G ( x r , x k ) .
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Affiliation(s)
- P. C. Bressloff
- Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT 84112, USA
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34
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Bressloff PC. Stochastic resetting and the mean-field dynamics of focal adhesions. Phys Rev E 2020; 102:022134. [PMID: 32942383 DOI: 10.1103/physreve.102.022134] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/20/2020] [Accepted: 08/12/2020] [Indexed: 11/07/2022]
Abstract
In this paper we investigate the effects of diffusion on the dynamics of a single focal adhesion at the leading edge of a crawling cell by considering a simplified model of sliding friction. Using a mean-field approximation, we derive an effective single-particle system that can be interpreted as an overdamped Brownian particle with spatially dependent stochastic resetting. We then use renewal and path-integral methods from the theory of stochastic resetting to calculate the mean sliding velocity under the combined action of diffusion, active forces, viscous drag, and elastic forces generated by the adhesive bonds. Our analysis suggests that the inclusion of diffusion can sharpen the response to changes in the effective stiffness of the adhesion bonds. This is consistent with the hypothesis that force fluctuations could play a role in mechanosensing of the local microenvironment.
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Affiliation(s)
- Paul C Bressloff
- Department of Mathematics, University of Utah Salt Lake City, Utah 84112, USA
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35
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Bressloff PC. Search processes with stochastic resetting and multiple targets. Phys Rev E 2020; 102:022115. [PMID: 32942430 DOI: 10.1103/physreve.102.022115] [Citation(s) in RCA: 16] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/09/2020] [Accepted: 07/22/2020] [Indexed: 11/07/2022]
Abstract
Search processes with stochastic resetting provide a general theoretical framework for understanding a wide range of naturally occurring phenomena. Most current models focus on the first-passage-time problem of finding a single target in a given search domain. Here we use a renewal method to derive general expressions for the splitting probabilities and conditional mean first passage times (MFPTs) in the case of multiple targets. Our analysis also incorporates the effects of delays arising from finite return times and refractory periods. Carrying out a small-r expansion, where r is the mean resetting rate, we obtain general conditions for when resetting increases the splitting probability or reduces the conditional MFPT to a particular target. This also depends on whether π_{tot}=1 or π_{tot}<1, where π_{tot} is the probability that the particle is eventually absorbed by one of the targets in the absence of resetting. We illustrate the theory by considering two distinct examples. The first consists of an actin-rich cell filament (cytoneme) searching along a one-dimensional array of target cells, a problem for which the splitting probabilities and MFPTs can be calculated explicitly. In particular, we highlight how the resetting rate plays an important role in shaping the distribution of splitting probabilities along the array. The second example involves a search process in a three-dimensional bounded domain containing a set of N small interior targets. We use matched asymptotics and Green's functions to determine the behavior of the splitting probabilities and MFPTs in the small-r regime. In particular, we show that the splitting probabilities and MFPTs depend on the "shape capacitance" of the targets.
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Affiliation(s)
- Paul C Bressloff
- Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, Utah 84112, USA
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36
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Tal-Friedman O, Pal A, Sekhon A, Reuveni S, Roichman Y. Experimental Realization of Diffusion with Stochastic Resetting. J Phys Chem Lett 2020; 11:7350-7355. [PMID: 32787296 PMCID: PMC7586404 DOI: 10.1021/acs.jpclett.0c02122] [Citation(s) in RCA: 61] [Impact Index Per Article: 15.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 05/13/2023]
Abstract
Stochastic resetting is prevalent in natural and man-made systems, giving rise to a long series of nonequilibrium 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 energetic cost of resetting in steady-state and first-passage scenarios. In both cases, we show that this cost cannot be made arbitrarily small because of fundamental constraints on realistic resetting protocols. The methods developed herein open the door to future experimental study of resetting phenomena beyond diffusion.
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Affiliation(s)
- Ofir Tal-Friedman
- School
of Physics & Astronomy, Raymond and Beverly Sackler Faculty of
Exact Sciences, Tel Aviv University, Tel Aviv 6997801, Israel
| | - Arnab Pal
- School
of Chemistry, The Center for Physics and Chemistry of Living Systems,
& The Mark Ratner Institute for Single Molecule Chemistry, Tel Aviv University, Tel Aviv 6997801, Israel
| | - Amandeep Sekhon
- School
of Chemistry, The Center for Physics and Chemistry of Living Systems,
& The Mark Ratner Institute for Single Molecule Chemistry, Tel Aviv University, Tel Aviv 6997801, Israel
| | - Shlomi Reuveni
- School
of Chemistry, The Center for Physics and Chemistry of Living Systems,
& The Mark Ratner Institute for Single Molecule Chemistry, Tel Aviv University, Tel Aviv 6997801, Israel
| | - Yael Roichman
- School
of Physics & Astronomy, Raymond and Beverly Sackler Faculty of
Exact Sciences, Tel Aviv University, Tel Aviv 6997801, Israel
- School
of Chemistry, The Center for Physics and Chemistry of Living Systems,
& The Mark Ratner Institute for Single Molecule Chemistry, Tel Aviv University, Tel Aviv 6997801, Israel
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37
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Bressloff PC. Queueing theory of search processes with stochastic resetting. Phys Rev E 2020; 102:032109. [PMID: 33075909 DOI: 10.1103/physreve.102.032109] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/25/2020] [Accepted: 08/21/2020] [Indexed: 06/11/2023]
Abstract
We use queueing theory to develop a general framework for analyzing search processes with stochastic resetting, under the additional assumption that following absorption by a target, the particle (searcher) delivers a packet of resources to the target and the search process restarts at the reset point x_{r}. This leads to a sequence of search-and-capture events, whereby resources accumulate in the target under the combined effects of resource supply and degradation. Combining the theory of G/M/∞ queues with a renewal method for analyzing resetting processes, we derive general expressions for the mean and variance of the number of resource packets within the target at steady state. These expressions apply to both exponential and nonexponential resetting protocols and take into account delays arising from various factors such as finite return times, refractory periods, and delays due to the loading or unloading of resources. In the case of exponential resetting, we show how the resource statistics can be expressed in terms of the MFPTs T_{r}(x_{r}) and T_{r+γ}(x_{r}), where r is the resetting rate and γ is the degradation rate. This allows us to derive various general results concerning the dependence of the mean and variance on the parameters r,γ. Our results are illustrated using several specific examples. Finally, we show how fluctuations can be reduced either by allowing the delivery of multiple packets that degrade independently or by having multiple independent searchers.
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Affiliation(s)
- Paul C Bressloff
- Department of Mathematics, University of Utah, Salt Lake City, Utah 84112, USA
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38
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Ahmad S, Das D. Role of dimensions in first passage of a diffusing particle under stochastic resetting and attractive bias. Phys Rev E 2020; 102:032145. [PMID: 33075998 DOI: 10.1103/physreve.102.032145] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/21/2020] [Accepted: 09/08/2020] [Indexed: 06/11/2023]
Abstract
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)=kR^{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 of the problem without resetting are known. We find the dimensionless critical strength κ_{c,n}(k_{c}) exactly when d/n and 2/n take positive integral values. Also for the limiting case of a box potential (representing n→∞), and the special case of a logarithmic potential kln(R/a), we find the corresponding transition points κ_{c,∞} and κ_{c,l} exactly for any dimension d. The asymptotic forms of the critical strengths at large dimensions d are interesting. We show that for the power law potential, for any n∈(0,∞), the dimensionless critical strength κ_{c,n}∼d^{1/n} at large d. For the box potential, asymptotically, κ_{c,∞}∼(1-ln(d/2)/d), while for the logarithmic potential, κ_{c,l}∼d.
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Affiliation(s)
- Saeed Ahmad
- Physics Department, Indian Institute of Technology Bombay, Mumbai 400076, India
| | - Dibyendu Das
- Physics Department, Indian Institute of Technology Bombay, Mumbai 400076, India
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39
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De Bruyne B, Randon-Furling J, Redner S. Optimization in First-Passage Resetting. PHYSICAL REVIEW LETTERS 2020; 125:050602. [PMID: 32794864 DOI: 10.1103/physrevlett.125.050602] [Citation(s) in RCA: 28] [Impact Index Per Article: 7.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/06/2020] [Accepted: 06/25/2020] [Indexed: 06/11/2023]
Abstract
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 nonstationary and its probability distribution exhibits rich features. In a finite domain, we define a nontrivial 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.
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Affiliation(s)
- B De Bruyne
- Perimeter Institute, 31 Caroline Street North, Waterloo, Ontario, N2L 2Y5, Canada
- CentraleSupélec, Université Paris-Saclay, 3 rue Joliot-Curie, 91190 Gif-sur-Yvette, France
| | - J Randon-Furling
- SAMM, Université Paris 1 Panthéon Sorbonne-FP2M (FR2036) CNRS, 90 rue de Tolbiac, 75013 Paris, France
- Department of Mathematics, Columbia University, 2990 Broadway, New York, New York 10027, USA
| | - S Redner
- Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA
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40
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Ray S, Reuveni S. Diffusion with resetting in a logarithmic potential. J Chem Phys 2020; 152:234110. [DOI: 10.1063/5.0010549] [Citation(s) in RCA: 41] [Impact Index Per Article: 10.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/10/2023] Open
Affiliation(s)
- Somrita Ray
- School of Chemistry, The Center for Physics and Chemistry of Living Systems, The Raymond and Beverly Sackler Center for Computational Molecular and Materials Science, and The Ratner Center for Single Molecule Science, Tel Aviv University, Tel Aviv 69978, Israel
| | - Shlomi Reuveni
- School of Chemistry, The Center for Physics and Chemistry of Living Systems, The Raymond and Beverly Sackler Center for Computational Molecular and Materials Science, and The Ratner Center for Single Molecule Science, Tel Aviv University, Tel Aviv 69978, Israel
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41
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Bodrova AS, Sokolov IM. Resetting processes with noninstantaneous return. Phys Rev E 2020; 101:052130. [PMID: 32575253 DOI: 10.1103/physreve.101.052130] [Citation(s) in RCA: 31] [Impact Index Per Article: 7.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/29/2019] [Accepted: 02/27/2020] [Indexed: 01/05/2023]
Abstract
We consider a random two-phase process which we call a reset-return one. The particle starts its motion at the origin. The first, displacement, phase corresponds to a stochastic motion of a particle and is finished at a resetting event. The second, return, phase corresponds to the particle's motion toward the origin from the position it attained at the end of the displacement phase. This motion toward the origin takes place according to a given equation of motion. The whole process is a renewal one. We provide general expressions for the stationary probability density function of the particle's position and for the mean hitting time in one dimension. We perform explicit analysis for the Brownian motion during the displacement phase and three different types of the return motion: return at a constant speed, return at a constant acceleration with zero initial speed, and return under the action of a harmonic force. We assume that the waiting times for resetting events follow an exponential distribution or that resetting takes place after a fixed waiting period. For the first two types of return motion and the exponential resetting, the stationary probability density function of the particle's position is invariant under return speed (acceleration), while no such invariance is found for deterministic resetting, and for exponential resetting with return under the action of the harmonic force. We discuss necessary conditions for such invariance of the stationary PDF of the positions with respect to the properties of the return process, and we demonstrate some additional examples when this invariance does or does not take place.
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Affiliation(s)
- Anna S Bodrova
- Humboldt University, Department of Physics, Newtonstrasse 15, 12489 Berlin, Germany.,Moscow Institute of Electronics and Mathematics, National Research University Higher School of Economics, 123458, Moscow, Russia.,Faculty of Physics, M.V.Lomonosov Moscow State University, 119991 Moscow, Russia
| | - Igor M Sokolov
- Humboldt University, Department of Physics, Newtonstrasse 15, 12489 Berlin, Germany.,IRIS Adlershof, Zum Großen Windkanal 6, 12489 Berlin, Germany
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42
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Masó-Puigdellosas A, Campos D, Méndez V. Transport properties of random walks under stochastic noninstantaneous resetting. Phys Rev E 2019; 100:042104. [PMID: 31770871 DOI: 10.1103/physreve.100.042104] [Citation(s) in RCA: 40] [Impact Index Per Article: 8.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/20/2019] [Indexed: 01/21/2023]
Abstract
Random walks with stochastic resetting provides a treatable framework to study interesting features about central-place motion. In this work, we introduce noninstantaneous resetting as a two-state model being a combination of an exploring state where the walker moves randomly according to a propagator and a returning state where the walker performs a ballistic motion with constant velocity towards the origin. We study the emerging transport properties for two types of reset time probability density functions (PDFs): exponential and Pareto. In the first case, we find the stationary distribution and a general expression for the stationary mean-square displacement (MSD) in terms of the propagator. We find that the stationary MSD may increase, decrease or remain constant with the returning velocity. This depends on the moments of the propagator. Regarding the Pareto resetting PDF we also study the stationary distribution and the asymptotic scaling of the MSD for diffusive motion. In this case, we see that the resetting modifies the transport regime, making the overall transport subdiffusive and even reaching a stationary MSD, i.e., a stochastic localization. This phenomena is also observed in diffusion under instantaneous Pareto resetting. We check the main results with stochastic simulations of the process.
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Affiliation(s)
- Axel Masó-Puigdellosas
- Grup de Física Estadística, Departament de Física, Facultat de Ciències, Edifici Cc, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
| | - Daniel Campos
- Grup de Física Estadística, Departament de Física, Facultat de Ciències, Edifici Cc, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
| | - Vicenç Méndez
- Grup de Física Estadística, Departament de Física, Facultat de Ciències, Edifici Cc, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
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43
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Pal A, Kuśmierz Ł, Reuveni S. Time-dependent density of diffusion with stochastic resetting is invariant to return speed. Phys Rev E 2019; 100:040101. [PMID: 31770943 DOI: 10.1103/physreve.100.040101] [Citation(s) in RCA: 25] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/22/2019] [Indexed: 01/07/2023]
Abstract
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 coupling between the particle's random position at the moment of resetting and its return time, and further gives rise to a nontrivial cross-talk between two separate phases of motion: the diffusive phase and the return phase. We show that each of these phases relaxes to the steady state in a unique manner; and while this could have also rendered the total relaxation dynamics extremely nontrivial, our analysis surprisingly reveals otherwise. Indeed, the time-dependent distribution describing the particle's position in our model is completely invariant to the speed of return. Thus, whether returns are slow or fast, we always recover the result originally obtained for diffusion with instantaneous returns to the origin.
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Affiliation(s)
- Arnab Pal
- School of Chemistry, The Center for Physics and Chemistry of Living Systems, The Raymond and Beverly Sackler Center for Computational Molecular and Materials Science, and The Mark Ratner Institute for Single Molecule Chemistry, Tel Aviv University, Tel Aviv 6997801, Israel
| | - Łukasz Kuśmierz
- Laboratory for Neural Computation and Adaptation, RIKEN Center for Brain Science, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
| | - Shlomi Reuveni
- School of Chemistry, The Center for Physics and Chemistry of Living Systems, The Raymond and Beverly Sackler Center for Computational Molecular and Materials Science, and The Mark Ratner Institute for Single Molecule Chemistry, Tel Aviv University, Tel Aviv 6997801, Israel
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44
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Basu U, Kundu A, Pal A. Symmetric exclusion process under stochastic resetting. Phys Rev E 2019; 100:032136. [PMID: 31639904 DOI: 10.1103/physreve.100.032136] [Citation(s) in RCA: 45] [Impact Index Per Article: 9.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/28/2019] [Indexed: 11/07/2022]
Abstract
We study the behavior of a symmetric exclusion process (SEP) in the presence of stochastic resetting where the configuration of the system is reset to a steplike profile with a fixed rate r. We show that the presence of resetting affects both the stationary and dynamical properties of SEPs strongly. We compute the exact time-dependent density profile and show that the stationary state is characterized by a nontrivial inhomogeneous profile in contrast to the flat one for r=0. We also show that for r>0 the average diffusive current grows linearly with time t, in stark contrast to the sqrt[t] growth for r=0. In addition to the underlying diffusive current, we identify the resetting current in the system which emerges due to the sudden relocation of the particles to the steplike configuration and is strongly correlated to the diffusive current. We show that the average resetting current is negative, but its magnitude also grows linearly with time t. We also compute the probability distributions of the diffusive current, resetting current, and total current (sum of the diffusive and the resetting currents) using the renewal approach. We demonstrate that while the typical fluctuations of both the diffusive and reset currents around the mean are typically Gaussian, the distribution of the total current shows a strong non-Gaussian behavior.
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Affiliation(s)
- Urna Basu
- Raman Research Institute, Bengaluru 560080, India
| | - Anupam Kundu
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
| | - Arnab Pal
- School of Chemistry, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 6997801, Israel.,Center for the Physics and Chemistry of Living Systems, Tel Aviv University, 6997801, Tel Aviv, Israel.,Sackler Center for Computational Molecular and Materials Science, Tel Aviv University, 6997801, Tel Aviv, Israel
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45
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Masoliver J, Montero M. Anomalous diffusion under stochastic resettings: A general approach. Phys Rev E 2019; 100:042103. [PMID: 31770932 DOI: 10.1103/physreve.100.042103] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/15/2019] [Indexed: 06/10/2023]
Abstract
We present a general formulation of the resetting problem which is valid for any distribution of resetting intervals and arbitrary underlying processes. We show that in such a general case, a stationary distribution may exist even if the reset-free process is not stationary, as well as a significant decreasing in the mean first-passage time. We apply the general formalism to anomalous diffusion processes which allow simple and explicit expressions for Poissonian resetting events.
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Affiliation(s)
- Jaume Masoliver
- Department of Condensed Matter Physics and Institute of Complex Systems (UBICS), University of Barcelona, Catalonia, Spain
| | - Miquel Montero
- Department of Condensed Matter Physics and Institute of Complex Systems (UBICS), University of Barcelona, Catalonia, Spain
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46
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Kuśmierz Ł, Toyoizumi T. Robust random search with scale-free stochastic resetting. Phys Rev E 2019; 100:032110. [PMID: 31639919 DOI: 10.1103/physreve.100.032110] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/30/2018] [Indexed: 06/10/2023]
Abstract
A new model of search based on stochastic resetting is introduced, wherein rate of resets depends explicitly on time elapsed since the beginning of the process. It is shown that rate inversely proportional to time leads to paradoxical diffusion which mixes self-similarity and linear growth of the mean-square displacement with nonlocality and non-Gaussian propagator. It is argued that such resetting protocol offers a general and efficient search-boosting method that does not need to be optimized with respect to the scale of the underlying search problem (e.g., distance to the goal) and is not very sensitive to other search parameters. Both subdiffusive and superdiffusive regimes of the mean-squared displacement scaling are demonstrated with more general rate functions.
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Affiliation(s)
- Łukasz Kuśmierz
- Laboratory for Neural Computation and Adaptation, RIKEN Center for Brain Science, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
| | - Taro Toyoizumi
- Laboratory for Neural Computation and Adaptation, RIKEN Center for Brain Science, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
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47
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Bodrova AS, Chechkin AV, Sokolov IM. Scaled Brownian motion with renewal resetting. Phys Rev E 2019; 100:012120. [PMID: 31499761 DOI: 10.1103/physreve.100.012120] [Citation(s) in RCA: 36] [Impact Index Per Article: 7.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/13/2018] [Indexed: 05/27/2023]
Abstract
We investigate an intermittent stochastic process in which the diffusive motion with time-dependent diffusion coefficient D(t)∼t^{α-1} with α>0 (scaled Brownian motion) is stochastically reset to its initial position, and starts anew. 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 the coordinate does not affect the diffusion coefficient's time dependence is considered in the other work of this series [A. S. Bodrova et al., Phys. Rev. E 100, 012119 (2019)10.1103/PhysRevE.100.012119]. We show that the properties of the probability densities in such processes (erasing or retaining the memory on the diffusion coefficient) are vastly different. In addition we discuss the first-passage properties of the scaled Brownian motion with renewal resetting and consider the dependence of the efficiency of search on the parameters of the process.
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Affiliation(s)
- Anna S Bodrova
- Department of Physics, Humboldt University, Newtonstrasse 15, 12489 Berlin, Germany
- Moscow Institute of Electronics and Mathematics, National Research University Higher School of Economics, 123458 Moscow, Russia
- Faculty of Physics, M. V. Lomonosov Moscow State University, Moscow 119991, Russia
| | - Aleksei V Chechkin
- Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany
- Akhiezer Institute for Theoretical Physics, Kharkov Institute of Physics and Technology, Kharkov 61108, Ukraine
| | - Igor M Sokolov
- Department of Physics, Humboldt University, Newtonstrasse 15, 12489 Berlin, Germany
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48
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Kuśmierz Ł, Gudowska-Nowak E. Subdiffusive continuous-time random walks with stochastic resetting. Phys Rev E 2019; 99:052116. [PMID: 31212503 DOI: 10.1103/physreve.99.052116] [Citation(s) in RCA: 21] [Impact Index Per Article: 4.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/20/2018] [Indexed: 06/09/2023]
Abstract
We analyze two models of subdiffusion with stochastic resetting. Each of them consists of two parts: subdiffusion based on the continuous-time random walk scheme and independent resetting events generated uniformly in time according to the Poisson point process. In the first model the whole process is reset to the initial state, whereas in the second model only the position is subject to resets. The distinction between these two models arises from the non-Markovian character of the subdiffusive process. We derive exact expressions for the two lowest moments of the full propagator, stationary distributions, and first hitting time statistics. We also show, with an example of a constant drift, how these models can be generalized to include external forces. Possible applications to data analysis and modeling of biological systems are also discussed.
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Affiliation(s)
- Łukasz Kuśmierz
- Laboratory for Neural Computation and Adaptation, RIKEN Center for Brain Science, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
| | - Ewa Gudowska-Nowak
- Marian Smoluchowski Institute of Physics, Jagiellonian University, ul. Łojasiewicza 11, 30-348 Kraków, Poland and Mark Kac Complex Systems Research Center, Jagiellonian University, ul. Łojasiewicza 11, 30-348 Kraków, Poland
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49
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Robin T, Hadany L, Urbakh M. Random search with resetting as a strategy for optimal pollination. Phys Rev E 2019; 99:052119. [PMID: 31212560 DOI: 10.1103/physreve.99.052119] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/28/2018] [Indexed: 06/09/2023]
Abstract
The problem of pollination is unique among a wide scope of search problems, since it requires optimization of benefits for both the searcher (pollinator) and its targets (plants). To address this challenge, we propose a pollination model which is based on a framework of first passage under stochastic restart. We derive equations for the search time and number of visited plants as functions of the distribution of nectar in the plant population and of the probability that a pollinator will leave the plant after examining a flower, thus effectively restarting the search. We demonstrate that nectar variation in plants serves as a driving force for pollination and establish conditions required for optimal pollination, which provides an efficient pollinator search strategy and the maximum number of plants visited by the pollinator.
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Affiliation(s)
- Tal Robin
- School of Chemistry, The Raymond and Beverly Sackler Faculty of Exact Sciences and The Sackler Center for Computational Molecular and Materials Science, Tel Aviv University, Tel Aviv 6997801, Israel
| | - Lilach Hadany
- School of Molecular Biology and Ecology of Plants, The George S. Wise Faculty of Life Sciences, Tel Aviv University, Tel Aviv 6997801, Israel
| | - Michael Urbakh
- School of Chemistry, The Raymond and Beverly Sackler Faculty of Exact Sciences and The Sackler Center for Computational Molecular and Materials Science, Tel Aviv University, Tel Aviv 6997801, Israel
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Pal A, Prasad VV. First passage under stochastic resetting in an interval. Phys Rev E 2019; 99:032123. [PMID: 30999497 DOI: 10.1103/physreve.99.032123] [Citation(s) in RCA: 41] [Impact Index Per Article: 8.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/16/2018] [Indexed: 05/27/2023]
Abstract
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 on 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 the probability that the particle will eventually be trapped on either of the boundaries without hitting the other one. Numerical studies are presented to support our analytic results.
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Affiliation(s)
- Arnab Pal
- School of Chemistry, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 6997801, Israel
- Center for the Physics and Chemistry of Living Systems. Tel Aviv University, 6997801, Tel Aviv, Israel
- The Sackler Center for Computational Molecular and Materials Science, Tel Aviv University, 6997801, Tel Aviv, Israel
| | - V V Prasad
- Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 7610001, Israel
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