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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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Singh RK, Sandev T, Singh S. Bernoulli trial under restarts: A comparative study of resetting transitions. Phys Rev E 2023; 108:L052106. [PMID: 38115400 DOI: 10.1103/physreve.108.l052106] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/02/2023] [Accepted: 10/23/2023] [Indexed: 12/21/2023]
Abstract
A Bernoulli trial describing the escape behavior of a lamb to a safe haven in pursuit by a lion is studied under restarts. The process ends in two ways: either the lamb makes it to the safe haven (success) or is captured by the lion (failure). We study the first passage properties of this Bernoulli trial and find that only mean first passage time exists. Considering Poisson and sharp resetting, we find that the success probability is a monotonically decreasing function of the restart rate. The mean time, however, exhibits a nonmonotonic dependence on the restart rate taking a minimal value at an optimal restart rate. Furthermore, for sharp restart, the mean time possesses a local and a global minima. As a result, the optimal restart rate exhibits a continuous transition for Poisson resetting while it exhibits a discontinuous transition for sharp resetting as a function of the relative separation of the lion and the lamb. We also find that the distribution of first passage times under sharp resetting exhibits a periodic behavior.
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Affiliation(s)
- R K Singh
- Department of Physics, Bar-Ilan University, Ramat-Gan 5290002, Israel
| | - T Sandev
- Research Center for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts, Bul. Krste Misirkov 2, 1000 Skopje, Macedonia
- Institute of Physics & Astronomy, University of Potsdam, D-14776 Potsdam-Golm, Germany
- Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss. Cyril and Methodius University, Arhimedova 3, 1000 Skopje, Macedonia
| | - Sadhana Singh
- The Avram and Stella Goldstein-Goren Department of Biotechnology Engineering, Ben-Gurion University of the Negev, Be'er Sheva 84105, Israel
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Roberts C, Zhen Z. Run-and-tumble motion in a linear ratchet potential: Analytic solution, power extraction, and first-passage properties. Phys Rev E 2023; 108:014139. [PMID: 37583167 DOI: 10.1103/physreve.108.014139] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/15/2023] [Accepted: 06/29/2023] [Indexed: 08/17/2023]
Abstract
We explore the properties of run-and-tumble particles moving in a piecewise-linear "ratchet" potential by deriving analytic results for the system's steady-state probability density, current, entropy production rate, extractable power, and thermodynamic efficiency. The ratchet's broken spatial symmetry rectifies the particles' self-propelled motion, resulting in a positive current that peaks at finite values of the diffusion strength, ratchet height, and particle self-propulsion speed. Similar nonmonotonic behavior is also observed for the extractable power and efficiency. We find the optimal apex position for generating maximum current varies with diffusion and that entropy production can have nonmonotonic dependence on diffusion. In particular, for vanishing diffusion, entropy production remains finite when particle self-propulsion is weaker than the ratchet force. Furthermore, power extraction with near-perfect efficiency is achievable in certain parameter regimes due to the simplifications afforded by modeling "dry" active particles. In the final part, we derive mean first-passage times and splitting probabilities for different boundary and initial conditions. This work connects the study of work extraction from active matter with exactly solvable active particle models and will therefore facilitate the design of active engines through these analytic results.
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Affiliation(s)
- Connor Roberts
- Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom and Centre for Complexity Science, Imperial College London SW7 2AZ, United Kingdom
| | - Zigan Zhen
- Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom and Centre for Complexity Science, Imperial College London SW7 2AZ, United Kingdom
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4
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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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Jain S, Boyer D, Pal A, Dagdug L. Fick-Jacobs description and first passage dynamics for diffusion in a channel under stochastic resetting. J Chem Phys 2023; 158:054113. [PMID: 36754825 DOI: 10.1063/5.0135249] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/09/2023] Open
Abstract
The transport of particles through channels is of paramount importance in physics, chemistry, and surface science due to its broad real world applications. Much insight can be gained by observing the transition paths of a particle through a channel and collecting statistics on the lifetimes in the channel or the escape probabilities from the channel. In this paper, we consider the diffusive transport through a narrow conical channel of a Brownian particle subject to intermittent dynamics, namely, stochastic resetting. As such, resetting brings the particle back to a desired location from where it resumes its diffusive phase. To this end, we extend the Fick-Jacobs theory of channel-facilitated diffusive transport to resetting-induced transport. Exact expressions for the conditional mean first passage times, escape probabilities, and the total average lifetime in the channel are obtained, and their behavior as a function of the resetting rate is highlighted. It is shown that resetting can expedite the transport through the channel-rigorous constraints for such conditions are then illustrated. Furthermore, we observe that a carefully chosen resetting rate can render the average lifetime of the particle inside the channel minimal. Interestingly, the optimal rate undergoes continuous and discontinuous transitions as some relevant system parameters are varied. The validity of our one-dimensional analysis and the corresponding theoretical predictions is supported by three-dimensional Brownian dynamics simulations. We thus believe that resetting can be useful to facilitate particle transport across biological membranes-a phenomenon that can spearhead further theoretical and experimental studies.
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Affiliation(s)
- Siddharth Jain
- Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad (Prayagraj), UP, 211019, India
| | - Denis Boyer
- Instituto de Física, Universidad Nacional Autónoma de México, Ciudad de México C.P. 04510, Mexico
| | - Arnab Pal
- The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India
| | - Leonardo Dagdug
- Physics Department, Universidad Autónoma Metropolitana-Iztapalapa, San Rafael Atlixco 186, Ciudad de México 09340, Mexico
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Singh RK, Singh S. Capture of a diffusing lamb by a diffusing lion when both return home. Phys Rev E 2022; 106:064118. [PMID: 36671194 DOI: 10.1103/physreve.106.064118] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/03/2022] [Accepted: 12/01/2022] [Indexed: 06/17/2023]
Abstract
A diffusing lion pursues a diffusing lamb when both of them are allowed to get back to their homes intermittently. Identifying the system with a pair of vicious random walkers, we study their dynamics under Poissonian and sharp resetting. In the absence of any resets, the location of intersection of the two walkers follows a Cauchy distribution. In the presence of resetting, the distribution of the location of annihilation is composed of two parts: one in which the trajectories cross without being reset (center) and the other where trajectories are reset at least once before they cross each other (tails). We find that the tail part decays exponentially for both the resetting protocols. The central part of the distribution, on the other hand, depends on the nature of the restart protocol, with Cauchy for Poisson resetting and Gaussian for sharp resetting. We find good agreement of the analytical results with numerical calculations.
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Affiliation(s)
- R K Singh
- Department of Physics, Bar-Ilan University, Ramat-Gan 5290002, Israel
| | - Sadhana Singh
- The Avram and Stella Goldstein-Goren Department of Biotechnology Engineering, Ben-Gurion University of the Negev, Be'er Sheva 84105, Israel
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7
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Benkhadaj Z, Grebenkov DS. Encounter-based approach to diffusion with resetting. Phys Rev E 2022; 106:044121. [PMID: 36397494 DOI: 10.1103/physreve.106.044121] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/10/2022] [Accepted: 09/27/2022] [Indexed: 06/16/2023]
Abstract
An encounter-based approach consists in using the boundary local time as a proxy for the number of encounters between a diffusing particle and a target to implement various surface reaction mechanisms on that target. In this paper, we investigate the effects of stochastic resetting onto diffusion-controlled reactions in bounded confining domains. We first discuss the effect of position resetting onto the propagator and related quantities; in this way, we retrieve a number of earlier results but also provide complementary insights into them. Second, we introduce boundary local time resetting and investigate its impact. Curiously, we find that this type of resetting does not alter the conventional propagator governing the diffusive dynamics in the presence of a partially reactive target with a constant reactivity. In turn, the generalized propagator for other surface reaction mechanisms can be significantly affected. Our general results are illustrated for diffusion on an interval with reactive end points. Further perspectives and some open problems are discussed.
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Affiliation(s)
| | - Denis S Grebenkov
- Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS-Ecole Polytechnique, IP Paris, 91128 Palaiseau, France
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