1
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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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2
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Lenzi EK, Zola RS, Rosseto MP, Mendes RS, Ribeiro HV, da Silva LR, Evangelista LR. Results for Nonlinear Diffusion Equations with Stochastic Resetting. ENTROPY (BASEL, SWITZERLAND) 2023; 25:1647. [PMID: 38136527 PMCID: PMC10742535 DOI: 10.3390/e25121647] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/06/2023] [Revised: 12/03/2023] [Accepted: 12/08/2023] [Indexed: 12/24/2023]
Abstract
In this study, we investigate a nonlinear diffusion process in which particles stochastically reset to their initial positions at a constant rate. The nonlinear diffusion process is modeled using the porous media equation and its extensions, which are nonlinear diffusion equations. We use analytical and numerical calculations to obtain and interpret the probability distribution of the position of the particles and the mean square displacement. These results are further compared and shown to agree with the results of numerical simulations. Our findings show that a system of this kind exhibits non-Gaussian distributions, transient anomalous diffusion (subdiffusion and superdiffusion), and stationary states that simultaneously depend on the nonlinearity and resetting rate.
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Affiliation(s)
- Ervin K. Lenzi
- Departamento de Física, Universidade Estadual de Ponta Grossa, Ponta Grossa 84030-900, PR, Brazil;
- National Institute of Science and Technology for Complex Systems, Centro Brasileiro de Pesquisas Físicas, Rio de Janeiro 22290-180, RJ, Brazil;
| | - Rafael S. Zola
- Departamento de Física, Universidade Tecnológica Federal do Paraná, Apucarana 86812-460, PR, Brazil;
| | - Michely P. Rosseto
- Departamento de Física, Universidade Estadual de Ponta Grossa, Ponta Grossa 84030-900, PR, Brazil;
| | - Renio S. Mendes
- Departamento de Física, Universidade Estadual de Maringá, Maringa 87020-900, PR, Brazil; (R.S.M.); (H.V.R.); (L.R.E.)
| | - Haroldo V. Ribeiro
- Departamento de Física, Universidade Estadual de Maringá, Maringa 87020-900, PR, Brazil; (R.S.M.); (H.V.R.); (L.R.E.)
| | - Luciano R. da Silva
- National Institute of Science and Technology for Complex Systems, Centro Brasileiro de Pesquisas Físicas, Rio de Janeiro 22290-180, RJ, Brazil;
- Departamento de Física, Universidade Federal do Rio Grande do Norte, Natal 59078-900, RN, Brazil
| | - Luiz R. Evangelista
- Departamento de Física, Universidade Estadual de Maringá, Maringa 87020-900, PR, Brazil; (R.S.M.); (H.V.R.); (L.R.E.)
- Istituto dei Sistemi Complessi (ISC–CNR), Via dei Taurini, 19, 00185 Rome, Italy
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3
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Trajanovski P, Jolakoski P, Zelenkovski K, Iomin A, Kocarev L, Sandev T. Ornstein-Uhlenbeck process and generalizations: Particle dynamics under comb constraints and stochastic resetting. Phys Rev E 2023; 107:054129. [PMID: 37328979 DOI: 10.1103/physreve.107.054129] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/27/2023] [Accepted: 05/01/2023] [Indexed: 06/18/2023]
Abstract
The Ornstein-Uhlenbeck process is interpreted as Brownian motion in a harmonic potential. This Gaussian Markov process has a bounded variance and admits a stationary probability distribution, in contrast to the standard Brownian motion. It also tends to a drift towards its mean function, and such a process is called mean reverting. Two examples of the generalized Ornstein-Uhlenbeck process are considered. In the first one, we study the Ornstein-Uhlenbeck process on a comb model, as an example of the harmonically bounded random motion in the topologically constrained geometry. The main dynamical characteristics (as the first and the second moments) and the probability density function are studied in the framework of both the Langevin stochastic equation and the Fokker-Planck equation. The second example is devoted to the study of the effects of stochastic resetting on the Ornstein-Uhlenbeck process, including stochastic resetting in the comb geometry. Here the nonequilibrium stationary state is the main question in task, where the two divergent forces, namely, the resetting and the drift towards the mean, lead to compelling results in the cases of both the Ornstein-Uhlenbeck process with resetting and its generalization on the two-dimensional comb structure.
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Affiliation(s)
- Pece Trajanovski
- Research Center for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts, Bul. Krste Misirkov 2, 1000 Skopje, Macedonia
| | - Petar Jolakoski
- Research Center for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts, Bul. Krste Misirkov 2, 1000 Skopje, Macedonia
| | - Kiril Zelenkovski
- Research Center for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts, Bul. Krste Misirkov 2, 1000 Skopje, Macedonia
| | - Alexander Iomin
- Department of Physics, Technion, Haifa 32000, Israel
- Max Planck Institute for the Physics of Complex Systems, D-01187 Dresden, Germany
| | - Ljupco Kocarev
- Research Center for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts, Bul. Krste Misirkov 2, 1000 Skopje, Macedonia
- Faculty of Computer Science and Engineering, Ss. Cyril and Methodius University, PO Box 393, 1000 Skopje, Macedonia
| | - Trifce 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
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4
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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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5
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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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6
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Tal-Friedman O, Roichman Y, Reuveni S. Diffusion with partial resetting. Phys Rev E 2022; 106:054116. [PMID: 36559492 DOI: 10.1103/physreve.106.054116] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/06/2022] [Accepted: 09/23/2022] [Indexed: 11/09/2022]
Abstract
Inspired by many examples in nature, stochastic resetting of random processes has been studied extensively in the past decade. In particular, various models of stochastic particle motion were considered where, upon resetting, the particle is returned to its initial position. Here we generalize the model of diffusion with resetting to account for situations where a particle is returned only a fraction of its distance to the origin, e.g., half way. We show that this model always attains a steady-state distribution which can be written as an infinite sum of independent, but not identical, Laplace random variables. As a result, we find that the steady-state transitions from the known Laplace form which is obtained in the limit of full resetting to a Gaussian form, which is obtained close to the limit of no resetting. A similar transition is shown to be displayed by drift diffusion whose steady state can also be expressed as an infinite sum of independent random variables. Finally, we extend our analysis to capture the temporal evolution of drift diffusion with partial resetting, providing a bottom-up probabilistic construction that yields a closed-form solution for the time-dependent distribution of this process in Fourier-Laplace space. Possible extensions and applications of diffusion with partial resetting are discussed.
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Affiliation(s)
- Ofir Tal-Friedman
- School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 6997801, Israel
| | - Yael Roichman
- School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 6997801, Israel.,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
| | - Shlomi Reuveni
- 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
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7
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Masó-Puigdellosas A, Campos D, Méndez V. Conditioned backward and forward times of diffusion with stochastic resetting: A renewal theory approach. Phys Rev E 2022; 106:034126. [PMID: 36266817 DOI: 10.1103/physreve.106.034126] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/03/2022] [Accepted: 09/05/2022] [Indexed: 06/16/2023]
Abstract
Stochastic resetting can be naturally understood as a renewal process governing the evolution of an underlying stochastic process. In this work, we formally derive well-known results of diffusion with resets from a renewal theory perspective. Parallel to the concepts from renewal theory, we introduce the conditioned backward B and forward F times being the times since the last and until the next reset, respectively, given that the current state of the system X(t) is known. These magnitudes are introduced with the paradigmatic case of diffusion under resetting, for which the backward and forward times are conditioned to the position of the walker. We find analytical expressions for the conditioned backward and forward time probability density functions (PDFs), and we compare them with numerical simulations. The general expressions allow us to study particular scenarios. For instance, for power-law reset time PDFs such that φ(t)∼t^{-1-α}, significant changes in the properties of the conditioned backward and forward times happen at half-integer values of α due to the composition between the long-time scaling of diffusion P(x,t)∼1/sqrt[t] and the reset time PDF.
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Affiliation(s)
- Axel Masó-Puigdellosas
- Grup de Física Estadística, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain
| | - Daniel Campos
- Grup de Física Estadística, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain
| | - Vicenç Méndez
- Grup de Física Estadística, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain
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8
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Singh RK, Górska K, Sandev T. General approach to stochastic resetting. Phys Rev E 2022; 105:064133. [PMID: 35854558 DOI: 10.1103/physreve.105.064133] [Citation(s) in RCA: 5] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/23/2022] [Accepted: 05/23/2022] [Indexed: 06/15/2023]
Abstract
We address the effect of stochastic resetting on diffusion and subdiffusion process. For diffusion we find that mean square displacement relaxes to a constant only when the distribution of reset times possess finite mean and variance. In this case, the leading order contribution to the probability density function (PDF) of a Gaussian propagator under resetting exhibits a cusp independent of the specific details of the reset time distribution. For subdiffusion we derive the PDF in Laplace space for arbitrary resetting protocol. Resetting at constant rate allows evaluation of the PDF in terms of H function. We analyze the steady state and derive the rate function governing the relaxation behavior. For a subdiffusive process the steady state could exist even if the distribution of reset times possesses only finite mean.
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Affiliation(s)
- R K Singh
- Department of Physics, Bar-Ilan University, Ramat-Gan 5290002, Israel
| | - K Górska
- Institute of Nuclear Physics, Polish Academy of Sciences, Radzikowskiego 152, PL-31342 Kraków, Poland
| | - 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
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9
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Méndez V, Masó-Puigdellosas A, Campos D. Nonstandard diffusion under Markovian resetting in bounded domains. Phys Rev E 2022; 105:054118. [PMID: 35706242 DOI: 10.1103/physreve.105.054118] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/28/2022] [Accepted: 04/13/2022] [Indexed: 06/15/2023]
Abstract
We consider a walker moving in a one-dimensional interval with absorbing boundaries under the effect of Markovian resettings to the initial position. The walker's motion follows a random walk characterized by a general waiting time distribution between consecutive short jumps. We investigate the existence of an optimal reset rate, which minimizes the mean exit passage time, in terms of the statistical properties of the waiting time probability. Generalizing previous results, we find that when the waiting time probability has first and second finite moments, resetting can be either (i) never beneficial, (ii) beneficial depending on the distance of the reset point to the boundary, or (iii) always beneficial. Instead, when the waiting time probability has the first or the two first moments diverging we find that resetting is always beneficial. Finally, we have also found that the optimal strategy to exit the domain depends on the reset rate. For low reset rates, walkers with exponential waiting times are found to be optimal, while for high reset rate, anomalous waiting times optimize the search process.
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Affiliation(s)
- Vicenç Méndez
- Grup de Física Estadística. Departament de Física. Facultat de Ciències. Edifici Cc. Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona) Spain
| | - 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
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10
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Zhou T, Xu P, Deng W. Gaussian process and Lévy walk under stochastic noninstantaneous resetting and stochastic rest. Phys Rev E 2021; 104:054124. [PMID: 34942832 DOI: 10.1103/physreve.104.054124] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/25/2021] [Accepted: 11/08/2021] [Indexed: 11/07/2022]
Abstract
A stochastic process with movement, return, and rest phases is considered in this paper. For the movement phase, the particles move following the dynamics of the Gaussian process or the ballistic type of Lévy walk, and the time of each movement is random. For the return phase, the particles will move back to the origin with a constant velocity or acceleration or under the action of a harmonic force after each movement, so that this phase can also be treated as a noninstantaneous resetting. After each return, a rest with a random time at the origin follows. The asymptotic behaviors of the mean-squared displacements with different kinds of movement dynamics, returning, and random resting time are discussed. The stationary distributions are also considered when the process is localized. In addition, the mean first passage time is considered when the dynamic of the movement phase is Brownian motion.
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Affiliation(s)
- Tian Zhou
- School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, People's Republic of China
| | - Pengbo Xu
- School of Mathematical Sciences, Peking University, Beijing 100871, People's Republic of China
| | - Weihua Deng
- School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, People's Republic of China
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11
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Stojkoski V, Sandev T, Kocarev L, Pal A. Geometric Brownian motion under stochastic resetting: A stationary yet nonergodic process. Phys Rev E 2021; 104:014121. [PMID: 34412255 DOI: 10.1103/physreve.104.014121] [Citation(s) in RCA: 16] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/07/2021] [Accepted: 06/16/2021] [Indexed: 01/19/2023]
Abstract
We study the effects of stochastic resetting on geometric Brownian motion with drift (GBM), a canonical stochastic multiplicative process for nonstationary and nonergodic dynamics. Resetting is a sudden interruption of a process, which consecutively renews its dynamics. We show that, although resetting renders GBM stationary, the resulting process remains nonergodic. Quite surprisingly, the effect of resetting is pivotal in manifesting the nonergodic behavior. In particular, we observe three different long-time regimes: a quenched state, an unstable state, and a stable annealed state depending on the resetting strength. Notably, in the last regime, the system is self-averaging and thus the sample average will always mimic ergodic behavior establishing a stand-alone feature for GBM under resetting. Crucially, the above-mentioned regimes are well separated by a self-averaging time period which can be minimized by an optimal resetting rate. Our results can be useful to interpret data emanating from stock market collapse or reconstitution of investment portfolios.
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Affiliation(s)
- Viktor Stojkoski
- Faculty of Economics, Ss. Cyril and Methodius University, 1000 Skopje, Macedonia.,Research Center for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts, Bul. Krste Misirkov 2, 1000 Skopje, Macedonia
| | - Trifce 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 and 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
| | - Ljupco Kocarev
- Research Center for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts, Bul. Krste Misirkov 2, 1000 Skopje, Macedonia.,Faculty of Computer Science and Engineering, Ss. Cyril and Methodius University, P.O. Box 393, 1000 Skopje, Macedonia
| | - Arnab Pal
- School of Chemistry, The Center for Physics and Chemistry of Living Systems, Tel Aviv University, Tel Aviv 6997801, Israel
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12
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Stanislavsky A, Weron A. Optimal non-Gaussian search with stochastic resetting. Phys Rev E 2021; 104:014125. [PMID: 34412216 DOI: 10.1103/physreve.104.014125] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/07/2021] [Accepted: 06/29/2021] [Indexed: 11/07/2022]
Abstract
In this paper we reveal that each subordinated Brownian process, leading to subdiffusion, under Poissonian resetting has a stationary state with the Laplace distribution. Its location parameter is defined only by the position to which the particle resets, and its scaling parameter is dependent on the Laplace exponent of the random process directing Brownian motion as a parent process. From the analysis of the scaling parameter the probability density function of the stochastic process, subject to reset, can be restored. In this case the mean time for the particle to reach a target is finite and has a minimum, optimal for the resetting rate. If the Brownian process is replaced by the Lévy motion (superdiffusion), then its stationary state obeys the Linnik distribution which belongs to the class of generalized Laplace distributions.
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Affiliation(s)
- Aleksander Stanislavsky
- Institute of Radio Astronomy, 4 Mystetstv Street, 61002 Kharkiv, Ukraine.,Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wrocław University of Science and Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland
| | - Aleksander Weron
- Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wrocław University of Science and Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland
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13
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González FH, Riascos AP, Boyer D. Diffusive transport on networks with stochastic resetting to multiple nodes. Phys Rev E 2021; 103:062126. [PMID: 34271672 DOI: 10.1103/physreve.103.062126] [Citation(s) in RCA: 9] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/01/2021] [Accepted: 05/27/2021] [Indexed: 01/30/2023]
Abstract
We study the diffusive transport of Markovian random walks on arbitrary networks with stochastic resetting to multiple nodes. We deduce analytical expressions for the stationary occupation probability and for the mean and global first passage times. This general approach allows us to characterize the effect of resetting on the capacity of random walk strategies to reach a particular target or to explore the network. Our formalism holds for ergodic random walks and can be implemented from the spectral properties of the random walk without resetting, providing a tool to analyze the efficiency of search strategies with resetting to multiple nodes. We apply the methods developed here to the dynamics with two reset nodes and derive analytical results for normal random walks and Lévy flights on rings. We also explore the effect of resetting to multiple nodes on a comb graph, Lévy flights that visit specific locations in a continuous space, and the Google random walk strategy on regular networks.
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Affiliation(s)
- Fernanda H González
- Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, 01000 Ciudad de México, México
| | - Alejandro P Riascos
- Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, 01000 Ciudad de México, México
| | - Denis Boyer
- Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, 01000 Ciudad de México, México
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14
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Dahlenburg M, Chechkin AV, Schumer R, Metzler R. Stochastic resetting by a random amplitude. Phys Rev E 2021; 103:052123. [PMID: 34134286 DOI: 10.1103/physreve.103.052123] [Citation(s) in RCA: 17] [Impact Index Per Article: 5.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/03/2020] [Accepted: 04/29/2021] [Indexed: 11/07/2022]
Abstract
Stochastic resetting, a diffusive process whose amplitude is reset to the origin at random times, is a vividly studied strategy to optimize encounter dynamics, e.g., in chemical reactions. Here we generalize the resetting step by introducing a random resetting amplitude such that the diffusing particle may be only partially reset towards the trajectory origin or even overshoot the origin in a resetting step. We introduce different scenarios for the random-amplitude stochastic resetting process and discuss the resulting dynamics. Direct applications are geophysical layering (stratigraphy) and population dynamics or financial markets, as well as generic search processes.
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Affiliation(s)
- Marcus Dahlenburg
- Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam, Germany.,Basque Center for Applied Mathematics, 48009 Bilbao, Basque Country, Spain
| | - Aleksei V Chechkin
- Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam, Germany.,Akhiezer Institute for Theoretical Physics, 61108 Kharkov, Ukraine
| | - Rina Schumer
- Desert Research Institute, Reno, Nevada 89512, USA
| | - Ralf Metzler
- Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam, Germany
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15
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Bonomo OL, Pal A. First passage under restart for discrete space and time: Application to one-dimensional confined lattice random walks. Phys Rev E 2021; 103:052129. [PMID: 34134266 DOI: 10.1103/physreve.103.052129] [Citation(s) in RCA: 15] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/03/2021] [Accepted: 05/04/2021] [Indexed: 11/07/2022]
Abstract
First passage under restart has recently emerged as a conceptual framework to study various stochastic processes under restart mechanism. Emanating from the canonical diffusion problem by Evans and Majumdar, restart has been shown to outperform the completion of many first-passage processes which otherwise would take longer time to finish. However, most of the studies so far assumed continuous time underlying first-passage time processes and moreover considered continuous time resetting restricting out restart processes broken up into synchronized time steps. To bridge this gap, in this paper, we study discrete space and time first-passage processes under discrete time resetting in a general setup without specifying their forms. We sketch out the steps to compute the moments and the probability density function which is often intractable in the continuous time restarted process. A criterion that dictates when restart remains beneficial is then derived. We apply our results to a symmetric and a biased random walker in one-dimensional lattice confined within two absorbing boundaries. Numerical simulations are found to be in excellent agreement with the theoretical results. Our method can be useful to understand the effect of restart on the spatiotemporal dynamics of confined lattice random walks in arbitrary dimensions.
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Affiliation(s)
- Ofek Lauber Bonomo
- School of Chemistry, Raymond and Beverly Sackler Faculty of Exact Sciences & The Center for Physics and Chemistry of Living Systems & The Ratner Center for Single Molecule Science, Tel Aviv University, Tel Aviv 6997801, Israel
| | - Arnab Pal
- School of Chemistry, Raymond and Beverly Sackler Faculty of Exact Sciences & The Center for Physics and Chemistry of Living Systems & The Ratner Center for Single Molecule Science, Tel Aviv University, Tel Aviv 6997801, Israel
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16
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Méndez V, Masó-Puigdellosas A, Sandev T, Campos D. Continuous time random walks under Markovian resetting. Phys Rev E 2021; 103:022103. [PMID: 33736111 DOI: 10.1103/physreve.103.022103] [Citation(s) in RCA: 12] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/02/2020] [Accepted: 01/15/2021] [Indexed: 06/12/2023]
Abstract
We investigate the effects of Markovian resetting events on continuous time random walks where the waiting times and the jump lengths are random variables distributed according to power-law probability density functions. We prove the existence of a nonequilibrium stationary state and finite mean first arrival time. However, the existence of an optimum reset rate is conditioned to a specific relationship between the exponents of both power-law tails. We also investigate the search efficiency by finding the optimal random walk which minimizes the mean first arrival time in terms of the reset rate, the distance of the initial position to the target, and the characteristic transport exponents.
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Affiliation(s)
- Vicenç Méndez
- Grup de Física Estadística, Departament de Física, Facultat de Ciències, Edifici Cc., Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
| | - Axel Masó-Puigdellosas
- Grup de Física Estadística, Departament de Física, Facultat de Ciències, Edifici Cc., Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
| | - Trifce 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 and 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
| | - Daniel Campos
- Grup de Física Estadística, Departament de Física, Facultat de Ciències, Edifici Cc., Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
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17
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Bodrova AS, Sokolov IM. Continuous-time random walks under power-law resetting. Phys Rev E 2020; 101:062117. [PMID: 32688585 DOI: 10.1103/physreve.101.062117] [Citation(s) in RCA: 13] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/20/2019] [Accepted: 05/20/2020] [Indexed: 06/11/2023]
Abstract
We study continuous-time random walks (CTRW) with power-law distribution of waiting times under resetting which brings the walker back to the origin, with a power-law distribution of times between the resetting events. Two situations are considered. Under complete resetting, the CTRW after the resetting event starts anew, with a new waiting time, independent of the prehistory. Under incomplete resetting, the resetting of the coordinate does not influence the waiting time until the next jump. We focus on the behavior of the mean-squared displacement (MSD) of the walker from its initial position, on the conditions under which the probability density functions of the walker's displacement show universal behavior, and on this universal behavior itself. We show, that the behavior of the MSD is the same as in the scaled Brownian motion (SBM), being the mean-field model of the CTRW. The intermediate asymptotics of the probability density functions (PDF) for CTRW under complete resetting (provided they exist) are also the same as in the corresponding case for SBM. For incomplete resetting, however, the behavior of the PDF for CTRW and SBM is vastly different.
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Affiliation(s)
- Anna S Bodrova
- Department of Physics, Humboldt University, Newtonstrasse 15, D-12489 Berlin, Germany
| | - Igor M Sokolov
- Department of Physics, Humboldt University, Newtonstrasse 15, D-12489 Berlin, Germany
- IRIS Adlershof, Zum Großen Windkanal 6, D-12489 Berlin, Germany
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18
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Riascos AP, Boyer D, Herringer P, Mateos JL. Random walks on networks with stochastic resetting. Phys Rev E 2020; 101:062147. [PMID: 32688619 DOI: 10.1103/physreve.101.062147] [Citation(s) in RCA: 23] [Impact Index Per Article: 5.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/25/2019] [Accepted: 06/12/2020] [Indexed: 06/11/2023]
Abstract
We study random walks with stochastic resetting to the initial position on arbitrary networks. We obtain the stationary probability distribution as well as the mean and global first passage times, which allow us to characterize the effect of resetting on the capacity of a random walker to reach a particular target or to explore a finite network. We apply the results to rings, Cayley trees, and random and complex networks. Our formalism holds for undirected networks and can be implemented from the spectral properties of the random walk without resetting, providing a tool to analyze the search efficiency in different structures with the small-world property or communities. In this way, we extend the study of resetting processes to the domain of networks.
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Affiliation(s)
- Alejandro P Riascos
- Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, 01000 Ciudad de México, México
| | - Denis Boyer
- Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, 01000 Ciudad de México, México
| | - Paul Herringer
- Department of Physics and Astronomy, University of Calgary, Calgary, Alberta, Canada T2N 1N4
| | - José L Mateos
- Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, 01000 Ciudad de México, México
- Centro de Ciencias de la Complejidad, Universidad Nacional Autónoma de México, 04510 Ciudad de México, México
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19
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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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20
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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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21
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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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22
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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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