1
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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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2
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Carollo F, Lesanovsky I, Garrahan JP. Universal and nonuniversal probability laws in Markovian open quantum dynamics subject to generalized reset processes. Phys Rev E 2024; 109:044129. [PMID: 38755866 DOI: 10.1103/physreve.109.044129] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/24/2023] [Accepted: 03/12/2024] [Indexed: 05/18/2024]
Abstract
We consider quantum-jump trajectories of Markovian open quantum systems subject to stochastic in time resets of their state to an initial configuration. The reset events provide a partitioning of quantum trajectories into consecutive time intervals, defining sequences of random variables from the values of a trajectory observable within each of the intervals. For observables related to functions of the quantum state, we show that the probability of certain orderings in the sequences obeys a universal law. This law does not depend on the chosen observable and, in the case of Poissonian reset processes, not even on the details of the dynamics. When considering (discrete) observables associated with the counting of quantum jumps, the probabilities in general lose their universal character. Universality is only recovered in cases when the probability of observing equal outcomes in the same sequence is vanishingly small, which we can achieve in a weak-reset-rate limit. Our results extend previous findings on classical stochastic processes [N. R. Smith et al., Europhys. Lett. 142, 51002 (2023)0295-507510.1209/0295-5075/acd79e] to the quantum domain and to state-dependent reset processes, shedding light on relevant aspects for the emergence of universal probability laws.
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Affiliation(s)
- Federico Carollo
- Institut für Theoretische Physik, Universität Tübingen, Auf der Morgenstelle 14, 72076 Tübingen, Germany
| | - Igor Lesanovsky
- Institut für Theoretische Physik, Universität Tübingen, Auf der Morgenstelle 14, 72076 Tübingen, Germany
- School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom
- Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham NG7 2RD, United Kingdom
| | - Juan P Garrahan
- School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom
- Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham NG7 2RD, United Kingdom
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3
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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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4
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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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5
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Santra S, Singh P. Exact fluctuation and long-range correlations in a single-file model under resetting. Phys Rev E 2024; 109:034123. [PMID: 38632800 DOI: 10.1103/physreve.109.034123] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/20/2023] [Accepted: 01/24/2024] [Indexed: 04/19/2024]
Abstract
Resetting is a renewal mechanism in which a process is intermittently repeated after a random or fixed time. This simple act of stop and repeat profoundly influences the behavior of a system as exemplified by the emergence of nonequilibrium properties and expedition of search processes. Herein we explore the ramifications of stochastic resetting in the context of a single-file system called random average process (RAP) in one dimension. In particular, we focus on the dynamics of tracer particles and analytically compute the variance, equal time correlation, autocorrelation, and unequal time correlation between the positions of different tracer particles. Our study unveils that resetting gives rise to rather different behaviors depending on whether the particles move symmetrically or asymmetrically. For the asymmetric case, the system for instance exhibits a long-range correlation which is not seen in absence of the resetting. Similarly, in contrast to the reset-free RAP, the variance shows distinct scalings for symmetric and asymmetric cases. While for the symmetric case, it decays (towards its steady value) as ∼e^{-rt}/sqrt[t], we find ∼te^{-rt} decay for the asymmetric case (r being the resetting rate). Finally, we examine the autocorrelation and unequal time correlation in the steady state and demonstrate that they obey interesting scaling forms at late times. All our analytical results are substantiated by extensive numerical simulations.
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Affiliation(s)
- Saikat Santra
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
| | - Prashant Singh
- Niels Bohr International Academy, Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark
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6
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Caraglio M, Kaur H, Fiderer LJ, López-Incera A, Briegel HJ, Franosch T, Muñoz-Gil G. Learning how to find targets in the micro-world: the case of intermittent active Brownian particles. SOFT MATTER 2024; 20:2008-2016. [PMID: 38328899 PMCID: PMC10900891 DOI: 10.1039/d3sm01680c] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/11/2023] [Accepted: 01/29/2024] [Indexed: 02/09/2024]
Abstract
Finding the best strategy to minimize the time needed to find a given target is a crucial task both in nature and in reaching decisive technological advances. By considering learning agents able to switch their dynamics between standard and active Brownian motion, here we focus on developing effective target-search behavioral policies for microswimmers navigating a homogeneous environment and searching for targets of unknown position. We exploit projective simulation, a reinforcement learning algorithm, to acquire an efficient stochastic policy represented by the probability of switching the phase, i.e. the navigation mode, in response to the type and the duration of the current phase. Our findings reveal that the target-search efficiency increases with the particle's self-propulsion during the active phase and that, while the optimal duration of the passive case decreases monotonically with the activity, the optimal duration of the active phase displays a non-monotonic behavior.
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Affiliation(s)
- Michele Caraglio
- Institut für Theoretische Physik, Universität Innsbruck, Technikerstraße 21A, A-6020, Innsbruck, Austria.
| | - Harpreet Kaur
- Institut für Theoretische Physik, Universität Innsbruck, Technikerstraße 21A, A-6020, Innsbruck, Austria.
| | - Lukas J Fiderer
- Institut für Theoretische Physik, Universität Innsbruck, Technikerstraße 21A, A-6020, Innsbruck, Austria.
| | - Andrea López-Incera
- Institut für Theoretische Physik, Universität Innsbruck, Technikerstraße 21A, A-6020, Innsbruck, Austria.
| | - Hans J Briegel
- Institut für Theoretische Physik, Universität Innsbruck, Technikerstraße 21A, A-6020, Innsbruck, Austria.
| | - Thomas Franosch
- Institut für Theoretische Physik, Universität Innsbruck, Technikerstraße 21A, A-6020, Innsbruck, Austria.
| | - Gorka Muñoz-Gil
- Institut für Theoretische Physik, Universität Innsbruck, Technikerstraße 21A, A-6020, Innsbruck, Austria.
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7
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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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8
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Cantisán J, Nieto AR, Seoane JM, Sanjuán MAF. Energy-based stochastic resetting can avoid noise-enhanced stability. Phys Rev E 2024; 109:024201. [PMID: 38491618 DOI: 10.1103/physreve.109.024201] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/27/2023] [Accepted: 01/18/2024] [Indexed: 03/18/2024]
Abstract
The theory of stochastic resetting asserts that restarting a stochastic process can expedite its completion. In this paper, we study the escape process of a Brownian particle in an open Hamiltonian system that suffers noise-enhanced stability. This phenomenon implies that under specific noise amplitudes the escape process is delayed. Here, we propose a protocol for stochastic resetting that can avoid the noise-enhanced stability effect. In our approach, instead of resetting the trajectories at certain time intervals, a trajectory is reset when a predefined energy threshold is reached. The trajectories that delay the escape process are the ones that lower their energy due to the stochastic fluctuations. Our resetting approach leverages this fact and avoids long transients by resetting trajectories before they reach low-energy levels. Finally, we show that the chaotic dynamics (i.e., the sensitive dependence on initial conditions) catalyzes the effectiveness of the resetting strategy.
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Affiliation(s)
- Julia Cantisán
- Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain
| | - Alexandre R Nieto
- Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain
| | - Jesús M Seoane
- Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain
| | - Miguel A F Sanjuán
- Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain
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9
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Bressloff PC. Truncated stochastically switching processes. Phys Rev E 2024; 109:024103. [PMID: 38491685 DOI: 10.1103/physreve.109.024103] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/31/2023] [Accepted: 01/09/2024] [Indexed: 03/18/2024]
Abstract
There are a large variety of hybrid stochastic systems that couple a continuous process with some form of stochastic switching mechanism. In many cases the system switches between different discrete internal states according to a finite-state Markov chain, and the continuous dynamics depends on the current internal state. The resulting hybrid stochastic differential equation (hSDE) could describe the evolution of a neuron's membrane potential, the concentration of proteins synthesized by a gene network, or the position of an active particle. Another major class of switching system is a search process with stochastic resetting, where the position of a diffusing or active particle is reset to a fixed position at a random sequence of times. In this case the system switches between a search phase and a reset phase, where the latter may be instantaneous. In this paper, we investigate how the behavior of a stochastically switching system is modified when the maximum number of switching (or reset) events in a given time interval is fixed. This is motivated by the idea that each time the system switches there is an additive energy cost. We first show that in the case of an hSDE, restricting the number of switching events is equivalent to truncating a Volterra series expansion of the particle propagator. Such a truncation significantly modifies the moments of the resulting renormalized propagator. We then investigate how restricting the number of reset events affects the diffusive search for an absorbing target. In particular, truncating a Volterra series expansion of the survival probability, we calculate the splitting probabilities and conditional MFPTs for the particle to be absorbed by the target or exceed a given number of resets, respectively.
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Affiliation(s)
- Paul C Bressloff
- Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom
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10
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Boyer D, Mercado-Vásquez G, Majumdar SN, Schehr G. Optimizing the random search of a finite-lived target by a Lévy flight. Phys Rev E 2024; 109:L022103. [PMID: 38491567 DOI: 10.1103/physreve.109.l022103] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/11/2023] [Accepted: 01/22/2024] [Indexed: 03/18/2024]
Abstract
In many random search processes of interest in chemistry, biology, or during rescue operations, an entity must find a specific target site before the latter becomes inactive, no longer available for reaction or lost. We present exact results on a minimal model system, a one-dimensional searcher performing a discrete time random walk, or Lévy flight. In contrast with the case of a permanent target, the capture probability and the conditional mean first passage time can be optimized. The optimal Lévy index takes a nontrivial value, even in the long lifetime limit, and exhibits an abrupt transition as the initial distance to the target is varied. Depending on the target lifetime, this transition is discontinuous or continuous, separated by a nonconventional tricritical point. These results pave the way to the optimization of search processes under time constraints.
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Affiliation(s)
- Denis Boyer
- Instituto de Física, Universidad Nacional Autónoma de México, Ciudad de México 04510, México
| | - Gabriel Mercado-Vásquez
- Pritzker School of Molecular Engineering, University of Chicago, Chicago, Illinois, 60637, USA
| | - 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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11
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Lahiri S, Gupta S. Efficiency of a microscopic heat engine subjected to stochastic resetting. Phys Rev E 2024; 109:014129. [PMID: 38366425 DOI: 10.1103/physreve.109.014129] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/29/2023] [Accepted: 12/15/2023] [Indexed: 02/18/2024]
Abstract
We explore the thermodynamics of stochastic heat engines in the presence of stochastic resetting. The setup comprises an engine whose working substance is a Brownian particle undergoing overdamped Langevin dynamics in a harmonic potential with a time-dependent stiffness, with the dynamics interrupted at random times with a resetting to a fixed location. The effect of resetting to the potential minimum is shown to enhance the efficiency of the engine, while the output work is shown to have a nonmonotonic dependence on the rate of resetting. The resetting events are found to drive the system out of the linear response regime, even for small differences in the bath temperatures. Shifting the reset point from the potential minimum is observed to reduce the engine efficiency. The experimental setup for the realization of such an engine is briefly discussed.
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Affiliation(s)
- Sourabh Lahiri
- Department of Physics, Birla Institute of Technology, Mesra, Ranchi, Jharkhand 835215, India
| | - Shamik Gupta
- Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
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12
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Biroli M, Larralde H, Majumdar SN, Schehr G. Exact extreme, order, and sum statistics in a class of strongly correlated systems. Phys Rev E 2024; 109:014101. [PMID: 38366495 DOI: 10.1103/physreve.109.014101] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/10/2023] [Accepted: 11/27/2023] [Indexed: 02/18/2024]
Abstract
Even though strongly correlated systems are abundant, only a few exceptional cases admit analytical solutions. In this paper we present a large class of solvable systems with strong correlations. We consider a set of N independent and identically distributed random variables {X_{1},X_{2},...,X_{N}} whose common distribution has a parameter Y (or a set of parameters) which itself is random with its own distribution. For a fixed value of this parameter Y, the X_{i} variables are independent and we call them conditionally independent and identically distributed. However, once integrated over the distribution of the parameter Y, the X_{i} variables get strongly correlated yet retain a solvable structure for various observables, such as for the sum and the extremes of X_{i}^{'}s. This provides a simple procedure to generate a class of solvable strongly correlated systems. We illustrate how this procedure works via three physical examples where N particles on a line perform independent (i) Brownian motions, (ii) ballistic motions with random initial velocities, and (iii) Lévy flights, but they get strongly correlated via simultaneous resetting to the origin. Our results are verified in numerical simulations. This procedure can be used to generate an endless variety of solvable strongly correlated systems.
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Affiliation(s)
- Marco Biroli
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Hernán Larralde
- Instituto de Ciencias Físicas, UNAM, CP 62210 Cuernavaca Morelos, México
| | - 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, 75252 Paris Cedex 05, France
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13
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Bebon R, Godec A. Controlling Uncertainty of Empirical First-Passage Times in the Small-Sample Regime. PHYSICAL REVIEW LETTERS 2023; 131:237101. [PMID: 38134782 DOI: 10.1103/physrevlett.131.237101] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/20/2023] [Revised: 10/18/2023] [Accepted: 10/31/2023] [Indexed: 12/24/2023]
Abstract
We derive general bounds on the probability that the empirical first-passage time τ[over ¯]_{n}≡∑_{i=1}^{n}τ_{i}/n of a reversible ergodic Markov process inferred from a sample of n independent realizations deviates from the true mean first-passage time by more than any given amount in either direction. We construct nonasymptotic confidence intervals that hold in the elusive small-sample regime and thus fill the gap between asymptotic methods and the Bayesian approach that is known to be sensitive to prior belief and tends to underestimate uncertainty in the small-sample setting. We prove sharp bounds on extreme first-passage times that control uncertainty even in cases where the mean alone does not sufficiently characterize the statistics. Our concentration-of-measure-based results allow for model-free error control and reliable error estimation in kinetic inference, and are thus important for the analysis of experimental and simulation data in the presence of limited sampling.
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Affiliation(s)
- Rick Bebon
- Mathematical bioPhysics Group, Max Planck Institute for Multidisciplinary Sciences, 37077 Göttingen, Germany
| | - Aljaž Godec
- Mathematical bioPhysics Group, Max Planck Institute for Multidisciplinary Sciences, 37077 Göttingen, Germany
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14
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Chen Y, Yuan Z, Gao L, Peng J. Optimizing search processes with stochastic resetting on the pseudofractal scale-free web. Phys Rev E 2023; 108:064109. [PMID: 38243504 DOI: 10.1103/physreve.108.064109] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/04/2023] [Accepted: 11/15/2023] [Indexed: 01/21/2024]
Abstract
The pseudofractal scale-free web (PSFW) is a well-known model for a scale-free network with small-world characteristics. Understanding the dynamic properties of this network can provide valuable insights into dynamic processes occurring in general scale-free and small-world networks. In this study we investigate search processes using discrete-time random walks on the PSFW to reveal the impact of the resetting position on optimizing search efficiency, as measured by the mean first-passage time (MFPT). At each step the walker has two options: with a probability of 1-γ, it moves to one of the neighboring sites, and with a probability of γ, it resets to the predefined resetting position. We explore various choices for the resetting position, present rigorous results for the MFPT to a given node of the network, determine the optimal resetting probability γ^{*} where the MFPT reaches its minimum, and evaluate the ratio of the minimum for MFPT to the MFPT without resetting for each case. Results show that, in large PSFWs, both the degree of the resetting position and the distance between the target and the resetting position significantly affect the search efficiency. A higher degree of the resetting position leads to a slower convergence of the walker to the target, while a greater distance between the target and the resetting position also results in a slower convergence. Additionally, we observe that resetting to a vertex randomly selected from the stationary distribution can significantly expedite the process of the walker reaching the target. The findings presented in this study shed light on optimizing stochastic search processes on large networks, offering valuable insights into improving search efficiency in real-world applications, where the target node's location is unknown.
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Affiliation(s)
- Yongjin Chen
- School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China; Guangdong Provincial Key Laboratory of Information Security Technology, Guangzhou University, Guangzhou 510006, China; and Guangzhou Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, China
| | - Zhenhua Yuan
- School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China; Guangdong Provincial Key Laboratory of Information Security Technology, Guangzhou University, Guangzhou 510006, China; and Guangzhou Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, China
| | - Long Gao
- School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China; Guangdong Provincial Key Laboratory of Information Security Technology, Guangzhou University, Guangzhou 510006, China; and Guangzhou Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, China
| | - Junhao Peng
- School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China; Guangdong Provincial Key Laboratory of Information Security Technology, Guangzhou University, Guangzhou 510006, China; and Guangzhou Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, China
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15
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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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16
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Pal PS, Pal A, Park H, Lee JS. Thermodynamic trade-off relation for first passage time in resetting processes. Phys Rev E 2023; 108:044117. [PMID: 37978646 DOI: 10.1103/physreve.108.044117] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/03/2023] [Accepted: 09/14/2023] [Indexed: 11/19/2023]
Abstract
Resetting is a strategy for boosting the speed of a target-searching process. Since its introduction over a decade ago, most studies have been carried out under the assumption that resetting takes place instantaneously. However, due to its irreversible nature, resetting processes incur a thermodynamic cost, which becomes infinite in the case of instantaneous resetting. Here, we take into consideration both the cost and the first passage time (FPT) required for a resetting process, in which the reset or return to the initial location is implemented using a trapping potential over a finite but random time period. An iterative generating function and a counting functional method à la Feynman and Kac are employed to calculate the FPT and the average work for this process. From these results, we obtain an explicit form of the time-cost trade-off relation, which provides the lower bound of the mean FPT for a given work input when the trapping potential is linear. This trade-off relation clearly shows that instantaneous resetting is achievable only when an infinite amount of work is provided. More surprisingly, the trade-off relation derived from the linear potential seems to be valid for a wide range of trapping potentials. In addition, we have also shown that the fixed-time or sharp resetting can further enhance the trade-off relation compared to that of the stochastic resetting.
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Affiliation(s)
- P S Pal
- School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea
| | - Arnab Pal
- The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India
- Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India
| | - Hyunggyu Park
- Quantum Universe Center, Korea Institute for Advanced Study, Seoul 02455, Korea
| | - Jae Sung Lee
- School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea
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17
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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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18
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Ghosh PK, Nayak S, Liu J, Li Y, Marchesoni F. Autonomous ratcheting by stochastic resetting. J Chem Phys 2023; 159:031101. [PMID: 37466227 DOI: 10.1063/5.0159148] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/21/2023] [Accepted: 06/30/2023] [Indexed: 07/20/2023] Open
Abstract
We propose a generalization of the stochastic resetting mechanism for a Brownian particle diffusing in a one-dimensional periodic potential: randomly in time, the particle gets reset at the bottom of the potential well it was in. Numerical simulations show that in mirror asymmetric potentials, stochastic resetting rectifies the particle's dynamics, with a maximum drift speed for an optimal average resetting time. Accordingly, an unbiased Brownian tracer diffusing on an asymmetric substrate can rectify its motion by adopting an adaptive stop-and-go strategy. Our proposed ratchet mechanism can model the directed autonomous motion of molecular motors and micro-organisms.
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Affiliation(s)
- Pulak K Ghosh
- Department of Chemistry, Presidency University, Kolkata 700073, India
| | - Shubhadip Nayak
- Department of Chemistry, Presidency University, Kolkata 700073, India
| | - Jianli Liu
- Shanghai Key Laboratory of Special Artificial Microstructure Materials and Technology, School of Physics Science and Engineering, Center for Phononics and Thermal Energy Science, Tongji University, Shanghai 200092, China
| | - Yunyun Li
- Shanghai Key Laboratory of Special Artificial Microstructure Materials and Technology, School of Physics Science and Engineering, Center for Phononics and Thermal Energy Science, Tongji University, Shanghai 200092, China
| | - Fabio Marchesoni
- Shanghai Key Laboratory of Special Artificial Microstructure Materials and Technology, School of Physics Science and Engineering, Center for Phononics and Thermal Energy Science, Tongji University, Shanghai 200092, China
- Dipartimento di Fisica, Università di Camerino, I-62032 Camerino, Italy
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19
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Biroli M, Majumdar SN, Schehr G. Critical number of walkers for diffusive search processes with resetting. Phys Rev E 2023; 107:064141. [PMID: 37464619 DOI: 10.1103/physreve.107.064141] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/13/2023] [Accepted: 05/30/2023] [Indexed: 07/20/2023]
Abstract
We consider N Brownian motions diffusing independently on a line, starting at x_{0}>0, in the presence of an absorbing target at the origin. The walkers undergo stochastic resetting under two protocols: (A) each walker resets independently to x_{0} with rate r and (B) all walkers reset simultaneously to x_{0} with rate r. We derive an explicit analytical expression for the mean first-passage time to the origin in terms of an integral which is evaluated numerically using Mathematica. We show that, as a function of r and for fixed x_{0}, it has a minimum at an optimal value r^{*}>0 as long as N<N_{c}. Thus resetting is beneficial for the search for N<N_{c}. When N>N_{c}, the optimal value occurs at r^{*}=0 indicating that resetting hinders search processes. We obtain different values of N_{c} for protocols A and B; indeed, for N≤7 resetting is beneficial in protocol A, while for N≤6 resetting is beneficial for protocol B. Our theoretical predictions are verified in numerical Langevin simulations.
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Affiliation(s)
- Marco Biroli
- 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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20
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Pogorzelec P, Dybiec B. Resetting induced multimodality. CHAOS (WOODBURY, N.Y.) 2023; 33:2894478. [PMID: 37276562 DOI: 10.1063/5.0143821] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/26/2023] [Accepted: 05/15/2023] [Indexed: 06/07/2023]
Abstract
Properties of stochastic systems are defined by the noise type and deterministic forces acting on the system. In out-of-equilibrium setups, e.g., for motions under action of Lévy noises, the existence of the stationary state is not only determined by the potential but also by the noise. Potential wells need to be steeper than parabolic in order to assure the existence of stationary states. The existence of stationary states, in sub-harmonic potential wells, can be restored by stochastic resetting, which is the protocol of starting over at random times. Herein, we demonstrate that the combined action of Lévy noise and Poissonian stochastic resetting can result in the phase transition between non-equilibrium stationary states of various multimodality in the overdamped system in super-harmonic potentials. Fine-tuned resetting rates can increase the modality of stationary states, while for high resetting rates, the multimodality is destroyed as the stochastic resetting limits the spread of particles.
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Affiliation(s)
- Przemysław Pogorzelec
- Doctoral School of Exact and Natural Sciences, 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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21
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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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22
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Biroli M, Larralde H, Majumdar SN, Schehr G. Extreme Statistics and Spacing Distribution in a Brownian Gas Correlated by Resetting. PHYSICAL REVIEW LETTERS 2023; 130:207101. [PMID: 37267543 DOI: 10.1103/physrevlett.130.207101] [Citation(s) in RCA: 11] [Impact Index Per Article: 11.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/04/2022] [Revised: 03/21/2023] [Accepted: 04/14/2023] [Indexed: 06/04/2023]
Abstract
We study a one-dimensional gas of N Brownian particles that diffuse independently, but are simultaneously reset to the origin at a constant rate r. The system approaches a nonequilibrium stationary state with long-range interactions induced by the simultaneous resetting. Despite the presence of strong correlations, we show that several observables can be computed exactly, which include the global average density, the distribution of the position of the kth rightmost particle, and the spacing distribution between two successive particles. Our analytical results are confirmed by numerical simulations. We also discuss a possible experimental realization of this resetting gas using optical traps.
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Affiliation(s)
- Marco Biroli
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Hernan Larralde
- Instituto de Ciencias Fisicas, UNAM, Avenida Universidad s/n, CP 62210 Cuernavaca, Morelos, Mexico
| | - 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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23
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Kumar A, Pal A. Universal Framework for Record Ages under Restart. PHYSICAL REVIEW LETTERS 2023; 130:157101. [PMID: 37115866 DOI: 10.1103/physrevlett.130.157101] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/07/2022] [Accepted: 03/16/2023] [Indexed: 06/19/2023]
Abstract
We propose a universal framework to compute record age statistics of a stochastic time series that undergoes random restarts. The proposed framework makes minimal assumptions on the underlying process and is furthermore suited to treat generic restart protocols going beyond the Markovian setting. After benchmarking the framework for classical random walks on the 1D lattice, we derive a universal criterion underpinning the impact of restart on the age of the nth record for generic time series with nearest-neighbor transitions. Crucially, the criterion contains a penalty of order n that puts strong constraints on restart expediting the creation of records, as compared to the simple first-passage completion. The applicability of our approach is further demonstrated on an aggregation-shattering process where we compute the typical growth rates of aggregate sizes. This unified framework paves the way to explore record statistics of time series under restart in a wide range of complex systems.
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Affiliation(s)
- Aanjaneya Kumar
- Department of Physics, Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pune 411008, India
| | - Arnab Pal
- The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India
- Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India
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24
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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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25
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Zelenkovski K, Sandev T, Metzler R, Kocarev L, Basnarkov L. Random Walks on Networks with Centrality-Based Stochastic Resetting. ENTROPY (BASEL, SWITZERLAND) 2023; 25:293. [PMID: 36832659 PMCID: PMC9955709 DOI: 10.3390/e25020293] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 12/26/2022] [Revised: 01/19/2023] [Accepted: 02/02/2023] [Indexed: 06/18/2023]
Abstract
We introduce a refined way to diffusely explore complex networks with stochastic resetting where the resetting site is derived from node centrality measures. This approach differs from previous ones, since it not only allows the random walker with a certain probability to jump from the current node to a deliberately chosen resetting node, rather it enables the walker to jump to the node that can reach all other nodes faster. Following this strategy, we consider the resetting site to be the geometric center, the node that minimizes the average travel time to all the other nodes. Using the established Markov chain theory, we calculate the Global Mean First Passage Time (GMFPT) to determine the search performance of the random walk with resetting for different resetting node candidates individually. Furthermore, we compare which nodes are better resetting node sites by comparing the GMFPT for each node. We study this approach for different topologies of generic and real-life networks. We show that, for directed networks extracted for real-life relationships, this centrality focused resetting can improve the search to a greater extent than for the generated undirected networks. This resetting to the center advocated here can minimize the average travel time to all other nodes in real networks as well. We also present a relationship between the longest shortest path (the diameter), the average node degree and the GMFPT when the starting node is the center. We show that, for undirected scale-free networks, stochastic resetting is effective only for networks that are extremely sparse with tree-like structures as they have larger diameters and smaller average node degrees. For directed networks, the resetting is beneficial even for networks that have loops. The numerical results are confirmed by analytic solutions. Our study demonstrates that the proposed random walk approach with resetting based on centrality measures reduces the memoryless search time for targets in the examined network topologies.
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Affiliation(s)
- Kiril Zelenkovski
- 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, Faculty of Natural Sciences and Mathematics, Ss. Cyril and Methodius University, Arhimedova 3, 1000 Skopje, Macedonia
- Institute of Physics & Astronomy, University of Potsdam, D-14776 Potsdam, Germany
| | - Ralf Metzler
- Institute of Physics & Astronomy, University of Potsdam, D-14776 Potsdam, Germany
- Asia Pacific Center for Theoretical Physics, Pohang 37673, Republic of Korea
| | - 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, 1000 Skopje, Macedonia
| | - Lasko Basnarkov
- 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, 1000 Skopje, Macedonia
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26
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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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27
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Zhang W, Li Y, Marchesoni F, Misko VR, Ghosh PK. Narrow Pore Crossing of Active Particles under Stochastic Resetting. ENTROPY (BASEL, SWITZERLAND) 2023; 25:271. [PMID: 36832639 PMCID: PMC9955654 DOI: 10.3390/e25020271] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 12/13/2022] [Revised: 01/17/2023] [Accepted: 01/30/2023] [Indexed: 06/18/2023]
Abstract
We propose a two-dimensional model of biochemical activation process, whereby self-propelling particles of finite correlation times are injected at the center of a circular cavity with constant rate equal to the inverse of their lifetime; activation is triggered when one such particle hits a receptor on the cavity boundary, modeled as a narrow pore. We numerically investigated this process by computing the particle mean-first exit times through the cavity pore as a function of the correlation and injection time constants. Due to the breach of the circular symmetry associated with the positioning of the receptor, the exit times may depend on the orientation of the self-propelling velocity at injection. Stochastic resetting appears to favor activation for large particle correlation times, where most of the underlying diffusion process occurs at the cavity boundary.
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Affiliation(s)
- Weitao Zhang
- Center for Phononics and Thermal Energy Science, Shanghai Key Laboratory of Special Artificial Microstructure Materials and Technology, School of Physics Science and Engineering, Tongji University, Shanghai 200092, China
| | - Yunyun Li
- Center for Phononics and Thermal Energy Science, Shanghai Key Laboratory of Special Artificial Microstructure Materials and Technology, School of Physics Science and Engineering, Tongji University, Shanghai 200092, China
| | - Fabio Marchesoni
- Center for Phononics and Thermal Energy Science, Shanghai Key Laboratory of Special Artificial Microstructure Materials and Technology, School of Physics Science and Engineering, Tongji University, Shanghai 200092, China
- Dipartimento di Fisica, Università di Camerino, I-62032 Camerino, Italy
| | - Vyacheslav R. Misko
- μFlow Group, Department of Chemical Engineering, Vrije Universiteit Brussel, 1050 Brussels, Belgium
| | - Pulak K. Ghosh
- Department of Chemistry, Presidency University, Kolkata 700073, India
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28
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Guéry-Odelin D, Jarzynski C, Plata CA, Prados A, Trizac E. Driving rapidly while remaining in control: classical shortcuts from Hamiltonian to stochastic dynamics. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2023; 86:035902. [PMID: 36535018 DOI: 10.1088/1361-6633/acacad] [Citation(s) in RCA: 8] [Impact Index Per Article: 8.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/12/2022] [Accepted: 12/19/2022] [Indexed: 06/17/2023]
Abstract
Stochastic thermodynamics lays down a broad framework to revisit the venerable concepts of heat, work and entropy production for individual stochastic trajectories of mesoscopic systems. Remarkably, this approach, relying on stochastic equations of motion, introduces time into the description of thermodynamic processes-which opens the way to fine control them. As a result, the field of finite-time thermodynamics of mesoscopic systems has blossomed. In this article, after introducing a few concepts of control for isolated mechanical systems evolving according to deterministic equations of motion, we review the different strategies that have been developed to realize finite-time state-to-state transformations in both over and underdamped regimes, by the proper design of time-dependent control parameters/driving. The systems under study are stochastic, epitomized by a Brownian object immersed in a fluid; they are thus strongly coupled to their environment playing the role of a reservoir. Interestingly, a few of those methods (inverse engineering, counterdiabatic driving, fast-forward) are directly inspired by their counterpart in quantum control. The review also analyzes the control through reservoir engineering. Besides the reachability of a given target state from a known initial state, the question of the optimal path is discussed. Optimality is here defined with respect to a cost function, a subject intimately related to the field of information thermodynamics and the question of speed limit. Another natural extension discussed deals with the connection between arbitrary states or non-equilibrium steady states. This field of control in stochastic thermodynamics enjoys a wealth of applications, ranging from optimal mesoscopic heat engines to population control in biological systems.
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Affiliation(s)
- David Guéry-Odelin
- Laboratoire Collisions, Agrégats, Réactivité, IRSAMC, Université de Toulouse, CNRS, Toulouse, France
| | - Christopher Jarzynski
- Department of Chemistry and Biochemistry, University of Maryland, College Park, MD, United States of America
- Institute for Physical Science and Technology, University of Maryland, College Park, MD, United States of America
- Department of Physics, University of Maryland, College Park, MD, United States of America
| | - Carlos A Plata
- Física Teórica, Universidad de Sevilla, Apartado de Correos 1065, E-41080 Sevilla, Spain
| | - Antonio Prados
- Física Teórica, Universidad de Sevilla, Apartado de Correos 1065, E-41080 Sevilla, Spain
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Mori F, Majumdar SN, Schehr G. Time to reach the maximum for a stationary stochastic process. Phys Rev E 2022; 106:054110. [PMID: 36559509 DOI: 10.1103/physreve.106.054110] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/27/2022] [Accepted: 10/11/2022] [Indexed: 11/06/2022]
Abstract
We consider a one-dimensional stationary time series of fixed duration T. We investigate the time t_{m} at which the process reaches the global maximum within the time interval [0,T]. By using a path-decomposition technique, we compute the probability density function P(t_{m}|T) of t_{m} for several processes, that are either at equilibrium (such as the Ornstein-Uhlenbeck process) or out of equilibrium (such as Brownian motion with stochastic resetting). We show that for equilibrium processes the distribution of P(t_{m}|T) is always symmetric around the midpoint t_{m}=T/2, as a consequence of the time-reversal symmetry. This property can be used to detect nonequilibrium fluctuations in stationary time series. Moreover, for a diffusive particle in a confining potential, we show that the scaled distribution P(t_{m}|T) becomes universal, i.e., independent of the details of the potential, at late times. This distribution P(t_{m}|T) becomes uniform in the "bulk" 1≪t_{m}≪T and has a nontrivial universal shape in the "edge regimes" t_{m}→0 and t_{m}→T. Some of these results have been announced in a recent letter [Europhys. Lett. 135, 30003 (2021)0295-507510.1209/0295-5075/ac19ee].
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Affiliation(s)
- Francesco Mori
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - 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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30
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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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31
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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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32
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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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33
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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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34
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Ray S. Expediting Feller process with stochastic resetting. Phys Rev E 2022; 106:034133. [PMID: 36266865 DOI: 10.1103/physreve.106.034133] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/07/2022] [Accepted: 09/07/2022] [Indexed: 06/16/2023]
Abstract
We explore the effect of stochastic resetting on the first-passage properties of the Feller process. The Feller process can be envisioned as space-dependent diffusion, with diffusion coefficient D(x)=x, in a potential U(x)=x(x/2-θ) that owns a minimum at θ. This restricts the process to the positive side of the origin and therefore, Feller diffusion can successfully model a vast array of phenomena in biological and social sciences, where realization of negative values is forbidden. In our analytically tractable model system, a particle that undergoes Feller diffusion is subject to Poissonian resetting, i.e., taken back to its initial position at a constant rate r, after random time epochs. We addressed the two distinct cases that arise when the relative position of the absorbing boundary (x_{a}) with respect to the initial position of the particle (x_{0}) differ, i.e., for (a) x_{0}<x_{a} and (b) x_{a}<x_{0}. Utilizing the Fokker-Planck description of the system, we obtained closed-form expressions for the Laplace transform of the survival probability and hence derived the exact expressions of the mean first-passage time 〈T_{r}〉. Performing a comprehensive analysis on the optimal resetting rate (r^{★}) that minimize 〈T_{r}〉 and the maximal speedup that r^{★} renders, we identify the phase space where Poissonian resetting facilitates first-passage for Feller diffusion. We observe that for x_{0}<x_{a}, resetting accelerates first-passage when θ<θ_{c}, where θ_{c} is a critical value of θ that decreases when x_{a} is moved away from the origin. In stark contrast, for x_{a}<x_{0}, resetting accelerates first-passage when θ>θ_{c}, where θ_{c} is a critical value of θ that increases when x_{0} is moved away from the origin. Our study opens up the possibility of a series of subsequent works with more case-specific models of Feller diffusion with resetting.
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Affiliation(s)
- Somrita Ray
- Department of Chemistry, Indian Institute of Technology Tirupati, Tirupati 517619, India
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35
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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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36
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Luo Y, Zeng C, Huang T, Ai BQ. Anomalous transport tuned through stochastic resetting in the rugged energy landscape of a chaotic system with roughness. Phys Rev E 2022; 106:034208. [PMID: 36266857 DOI: 10.1103/physreve.106.034208] [Citation(s) in RCA: 5] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/03/2022] [Accepted: 08/15/2022] [Indexed: 06/16/2023]
Abstract
Stochastic resetting causes kinetic phase transitions, whereas its underlying physical mechanism remains to be elucidated. We here investigate the anomalous transport of a particle moving in a chaotic system with a stochastic resetting and a rough potential and focus on how the stochastic resetting, roughness, and nonequilibrium noise affect the transports of the particle. We uncover the physical mechanism for stochastic resetting resulting in the anomalous transport in a nonlinear chaotic system: The particle is reset to a new basin of attraction which may be different from the initial basin of attraction from the view of dynamics. From the view of the energy landscape, the particle is reset to a new energy state of the energy landscape which may be different from the initial energy state. This resetting can lead to a kinetic phase transition between no transport and a finite net transport or between negative mobility and positive mobility. The roughness and noise also lead to the transition. Based on the mechanism, the transport of the particle can be tuned by these parameters. For example, the combination of the stochastic resetting, roughness, and noise can enhance the transport and tune negative mobility, the enhanced stability of the system, and the resonant-like activity. We analyze these results through variances (e.g., mean-squared velocity, etc.) and correlation functions (i.e., velocity autocorrelation function, position-velocity correlation function, etc.). Our results can be extensively applied in the biology, physics, and chemistry, even social system.
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Affiliation(s)
- Yuhui Luo
- Faculty of Civil Engineering and Mechanics/Faculty of Science, Kunming University of Science and Technology, Kunming 650500, China
- School of Physics and Information Engineering, Zhaotong University, Zhaotong 657000, China
| | - Chunhua Zeng
- Faculty of Civil Engineering and Mechanics/Faculty of Science, Kunming University of Science and Technology, Kunming 650500, China
| | - Tao Huang
- Faculty of Civil Engineering and Mechanics/Faculty of Science, Kunming University of Science and Technology, Kunming 650500, China
| | - Bao-Quan Ai
- Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, GPETR Center for Quantum Precision Measurement, SPTE, South China Normal University, Guangzhou 510006, China
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37
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Sarkar M, Gupta S. Synchronization in the Kuramoto model in presence of stochastic resetting. CHAOS (WOODBURY, N.Y.) 2022; 32:073109. [PMID: 35907730 DOI: 10.1063/5.0090861] [Citation(s) in RCA: 10] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/09/2022] [Accepted: 06/10/2022] [Indexed: 06/15/2023]
Abstract
What happens when the paradigmatic Kuramoto model involving interacting oscillators of distributed natural frequencies and showing spontaneous collective synchronization in the stationary state is subject to random and repeated interruptions of its dynamics with a reset to the initial condition? While resetting to a synchronized state, it may happen between two successive resets that the system desynchronizes, which depends on the duration of the random time interval between the two resets. Here, we unveil how such a protocol of stochastic resetting dramatically modifies the phase diagram of the bare model, allowing, in particular, for the emergence of a synchronized phase even in parameter regimes for which the bare model does not support such a phase. Our results are based on an exact analysis invoking the celebrated Ott-Antonsen ansatz for the case of the Lorentzian distribution of natural frequencies and numerical results for Gaussian frequency distribution. Our work provides a simple protocol to induce global synchrony in the system through stochastic resetting.
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Affiliation(s)
- Mrinal Sarkar
- Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India
| | - Shamik Gupta
- Department of Physics, Ramakrishna Mission Vivekananda Educational and Research Institute, Belur Math, Howrah 711202, India
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38
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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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39
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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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40
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Szarek D, Maraj-Zygmąt K, Sikora G, Krapf D, Wyłomańska A. Statistical test for anomalous diffusion based on empirical anomaly measure for Gaussian processes. Comput Stat Data Anal 2022. [DOI: 10.1016/j.csda.2021.107401] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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41
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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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42
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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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43
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Klinger J, Barbier-Chebbah A, Voituriez R, Bénichou O. Joint statistics of space and time exploration of one-dimensional random walks. Phys Rev E 2022; 105:034116. [PMID: 35428155 DOI: 10.1103/physreve.105.034116] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/15/2021] [Accepted: 02/09/2022] [Indexed: 06/14/2023]
Abstract
The statistics of first-passage times of random walks to target sites has proved to play a key role in determining the kinetics of space exploration in various contexts. In parallel, the number of distinct sites visited by a random walker and related observables has been introduced to characterize the geometry of space exploration. Here, we address the question of the joint distribution of the first-passage time to a target and the number of distinct sites visited when the target is reached, which fully quantifies the coupling between the kinetics and geometry of search trajectories. Focusing on one-dimensional systems, we present a general method and derive explicit expressions of this joint distribution for several representative examples of Markovian search processes. In addition, we obtain a general scaling form, which holds also for non-Markovian processes and captures the general dependence of the joint distribution on its space and time variables. We argue that the joint distribution has important applications to various problems, such as a conditional form of the Rosenstock trapping model, and the persistence properties of self-interacting random walks.
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Affiliation(s)
- J Klinger
- Laboratoire de Physique Théorique de la Matière Condensée, CNRS/Sorbonne Université, 4 Place Jussieu, 75005 Paris, France
- Laboratoire Jean Perrin, CNRS/Sorbonne Université, 4 Place Jussieu, 75005 Paris, France
| | - A Barbier-Chebbah
- Laboratoire de Physique Théorique de la Matière Condensée, CNRS/Sorbonne Université, 4 Place Jussieu, 75005 Paris, France
| | - R Voituriez
- Laboratoire de Physique Théorique de la Matière Condensée, CNRS/Sorbonne Université, 4 Place Jussieu, 75005 Paris, France
- Laboratoire Jean Perrin, CNRS/Sorbonne Université, 4 Place Jussieu, 75005 Paris, France
| | - O Bénichou
- Laboratoire de Physique Théorique de la Matière Condensée, CNRS/Sorbonne Université, 4 Place Jussieu, 75005 Paris, France
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44
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Bernardi D, Lindner B. Run with the Brownian Hare, Hunt with the Deterministic Hounds. PHYSICAL REVIEW LETTERS 2022; 128:040601. [PMID: 35148130 DOI: 10.1103/physrevlett.128.040601] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/13/2021] [Accepted: 12/24/2021] [Indexed: 06/14/2023]
Abstract
We present analytic results for mean capture time and energy expended by a pack of deterministic hounds actively chasing a randomly diffusing prey. Depending on the number of chasers, the mean capture time as a function of the prey's diffusion coefficient can be monotonically increasing, decreasing, or attain a minimum at a finite value. Optimal speed and number of chasing hounds exist and depend on each chaser's baseline power consumption. The model can serve as an analytically tractable basis for further studies with bearing on the growing field of smart microswimmers and autonomous robots.
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Affiliation(s)
- Davide Bernardi
- Center for Translational Neurophysiology of Speech and Communication, Fondazione Istituto Italiano di Tecnologia, via Fossato di Mortara 19, 44121 Ferrara, Italy
| | - Benjamin Lindner
- Bernstein Center for Computational Neuroscience Berlin, Philippstraße 13, Haus 2, 10115 Berlin, Germany and Physics Department of Humboldt University Berlin, Newtonstraße 15, 12489 Berlin, Germany
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45
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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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46
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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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47
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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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48
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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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49
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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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50
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Maraj K, Szarek D, Sikora G, Wyłomańska A. Time-averaged mean squared displacement ratio test for Gaussian processes with unknown diffusion coefficient. CHAOS (WOODBURY, N.Y.) 2021; 31:073120. [PMID: 34340341 DOI: 10.1063/5.0054119] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/14/2021] [Accepted: 06/23/2021] [Indexed: 06/13/2023]
Abstract
The time-averaged mean squared displacement (TAMSD) is one of the most common statistics used for the analysis of anomalous diffusion processes. Anomalous diffusion is manifested by non-linear (mostly power-law) characteristics of the process in contrast to normal diffusion where linear characteristics are expected. One can distinguish between sub- and super-diffusive processes. We consider Gaussian anomalous diffusion models and propose a new approach used for their testing. This approach is based on the TAMSD ratio statistic for different time lags. Similar to the TAMSD, this statistic exhibits a specific behavior in the anomalous diffusion regime. Through its structure, it is independent of the diffusion coefficient, which, in general, does not influence anomalous diffusion behavior. Thus, the TAMSD ratio-based approach does not require preliminary knowledge of the diffusion coefficient's value, in contrast to the TAMSD-approach, where this value is crucial in the testing procedure. Based on the quadratic form representation of the TAMSD ratio, we calculate its main characteristics and propose a step-by-step testing procedure that can be applied for any Gaussian process. For the anomalous diffusion model used here, namely, the fractional Brownian motion, we demonstrate the effectiveness of the proposed methodology. We show that the new approach outperforms the TAMSD-based one, especially for small sample sizes. Finally, the methodology is applied to the real data from the financial market.
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Affiliation(s)
- Katarzyna Maraj
- Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wroclaw University of Science and Technology, 50-370 Wroclaw, Poland
| | - Dawid Szarek
- Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wroclaw University of Science and Technology, 50-370 Wroclaw, Poland
| | - Grzegorz Sikora
- Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wroclaw University of Science and Technology, 50-370 Wroclaw, Poland
| | - Agnieszka Wyłomańska
- Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wroclaw University of Science and Technology, 50-370 Wroclaw, Poland
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