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Wiśniewski M, Łuczka J, Spiechowicz J. Memory Corrections to Markovian Langevin Dynamics. ENTROPY (BASEL, SWITZERLAND) 2024; 26:425. [PMID: 38785674 PMCID: PMC11120160 DOI: 10.3390/e26050425] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/30/2024] [Revised: 05/10/2024] [Accepted: 05/14/2024] [Indexed: 05/25/2024]
Abstract
Analysis of non-Markovian systems and memory-induced phenomena poses an everlasting challenge in the realm of physics. As a paradigmatic example, we consider a classical Brownian particle of mass M subjected to an external force and exposed to correlated thermal fluctuations. We show that the recently developed approach to this system, in which its non-Markovian dynamics given by the Generalized Langevin Equation is approximated by its memoryless counterpart but with the effective particle mass M∗
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Wiśniewski M, Łuczka J, Spiechowicz J. Effective mass approach to memory in non-Markovian systems. Phys Rev E 2024; 109:044116. [PMID: 38755811 DOI: 10.1103/physreve.109.044116] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/13/2023] [Accepted: 03/14/2024] [Indexed: 05/18/2024]
Abstract
Recent pioneering experiments on non-Markovian dynamics done, e.g., for active matter have demonstrated that our theoretical understanding of this challenging yet hot topic is rather incomplete and there is a wealth of phenomena still awaiting discovery. It is related to the fact that typically for simplification the Markovian approximation is employed and as a consequence the memory is neglected. Therefore, methods allowing to study memory effects are extremely valuable. We demonstrate that a non-Markovian system described by the Generalized Langevin Equation (GLE) for a Brownian particle of mass M can be approximated by the memoryless Langevin equation in which the memory effects are correctly reproduced solely via the effective mass M^{*} of the Brownian particle which is determined only by the form of the memory kernel. Our work lays the foundation for an impactful approach which allows one to readily study memory-related corrections to Markovian dynamics.
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Affiliation(s)
- M Wiśniewski
- Institute of Physics, University of Silesia, 41-500 Chorzów, Poland
| | - J Łuczka
- Institute of Physics, University of Silesia, 41-500 Chorzów, Poland
| | - J Spiechowicz
- Institute of Physics, University of Silesia, 41-500 Chorzów, Poland
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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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Flaquer-Galmés R, Campos D, Méndez V. Intermittent random walks under stochastic resetting. Phys Rev E 2024; 109:034103. [PMID: 38632743 DOI: 10.1103/physreve.109.034103] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/06/2023] [Accepted: 02/06/2024] [Indexed: 04/19/2024]
Abstract
We analyze a one-dimensional intermittent random walk on an unbounded domain in the presence of stochastic resetting. In this process, the walker alternates between local intensive search, diffusion, and rapid ballistic relocations in which it does not react to the target. We demonstrate that Poissonian resetting leads to the existence of a non-equilibrium steady state. We calculate the distribution of the first arrival time to a target along with its mean and show the existence of an optimal reset rate. In particular, we prove that the initial condition of the walker, i.e., either starting diffusely or relocating, can significantly affect the long-time properties of the search process. Moreover, we demonstrate the presence of distinct parameter regimes for the global optimization of the mean first arrival time when ballistic and diffusive movements are in direct competition.
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Affiliation(s)
- Rosa Flaquer-Galmés
- Grupo de Física Estadística, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain
| | - Daniel Campos
- Grupo de Física Estadística, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain
| | - Vicenç Méndez
- Grupo de Física Estadística, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain
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Requena B, Masó-Orriols S, Bertran J, Lewenstein M, Manzo C, Muñoz-Gil G. Inferring pointwise diffusion properties of single trajectories with deep learning. Biophys J 2023; 122:4360-4369. [PMID: 37853693 PMCID: PMC10698275 DOI: 10.1016/j.bpj.2023.10.015] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/25/2023] [Revised: 09/14/2023] [Accepted: 10/13/2023] [Indexed: 10/20/2023] Open
Abstract
To characterize the mechanisms governing the diffusion of particles in biological scenarios, it is essential to accurately determine their diffusive properties. To do so, we propose a machine-learning method to characterize diffusion processes with time-dependent properties at the experimental time resolution. Our approach operates at the single-trajectory level predicting the properties of interest, such as the diffusion coefficient or the anomalous diffusion exponent, at every time step of the trajectory. In this way, changes in the diffusive properties occurring along the trajectory emerge naturally in the prediction and thus allow the characterization without any prior knowledge or assumption about the system. We first benchmark the method on synthetic trajectories simulated under several conditions. We show that our approach can successfully characterize both abrupt and continuous changes in the diffusion coefficient or the anomalous diffusion exponent. Finally, we leverage the method to analyze experiments of single-molecule diffusion of two membrane proteins in living cells: the pathogen-recognition receptor DC-SIGN and the integrin α5β1. The analysis allows us to characterize physical parameters and diffusive states with unprecedented accuracy, shedding new light on the underlying mechanisms.
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Affiliation(s)
- Borja Requena
- ICFO - Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, Castelldefels (Barcelona), Spain
| | - Sergi Masó-Orriols
- Facultat de Ciències, Tecnologia I Enginyeries, Universitat de Vic - Universitat Central de Catalunya (UVic-UCC), Vic, Spain; Institut de Recerca i Innovació en Ciències de la Vida i de la Salut a la Catalunya Central (IRIS-CC), Vic, Barcelona, Spain
| | - Joan Bertran
- Facultat de Ciències, Tecnologia I Enginyeries, Universitat de Vic - Universitat Central de Catalunya (UVic-UCC), Vic, Spain; Institut de Recerca i Innovació en Ciències de la Vida i de la Salut a la Catalunya Central (IRIS-CC), Vic, Barcelona, Spain
| | - Maciej Lewenstein
- ICFO - Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, Castelldefels (Barcelona), Spain; ICREA, Pg. Lluís Companys 23, Barcelona, Spain
| | - Carlo Manzo
- Facultat de Ciències, Tecnologia I Enginyeries, Universitat de Vic - Universitat Central de Catalunya (UVic-UCC), Vic, Spain; Institut de Recerca i Innovació en Ciències de la Vida i de la Salut a la Catalunya Central (IRIS-CC), Vic, Barcelona, Spain.
| | - Gorka Muñoz-Gil
- Institute for Theoretical Physics, University of Innsbruck, Innsbruck, Austria.
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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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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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Chen H, Ye Y. Random walks on complex networks under time-dependent stochastic resetting. Phys Rev E 2022; 106:044139. [PMID: 36397577 DOI: 10.1103/physreve.106.044139] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/31/2022] [Accepted: 10/05/2022] [Indexed: 06/16/2023]
Abstract
We study discrete-time random walks on networks subject to a time-dependent stochastic resetting, where the walker either hops randomly between neighboring nodes with a probability 1-ϕ(a) or is reset to a given node with a complementary probability ϕ(a). The resetting probability ϕ(a) depends on the time a since the last reset event (also called a, the age of the walker). Using the renewal approach and spectral decomposition of the transition matrix, we formulate the stationary occupation probability of the walker at each node and the mean first passage time between two arbitrary nodes. Concretely, we consider two different time-dependent resetting protocols that are both exactly solvable. One is that ϕ(a) is a step-shaped function of a and the other one is that ϕ(a) is a rational function of a. We demonstrate the theoretical results on several different networks, also validated by numerical simulations, and find that the time-modulated resetting protocols can be more advantageous than the constant-probability resetting in accelerating the completion of a target search process.
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Affiliation(s)
- Hanshuang Chen
- School of Physics and Optoelectronic Engineering, Anhui University, Hefei 230601, China
| | - Yanfei Ye
- School of Physics and Optoelectronic Engineering, Anhui University, Hefei 230601, China
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McCormick K. Memory Plays a Critical Role in Animal Behavior. PHYSICS 2022. [DOI: 10.1103/physics.15.s39] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/27/2023]
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