1
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Burenev IN, Majumdar SN, Rosso A. Occupation time of a system of Brownian particles on the line with steplike initial condition. Phys Rev E 2024; 109:044150. [PMID: 38755944 DOI: 10.1103/physreve.109.044150] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/30/2023] [Accepted: 02/29/2024] [Indexed: 05/18/2024]
Abstract
We consider a system of noninteracting Brownian particles on the line with steplike initial condition and study the statistics of the occupation time on the positive half-line. We demonstrate that even at large times, the behavior of the occupation time exhibits long-lasting memory effects of the initialization. Specifically, we calculate the mean and the variance of the occupation time, demonstrating that the memory effects in the variance are determined by a generalized compressibility (or Fano factor), associated with the initial condition. In the particular case of the uncorrelated uniform initial condition we conduct a detailed study of two probability distributions of the occupation time: annealed (averaged over all possible initial configurations) and quenched (for a typical configuration). We show that at large times both the annealed and the quenched distributions admit large deviation form and we compute analytically the associated rate functions. We verify our analytical predictions via numerical simulations using importance sampling Monte Carlo strategy.
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Affiliation(s)
- Ivan N Burenev
- LPTMS, CNRS, Université Paris-Saclay, 91405 Orsay, France
| | | | - Alberto Rosso
- LPTMS, CNRS, Université Paris-Saclay, 91405 Orsay, France
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2
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Burenev IN, Majumdar SN, Rosso A. Local time of a system of Brownian particles on the line with steplike initial condition. Phys Rev E 2023; 108:064113. [PMID: 38243455 DOI: 10.1103/physreve.108.064113] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/30/2023] [Accepted: 10/30/2023] [Indexed: 01/21/2024]
Abstract
We consider a system of noninteracting Brownian particles on a line with a steplike initial condition, and we investigate the behavior of the local time at the origin at large times. We compute the mean and the variance of the local time, and we show that the memory effects are governed by the Fano factor associated with the initial condition. For the uniform initial condition, we show that the probability distribution of the local time admits a large deviation form, and we compute the corresponding large deviation functions for the annealed and quenched averaging schemes. The two resulting large deviation functions are very different. Our analytical results are supported by extensive numerical simulations.
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Affiliation(s)
- Ivan N Burenev
- LPTMS, CNRS, Université Paris-Saclay, 91405 Orsay, France
| | | | - Alberto Rosso
- LPTMS, CNRS, Université Paris-Saclay, 91405 Orsay, France
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3
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Mukherjee S, Smith NR. Dynamical phase transition in the occupation fraction statistics for noncrossing Brownian particles. Phys Rev E 2023; 107:064133. [PMID: 37464710 DOI: 10.1103/physreve.107.064133] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/10/2023] [Accepted: 06/09/2023] [Indexed: 07/20/2023]
Abstract
We consider a system of N noncrossing Brownian particles in one dimension. We find the exact rate function that describes the long-time large deviation statistics of their occupation fraction in a finite interval in space. Remarkably, we find that, for any general N≥2, the system undergoes N-1 dynamical phase transitions of second order. The N-1 transitions are the boundaries of N phases that correspond to different numbers of particles which are in the vicinity of the interval throughout the dynamics. We achieve this by mapping the problem to that of finding the ground-state energy for N noninteracting spinless fermions in a square-well potential. The phases correspond to different numbers of single-body bound states for the quantum problem. We also study the process conditioned on a given occupation fraction and the large-N limiting behavior.
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Affiliation(s)
- Soheli Mukherjee
- Department of Solar Energy and Environmental Physics, Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Sede Boqer Campus 8499000, Israel
| | - Naftali R Smith
- Department of Solar Energy and Environmental Physics, Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Sede Boqer Campus 8499000, Israel
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4
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Kay T, Giuggioli L. Subdiffusion in the Presence of Reactive Boundaries: A Generalized Feynman-Kac Approach. JOURNAL OF STATISTICAL PHYSICS 2023; 190:92. [PMID: 37128546 PMCID: PMC10140114 DOI: 10.1007/s10955-023-03105-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 12/24/2022] [Accepted: 04/03/2023] [Indexed: 05/03/2023]
Abstract
We derive, through subordination techniques, a generalized Feynman-Kac equation in the form of a time fractional Schrödinger equation. We relate such equation to a functional which we name the subordinated local time. We demonstrate through a stochastic treatment how this generalized Feynman-Kac equation describes subdiffusive processes with reactions. In this interpretation, the subordinated local time represents the number of times a specific spatial point is reached, with the amount of time spent there being immaterial. This distinction provides a practical advance due to the potential long waiting time nature of subdiffusive processes. The subordinated local time is used to formulate a probabilistic understanding of subdiffusion with reactions, leading to the well known radiation boundary condition. We demonstrate the equivalence between the generalized Feynman-Kac equation with a reflecting boundary and the fractional diffusion equation with a radiation boundary. We solve the former and find the first-reaction probability density in analytic form in the time domain, in terms of the Wright function. We are also able to find the survival probability and subordinated local time density analytically. These results are validated by stochastic simulations that use the subordinated local time description of subdiffusion in the presence of reactions.
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Affiliation(s)
- Toby Kay
- Department of Engineering Mathematics, University of Bristol, Bristol, BS8 1UB UK
| | - Luca Giuggioli
- Department of Engineering Mathematics, University of Bristol, Bristol, BS8 1UB UK
- Bristol Centre for Complexity Sciences, University of Bristol, Bristol, BS8 1UB UK
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5
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Kimura M, Akimoto T. Occupation time statistics of the fractional Brownian motion in a finite domain. Phys Rev E 2022; 106:064132. [PMID: 36671174 DOI: 10.1103/physreve.106.064132] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/09/2022] [Accepted: 12/09/2022] [Indexed: 06/17/2023]
Abstract
We study statistics of occupation times for a fractional Brownian motion (fBm), which is a typical model of a non-Markov process. Due to the non-Markovian nature, recurrence times to the origin depend on the history. Numerical simulations indicate that dependence on the sum of successive recurrence times becomes weak. As a result, the distribution of the occupation time in a finite domain follows the Mittag-Leffler distribution when the Hurst exponent of the fBm is close to 1/2. We show this distributional behavior of a time-averaged observable by renewal theory. This result is an extension of the distributional limit theorem known as the Darling-Kac theorem in general Markov processes to non-Markov processes.
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Affiliation(s)
- Mutsumi Kimura
- Department of Physics, Tokyo University of Science, Noda, Chiba 278-8510, Japan
| | - Takuma Akimoto
- Department of Physics, Tokyo University of Science, Noda, Chiba 278-8510, Japan
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6
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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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7
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Dieball C, Godec A. Mathematical, Thermodynamical, and Experimental Necessity for Coarse Graining Empirical Densities and Currents in Continuous Space. PHYSICAL REVIEW LETTERS 2022; 129:140601. [PMID: 36240401 DOI: 10.1103/physrevlett.129.140601] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/24/2021] [Revised: 07/19/2022] [Accepted: 07/28/2022] [Indexed: 06/16/2023]
Abstract
We present general results on fluctuations and spatial correlations of the coarse-grained empirical density and current of Markovian diffusion in equilibrium or nonequilibrium steady states on all timescales. We unravel a deep connection between current fluctuations and generalized time-reversal symmetry, providing new insight into time-averaged observables. We highlight the essential role of coarse graining in space from mathematical, thermodynamical, and experimental points of view. Spatial coarse graining is required to uncover salient features of currents that break detailed balance, and a thermodynamically "optimal" coarse graining ensures the most precise inference of dissipation. Defined without coarse graining, the fluctuations of empirical density and current are proven to diverge on all timescales in dimensions higher than one, which has far-reaching consequences for the central-limit regime in continuous space. We apply the results to examples of irreversible diffusion. Our findings provide new intuition about time-averaged observables and allow for a more efficient analysis of single-molecule experiments.
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Affiliation(s)
- Cai Dieball
- Mathematical bioPhysics Group, Max Planck Institute for Multidisciplinary Sciences, Am Faßberg 11, 37077 Göttingen
| | - Aljaž Godec
- Mathematical bioPhysics Group, Max Planck Institute for Multidisciplinary Sciences, Am Faßberg 11, 37077 Göttingen
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8
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Grebenkov DS. Depletion of resources by a population of diffusing species. Phys Rev E 2022; 105:054402. [PMID: 35706291 DOI: 10.1103/physreve.105.054402] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/19/2022] [Accepted: 04/19/2022] [Indexed: 06/15/2023]
Abstract
Depletion of natural and artificial resources is a fundamental problem and a potential cause of economic crises, ecological catastrophes, and death of living organisms. Understanding the depletion process is crucial for its further control and optimized replenishment of resources. In this paper, we investigate a stock depletion by a population of species that undergo an ordinary diffusion and consume resources upon each encounter with the stock. We derive the exact form of the probability density of the random depletion time, at which the stock is exhausted. The dependence of this distribution on the number of species, the initial amount of resources, and the geometric setting is analyzed. Future perspectives and related 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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9
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Kaldasch S, Engel A. Stiffness of random walks with reflecting boundary conditions. Phys Rev E 2022; 105:034132. [PMID: 35428115 DOI: 10.1103/physreve.105.034132] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/18/2021] [Accepted: 01/05/2022] [Indexed: 06/14/2023]
Abstract
We study the distribution of occupation times for a one-dimensional random walk restricted to a finite interval by reflecting boundary conditions. At short times the classical bimodal distribution due to Lévy is reproduced with walkers staying mostly either to the left or right of the initial point. With increasing time, however, the boundaries suppress large excursions from the starting point, and the distribution becomes unimodal, converging to a δ distribution in the long-time limit. An approximate spectral analysis of the underlying Fokker-Planck equation yields results in excellent agreement with numerical simulations.
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Affiliation(s)
- Sascha Kaldasch
- Institut für Physik, Universität Oldenburg, 26111 Oldenburg, Germany
| | - Andreas Engel
- Institut für Physik, Universität Oldenburg, 26111 Oldenburg, Germany
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10
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Singh P. Extreme value statistics and arcsine laws for heterogeneous diffusion processes. Phys Rev E 2022; 105:024113. [PMID: 35291128 DOI: 10.1103/physreve.105.024113] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/22/2021] [Accepted: 01/26/2022] [Indexed: 06/14/2023]
Abstract
Heterogeneous diffusion with a spatially changing diffusion coefficient arises in many experimental systems such as protein dynamics in the cell cytoplasm, mobility of cajal bodies, and confined hard-sphere fluids. Here, we showcase a simple model of heterogeneous diffusion where the diffusion coefficient D(x) varies in a power-law way, i.e., D(x)∼|x|^{-α} with the exponent α>-1. This model is known to exhibit anomalous scaling of the mean-squared displacement (MSD) of the form ∼t^{2/2+α} and weak ergodicity breaking in the sense that ensemble averaged and time averaged MSDs do not converge. In this paper, we look at the extreme value statistics of this model and derive, for all α, the exact probability distributions of the maximum spatial displacement M(t) and arg-maximum t_{m}(t) (i.e., the time at which this maximum is reached) till duration t. In the second part of our paper, we analyze the statistical properties of the residence time t_{r}(t) and the last-passage time t_{ℓ}(t) and compute their distributions exactly for all values of α. Our study unravels that the heterogeneous version (α≠0) displays many rich and contrasting features compared to that of the standard Brownian motion (BM). For example, while for BM (α=0), the distributions of t_{m}(t),t_{r}(t), and t_{ℓ}(t) are all identical (á la "arcsine laws" due to Lévy), they turn out to be significantly different for nonzero α. Another interesting property of t_{r}(t) is the existence of a critical α (which we denote by α_{c}=-0.3182) such that the distribution exhibits a local maximum at t_{r}=t/2 for α<α_{c} whereas it has minima at t_{r}=t/2 for α≥α_{c}. The underlying reasoning for this difference hints at the very contrasting natures of the process for α≥α_{c} and α<α_{c} which we thoroughly examine in our paper. All our analytical results are backed by extensive numerical simulations.
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Affiliation(s)
- Prashant Singh
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
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11
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Singh P, Kundu A. Local time for run and tumble particle. Phys Rev E 2021; 103:042119. [PMID: 34005947 DOI: 10.1103/physreve.103.042119] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/15/2020] [Accepted: 03/03/2021] [Indexed: 11/07/2022]
Abstract
We investigate the local time T_{loc} statistics for a run and tumble particle (RTP) in one dimension, which is the quintessential model for the motion of bacteria. In random walk literature, the RTP dynamics is studied as the persistent Brownian motion. We consider the inhomogeneous version of this model where the inhomogeneity is introduced by considering the position-dependent rate of the form R(x)=γ|x|^{α}/l^{α} with α≥0. For α=0, we derive the probability distribution of T_{loc} exactly, which is expressed as a series of δ functions in which the coefficients can be interpreted as the probability of multiple revisits of the RTP to the origin starting from the origin. For general α, we show that the typical fluctuations of T_{loc} scale with time as T_{loc}∼t^{1+α/2+α} for large t and their probability distribution possesses a scaling behavior described by a scaling function which we have computed analytically. Second, we study the statistics of T_{loc} until the RTP makes a first passage to x=M(>0). In this case, we also show that the probability distribution can be expressed as a series sum of δ functions for all values of α(≥0) with coefficients originating from appropriate exit problems. All our analytical findings are supported with numerical simulations.
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Affiliation(s)
- Prashant Singh
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
| | - Anupam Kundu
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
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12
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Pronin KA. Fluctuations and self-averaging in random trapping transport: The diffusion coefficient. Phys Rev E 2020; 101:022132. [PMID: 32168707 DOI: 10.1103/physreve.101.022132] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/28/2018] [Accepted: 02/07/2020] [Indexed: 11/07/2022]
Abstract
On the basis of a self-consistent cluster effective-medium approximation for random trapping transport, we study the problem of self-averaging of the diffusion coefficient in a nonstationary formulation. In the long-time domain, we investigate different cases that correspond to the increasing degree of disorder. In the regular and subregular cases the diffusion coefficient is found to be a self-averaging quantity-its relative fluctuations (relative standard deviation) decay in time in a power-law fashion. In the subdispersive case the diffusion coefficient is self-averaging in three dimensions (3D) and weakly self-averaging in two dimensions (2D) and one dimension (1D), when its relative fluctuations decay anomalously slowly logarithmically. In the dispersive case, the diffusion coefficient is self-averaging in 3D, weakly self-averaging in 2D, and non-self-averaging in 1D. When non-self-averaging, its fluctuations remain of the same order as, or larger than, its average value. In the irreversible case, the diffusion coefficient is non-self-averaging in any dimension. In general, with the decreasing dimension and/or increasing disorder, the self-averaging worsens and eventually disappears. In the cases of weak self-averaging and, especially, non-self-averaging, the reliable reproducible experimental measurements are highly problematic. In all the cases under consideration, asymptotics with prefactors are obtained beyond the scaling laws. Transition between all cases is analyzed as the disorder increases.
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Affiliation(s)
- K A Pronin
- Institute of Biochemical Physics, Russian Academy of Sciences, Moscow, Kosygin Street 4, 119 334, Russia
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13
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Pronin KA. Non-self-averaging in random trapping transport: The diffusion coefficient in the fluctuation regime. Phys Rev E 2019; 100:052144. [PMID: 31869891 DOI: 10.1103/physreve.100.052144] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/05/2018] [Indexed: 11/07/2022]
Abstract
The nonstationary diffusion of particles in a medium with static random traps or sinks is considered. The question of the self-averaging of the diffusion coefficient (or, equivalently, of the mean-square displacement) is addressed for the fluctuation regime in the long-time limit. The property of self-averaging is needed for the result of a single measurement to be representative and reproducible. It is demonstrated that the diffusion coefficient of the surviving particles is a strongly non-self-averaging quantity: In a d-dimensional system its reciprocal standard deviation grows with time exponentially ≈exp[const_{d,1}t^{d/(d+2)}]. The same result is reproduced in the "normalized" formulation "per one survivor on average." The case when all the particles, both the survivors and the trapped ones, are contributing to the diffusion coefficient and its variance is considered also. Non-self-averaging is demonstrated for this case as well, the fluctuations of the diffusion coefficient being of the same order as its average value. The critical dimension, above which the mean-field result becomes exact, is infinite-due to the drastic difference between the classes of trajectories, upon which the corresponding results are being built. In high dimensions the strong non-self-averaging of survivors is preserved. For the case of all the particles taken into account, the nonstrong non-self-averaging is retained for any finite dimension. However, for d→∞ the limiting value of the reciprocal standard deviation, calculated for all the particles, decreases to zero. This signifies restoration of the self-averaging in some sense. In all the cases, the time evolution of the average characteristics and of their variances is governed by the decaying concentration of the survivors in fluctuational cavities.
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Affiliation(s)
- K A Pronin
- Institute of Biochemical Physics, Russian Academy of Sciences, Kosygin Street 4, Moscow 119 334, Russia
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14
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Agranov T, Krapivsky PL, Meerson B. Occupation-time statistics of a gas of interacting diffusing particles. Phys Rev E 2019; 99:052102. [PMID: 31212513 DOI: 10.1103/physreve.99.052102] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/01/2019] [Indexed: 11/07/2022]
Abstract
The time that a diffusing particle spends in a certain region of space is known as the occupation time, or the residence time. Recently, the joint occupation-time statistics of an ensemble of noninteracting particles was addressed using the single-particle statistics. Here we employ the macroscopic fluctuation theory (MFT) to study the occupation-time statistics of many interacting particles. We find that interactions can significantly change the statistics and, in some models, even cause a singularity of the large-deviation function describing these statistics. This singularity can be interpreted as a dynamical phase transition. We also point out to a close relation between the MFT description of the occupation-time statistics of noninteracting particles and the level 2 large deviation formalism which describes the occupation-time statistics of a single particle.
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Affiliation(s)
- Tal Agranov
- Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel
| | - P L Krapivsky
- Department of Physics, Boston University, Boston, Massachusetts 02215, USA
| | - Baruch Meerson
- Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel
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15
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Wu X, Deng W, Barkai E. Tempered fractional Feynman-Kac equation: Theory and examples. Phys Rev E 2016; 93:032151. [PMID: 27078336 DOI: 10.1103/physreve.93.032151] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/30/2016] [Indexed: 06/05/2023]
Abstract
Functionals of Brownian and non-Brownian motions have diverse applications and attracted a lot of interest among scientists. This paper focuses on deriving the forward and backward fractional Feynman-Kac equations describing the distribution of the functionals of the space and time-tempered anomalous diffusion, belonging to the continuous time random walk class. Several examples of the functionals are explicitly treated, including the occupation time in half-space, the first passage time, the maximal displacement, the fluctuations of the occupation fraction, and the fluctuations of the time-averaged position.
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Affiliation(s)
- Xiaochao Wu
- School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, P.R. China
| | - Weihua Deng
- School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, P.R. China
| | - Eli Barkai
- Department of Physics, Advanced Materials and Nanotechnology Institute, Bar Ilan University, Ramat-Gan 52900, Israel
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16
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Menzel AM. Velocity and displacement statistics in a stochastic model of nonlinear friction showing bounded particle speed. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:052302. [PMID: 26651690 DOI: 10.1103/physreve.92.052302] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/21/2015] [Indexed: 06/05/2023]
Abstract
Diffusion of colloidal particles in a complex environment such as polymer networks or biological cells is a topic of high complexity with significant biological and medical relevance. In such situations, the interaction between the surroundings and the particle motion has to be taken into account. We analyze a simplified diffusion model that includes some aspects of a complex environment in the framework of a nonlinear friction process: at low particle speeds, friction grows linearly with the particle velocity as for regular viscous friction; it grows more than linearly at higher particle speeds; finally, at a maximum of the possible particle speed, the friction diverges. In addition to bare diffusion, we study the influence of a constant drift force acting on the diffusing particle. While the corresponding stationary velocity distributions can be derived analytically, the displacement statistics generally must be determined numerically. However, as a benefit of our model, analytical progress can be made in one case of a special maximum particle speed. The effect of a drift force in this case is analytically determined by perturbation theory. It will be interesting in the future to compare our results to real experimental systems. One realization could be magnetic colloidal particles diffusing through a shear-thickening environment such as starch suspensions, possibly exposed to an external magnetic field gradient.
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Affiliation(s)
- Andreas M Menzel
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, D-40225 Düsseldorf, Germany
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17
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Schulz JHP, Barkai E. Fluctuations around equilibrium laws in ergodic continuous-time random walks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:062129. [PMID: 26172683 DOI: 10.1103/physreve.91.062129] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/17/2014] [Indexed: 06/04/2023]
Abstract
We study occupation time statistics in ergodic continuous-time random walks. Under thermal detailed balance conditions, the average occupation time is given by the Boltzmann-Gibbs canonical law. But close to the nonergodic phase, the finite-time fluctuations around this mean are large and nontrivial. They exhibit dual time scaling and distribution laws: the infinite density of large fluctuations complements the Lévy-stable density of bulk fluctuations. Neither of the two should be interpreted as a stand-alone limiting law, as each has its own deficiency: the infinite density has an infinite norm (despite particle conservation), while the stable distribution has an infinite variance (although occupation times are bounded). These unphysical divergences are remedied by consistent use and interpretation of both formulas. Interestingly, while the system's canonical equilibrium laws naturally determine the mean occupation time of the ergodic motion, they also control the infinite and Lévy-stable densities of fluctuations. The duality of stable and infinite densities is in fact ubiquitous for these dynamics, as it concerns the time averages of general physical observables.
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Affiliation(s)
- Johannes H P Schulz
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat Gan 52900, Israel
| | - Eli Barkai
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat Gan 52900, Israel
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18
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Chen Y, Just W. Large-deviation properties of Brownian motion with dry friction. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:042102. [PMID: 25375433 DOI: 10.1103/physreve.90.042102] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/28/2014] [Indexed: 06/04/2023]
Abstract
We investigate piecewise-linear stochastic models with regard to the probability distribution of functionals of the stochastic processes, a question that occurs frequently in large deviation theory. The functionals that we are looking into in detail are related to the time a stochastic process spends at a phase space point or in a phase space region, as well as to the motion with inertia. For a Langevin equation with discontinuous drift, we extend the so-called backward Fokker-Planck technique for non-negative support functionals to arbitrary support functionals, to derive explicit expressions for the moments of the functional. Explicit solutions for the moments and for the distribution of the so-called local time, the occupation time, and the displacement are derived for the Brownian motion with dry friction, including quantitative measures to characterize deviation from Gaussian behavior in the asymptotic long time limit.
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Affiliation(s)
- Yaming Chen
- School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom
| | - Wolfram Just
- School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom
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Bénichou O, Meunier N, Redner S, Voituriez R. Non-Gaussianity and dynamical trapping in locally activated random walks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:021137. [PMID: 22463182 DOI: 10.1103/physreve.85.021137] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/03/2011] [Indexed: 05/31/2023]
Abstract
We propose a minimal model of locally activated diffusion, in which the diffusion coefficient of a one-dimensional Brownian particle is modified in a prescribed way-either increased or decreased-upon each crossing of the origin. Such a local mobility decrease arises in the formation of atherosclerotic plaques due to diffusing macrophage cells accumulating lipid particles. We show that spatially localized mobility perturbations have remarkable consequences on diffusion at all scales, such as the emergence of a non-Gaussian multipeaked probability distribution and a dynamical transition to an absorbing static state. In the context of atherosclerosis, this dynamical transition can be viewed as a minimal mechanism that causes macrophages to aggregate in lipid-enriched regions and thereby to the formation of atherosclerotic plaques.
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Affiliation(s)
- O Bénichou
- Laboratoire de Physique Théorique de la Matière Condensée, CNRS UMR 7600, case courrier 121, Université Paris 6, 4 Place Jussieu, FR-75255 Paris Cedex, France
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20
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Zoia A, Dumonteil E, Mazzolo A. Counting statistics: a Feynman-Kac perspective. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:011132. [PMID: 22400537 DOI: 10.1103/physreve.85.011132] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/14/2011] [Indexed: 05/31/2023]
Abstract
By building upon a Feynman-Kac formalism, we assess the distribution of the number of collisions in a given region for a broad class of discrete-time random walks in absorbing and nonabsorbing media. We derive the evolution equation for the generating function of the number of collisions, and we complete our analysis by examining the moments of the distribution and their relation to the walker equilibrium density. Some significant applications are discussed in detail: in particular, we revisit the gambler's ruin problem and generalize to random walks with absorption the arcsine law for the number of collisions on the half-line.
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Affiliation(s)
- A Zoia
- CEA/Saclay, DEN/DANS/DM2S/SERMA/LTSD, F-91191 Gif-sur-Yvette, France.
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21
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Carmi S, Barkai E. Fractional Feynman-Kac equation for weak ergodicity breaking. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:061104. [PMID: 22304037 DOI: 10.1103/physreve.84.061104] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/24/2011] [Indexed: 05/31/2023]
Abstract
The continuous-time random walk (CTRW) is a model of anomalous subdiffusion in which particles are immobilized for random times between successive jumps. A power-law distribution of the waiting times, ψ(τ) ~ τ(-(1+α)), leads to subdiffusion (x(2) ~ t(α)) for 0 < α < 1. In closed systems, the long stagnation periods cause time averages to divert from the corresponding ensemble averages, which is a manifestation of weak ergodicity breaking. The time average of a general observable U(t) = 1/t ∫(0)(t) U[x(τ)]dτ is a functional of the path and is described by the well-known Feynman-Kac equation if the motion is Brownian. Here, we derive forward and backward fractional Feynman-Kac equations for functionals of CTRW in a binding potential. We use our equations to study two specific time averages: the fraction of time spent by a particle in half-box, and the time average of the particle's position in a harmonic field. In both cases, we obtain the probability density function of the time averages for t → ∞ and the first two moments. Our results show that both the occupation fraction and the time-averaged position are random variables even for long times, except for α = 1, when they are identical to their ensemble averages. Using our fractional Feynman-Kac equation, we also study the dynamics leading to weak ergodicity breaking, namely the convergence of the fluctuations to their asymptotic values.
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Affiliation(s)
- Shai Carmi
- Department of Physics & Advanced Materials and Nanotechnology Institute, Bar-Ilan University, Ramat Gan 52900, Israel
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22
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Burov S, Barkai E. Residence time statistics for N renewal processes. PHYSICAL REVIEW LETTERS 2011; 107:170601. [PMID: 22107497 DOI: 10.1103/physrevlett.107.170601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/27/2011] [Indexed: 05/31/2023]
Abstract
We present a study of residence time statistics for N renewal processes with a long tailed distribution of the waiting time. Such processes describe many nonequilibrium systems ranging from the intensity of N blinking quantum dots to the residence time of N Brownian particles. With numerical simulations and exact calculations, we show sharp transitions for a critical number of degrees of freedom N. In contrast to the expectation, the fluctuations in the limit of N→∞ are nontrivial. We briefly discuss how our approach can be used to detect nonergodic kinetics from the measurements of many blinking chromophores, without the need to reach the single molecule limit.
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Affiliation(s)
- S Burov
- Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar Ilan University, Ramat-Gan 52900, Israel
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23
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Agmon N. Single molecule diffusion and the solution of the spherically symmetric residence time equation. J Phys Chem A 2011; 115:5838-46. [PMID: 21306174 DOI: 10.1021/jp1099877] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/12/2022]
Abstract
The residence time of a single dye molecule diffusing within a laser spot is propotional to the total number of photons emitted by it. With this application in mind, we solve the spherically symmetric "residence time equation" (RTE) to obtain the solution for the Laplace transform of the mean residence time (MRT) within a d-dimensional ball, as a function of the initial location of the particle and the observation time. The solutions for initial conditions of potential experimental interest, starting in the center, on the surface or uniformly within the ball, are explicitly presented. Special cases for dimensions 1, 2, and 3 are obtained, which can be Laplace inverted analytically for d = 1 and 3. In addition, the analytic short- and long-time asymptotic behaviors of the MRT are derived and compared with the exact solutions for d = 1, 2, and 3. As a demonstration of the simplification afforded by the RTE, the Appendix obtains the residence time distribution by solving the Feynman-Kac equation, from which the MRT is obtained by differentiation. Single-molecule diffusion experiments could be devised to test the results for the MRT presented in this work.
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Affiliation(s)
- Noam Agmon
- The Fritz Haber Research Center, Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel.
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Monthus C, Garel T. Random walk in a two-dimensional self-affine random potential: Properties of the anomalous diffusion phase at small external force. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:021125. [PMID: 20866793 DOI: 10.1103/physreve.82.021125] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/08/2010] [Indexed: 05/29/2023]
Abstract
We study the dynamical response to an external force F for a particle performing a random walk in a two-dimensional quenched random potential of Hurst exponent H=1/2 . We present numerical results on the statistics of first-passage times that satisfy closed backward master equations. We find that there exists a zero-velocity phase in a finite region of the external force 0<F<F{c} , where the dynamics follows the anomalous diffusion law x(t)∼ξ(F)t^{μ(F)} . The anomalous exponent 0<μ(F)<1 and the correlation length ξ(F) vary continuously with F . In the limit of vanishing force F→0 , we measure the following power laws: the anomalous exponent vanishes as μ(F)∝F^{a} with a≃0.6 (instead of a=1 in dimension d=1 ), and the correlation length diverges as ξ(F)∝F^{-ν} with ν≃1.29 (instead of ν=2 in dimension d=1 ). Our main conclusion is thus that the dynamics renormalizes onto an effective directed trap model, where the traps are characterized by a typical length ξ(F) along the direction of the force, and by a typical barrier 1/μ(F) . The fact that these traps are "smaller" in linear size and in depth than in dimension d=1 , means that the particle uses the transverse direction to find lower barriers.
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Affiliation(s)
- Cécile Monthus
- Institut de Physique Théorique, CNRS and CEA Saclay, Gif-sur-Yvette, France
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26
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Bénichou O, Voituriez R. Optimization of the residence time of a Brownian particle in a spherical subdomain. J Chem Phys 2010; 131:181104. [PMID: 19916589 DOI: 10.1063/1.3264122] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
In this communication, we show that the residence time of a Brownian particle, defined as the cumulative time spent in a given region of space, can be optimized as a function of the diffusion coefficient. We discuss the relevance of this effect to several schematic experimental situations classified in nature--random or deterministic--both of the observation time and of the starting position of the Brownian particle.
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Affiliation(s)
- O Bénichou
- Laboratoire de Physique Théorique de la Matière Condensée (UMR 7600), Case Courrier 121, UniversitéParis 6, 4 Place Jussieu, 75255 Paris Cedex, France.
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27
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Turgeman L, Carmi S, Barkai E. Fractional Feynman-Kac equation for non-brownian functionals. PHYSICAL REVIEW LETTERS 2009; 103:190201. [PMID: 20365911 DOI: 10.1103/physrevlett.103.190201] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/08/2009] [Indexed: 05/29/2023]
Abstract
We derive backward and forward fractional Feynman-Kac equations for the distribution of functionals of the path of a particle undergoing anomalous diffusion. Fractional substantial derivatives introduced by Friedrich and co-workers [Phys. Rev. Lett. 96, 230601 (2006)10.1103/PhysRevLett.96.230601] provide the correct fractional framework for the problem. For applications, we calculate the distribution of occupation times in half space and show how the statistics of anomalous functionals is related to weak ergodicity breaking.
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Affiliation(s)
- Lior Turgeman
- Department of Physics, Bar Ilan University, Ramat-Gan 52900 Israel
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28
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Condamin S, Tejedor V, Bénichou O. Occupation times of random walks in confined geometries: from random trap model to diffusion-limited reactions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:050102. [PMID: 18233611 DOI: 10.1103/physreve.76.050102] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/06/2007] [Indexed: 05/25/2023]
Abstract
We consider a random walk in confined geometry, starting from a site and eventually reaching a target site. We calculate analytically the distribution of the occupation time on a third site, before reaching the target site. The obtained distribution is exact and completely explicit in the case or parallelepipedic confining domains. We discuss implications of these results in two different fields: The mean first passage time for the random trap model is computed in dimensions greater than 1 and is shown to display a nontrivial dependence with the source and target positions. The probability of reaction with a given imperfect center before being trapped by another one is also explicitly calculated, revealing a complex dependence both in geometrical and chemical parameters.
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Affiliation(s)
- S Condamin
- Laboratoire de Physique Théorique de la Matière Condensée (UMR 7600), Case Courrier 121, Université Paris 6, 4 Place Jussieu, 75255 Paris Cedex, France
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29
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Burov S, Barkai E. Occupation time statistics in the quenched trap model. PHYSICAL REVIEW LETTERS 2007; 98:250601. [PMID: 17678005 DOI: 10.1103/physrevlett.98.250601] [Citation(s) in RCA: 49] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/08/2007] [Revised: 03/19/2007] [Indexed: 05/16/2023]
Abstract
We investigate the distribution of the occupation time of a particle undergoing a random walk among random energy traps and in the presence of a deterministic potential field. When the distribution of energy traps is exponential with a width T(g), we find in thermal equilibrium a transition between Boltzmann statistics when T>T(g) to Lamperti statistics when T < T(g). We explain why our main results are valid for other models of quenched disorder, and discuss briefly implications on single particle experiments.
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Affiliation(s)
- S Burov
- Department of Physics, Bar Ilan University, Ramat-Gan 52900, Israel
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30
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Sabhapandit S, Majumdar SN, Comtet A. Statistical properties of functionals of the paths of a particle diffusing in a one-dimensional random potential. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:051102. [PMID: 16802913 DOI: 10.1103/physreve.73.051102] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/19/2006] [Indexed: 05/10/2023]
Abstract
We present a formalism for obtaining the statistical properties of functionals and inverse functionals of the paths of a particle diffusing in a one-dimensional quenched random potential. We demonstrate the implementation of the formalism in two specific examples: (1) where the functional corresponds to the local time spent by the particle around the origin and (2) where the functional corresponds to the occupation time spent by the particle on the positive side of the origin, within an observation time window of size t. We compute the disorder average distributions of the local time, the inverse local time, the occupation time, and the inverse occupation time and show that in many cases disorder modifies the behavior drastically.
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Affiliation(s)
- Sanjib Sabhapandit
- Laboratoire de Physique Théorique et Modèles Statistiques, Université Paris-Sud, Bâtiment 100, 91405 Orsay Cedex, France
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31
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Bel G, Barkai E. Random walk to a nonergodic equilibrium concept. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:016125. [PMID: 16486234 DOI: 10.1103/physreve.73.016125] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/19/2005] [Indexed: 05/06/2023]
Abstract
Random walk models, such as the trap model, continuous time random walks, and comb models, exhibit weak ergodicity breaking, when the average waiting time is infinite. The open question is, what statistical mechanical theory replaces the canonical Boltzmann-Gibbs theory for such systems? In this paper a nonergodic equilibrium concept is investigated, for a continuous time random walk model in a potential field. In particular we show that in the nonergodic phase the distribution of the occupation time of the particle in a finite region of space approaches U- or W-shaped distributions related to the arcsine law. We show that when conditions of detailed balance are applied, these distributions depend on the partition function of the problem, thus establishing a relation between the nonergodic dynamics and canonical statistical mechanics. In the ergodic phase the distribution function of the occupation times approaches a delta function centered on the value predicted based on standard Boltzmann-Gibbs statistics. The relation of our work to single-molecule experiments is briefly discussed.
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Affiliation(s)
- G Bel
- Department of Physics, Bar Ilan University, Ramat-Gan 52900, Israel.
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Ehrhardt GCMA, Majumdar SN, Bray AJ. Persistence exponents and the statistics of crossings and occupation times for Gaussian stationary processes. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:016106. [PMID: 14995666 DOI: 10.1103/physreve.69.016106] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/05/2003] [Indexed: 05/24/2023]
Abstract
We consider the persistence probability, the occupation-time distribution, and the distribution of the number of zero crossings for discrete or (equivalently) discretely sampled Gaussian stationary processes (GSPs) of zero mean. We first consider the Ornstein-Uhlenbeck process, finding expressions for the mean and variance of the number of crossings and the "partial survival" probability. We then elaborate on the correlator expansion developed in an earlier paper [G. C. M. A. Ehrhardt and A. J. Bray, Phys. Rev. Lett. 88, 070602 (2002)] to calculate discretely sampled persistence exponents of GSPs of known correlator by means of a series expansion in the correlator. We apply this method to the processes d(n)x/dt(n)=eta(t) with n>/=3, incorporating an extrapolation of the series to the limit of continuous sampling. We then extend the correlator method to calculate the occupation-time and crossing-number distributions, as well as their partial-survival distributions and the means and variances of the occupation time and number of crossings. We apply these general methods to the d(n)x/dt(n)=eta(t) processes for n=1 (random walk), n=2 (random acceleration), and larger n, and to simple diffusion from random initial conditions in one to three dimensions. The results for discrete sampling are extrapolated to the continuum limit where possible.
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Affiliation(s)
- G C M A Ehrhardt
- Department of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom
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33
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Comtet A, Desbois J, Majumdar SN. The local time distribution of a particle diffusing on a graph. ACTA ACUST UNITED AC 2002. [DOI: 10.1088/0305-4470/35/47/102] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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34
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Majumdar SN, Dean DS. Exact occupation time distribution in a non-Markovian sequence and its relation to spin glass models. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:041102. [PMID: 12443172 DOI: 10.1103/physreve.66.041102] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/09/2002] [Indexed: 05/24/2023]
Abstract
We compute exactly the distribution of the occupation time in a discrete non-Markovian toy sequence that appears in various physical contexts such as the diffusion processes and Ising spin glass chains. The non-Markovian property makes the results nontrivial even for this toy sequence. The distribution is shown to have non-Gaussian tails characterized by a nontrivial large deviation function that is computed explicitly. An exact mapping of this sequence to an Ising spin glass chain via a gauge transformation raises an interesting question for a generic finite sized spin glass model; at a given temperature, what is the distribution (over disorder) of the thermally averaged number of spins that are aligned to their local fields? We show that this distribution remains nontrivial even at infinite temperature and can be computed explicitly in few cases such as in the Sherrington-Kirkpatrick model with Gaussian disorder.
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Affiliation(s)
- Satya N Majumdar
- Laboratoire de Physique Quantique (UMR C5626 du CNRS), Université Paul Sabatier, 31062 Toulouse Cedex, France
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