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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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Malik O, Varga M, Moussawi A, Hunt D, Szymanski BK, Toroczkai Z, Korniss G. Diffusive persistence on disordered lattices and random networks. Phys Rev E 2024; 109:024113. [PMID: 38491611 DOI: 10.1103/physreve.109.024113] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/25/2023] [Accepted: 12/22/2023] [Indexed: 03/18/2024]
Abstract
To better understand the temporal characteristics and the lifetime of fluctuations in stochastic processes in networks, we investigated diffusive persistence in various graphs. Global diffusive persistence is defined as the fraction of nodes for which the diffusive field at a site (or node) has not changed sign up to time t (or, in general, that the node remained active or inactive in discrete models). Here we investigate disordered and random networks and show that the behavior of the persistence depends on the topology of the network. In two-dimensional (2D) disordered networks, we find that above the percolation threshold diffusive persistence scales similarly as in the original 2D regular lattice, according to a power law P(t,L)∼t^{-θ} with an exponent θ≃0.186, in the limit of large linear system size L. At the percolation threshold, however, the scaling exponent changes to θ≃0.141, as the result of the interplay of diffusive persistence and the underlying structural transition in the disordered lattice at the percolation threshold. Moreover, studying finite-size effects for 2D lattices at and above the percolation threshold, we find that at the percolation threshold, the long-time asymptotic value obeys a power law P(t,L)∼L^{-zθ} with z≃2.86 instead of the value of z=2 normally associated with finite-size effects on 2D regular lattices. In contrast, we observe that in random networks without a local regular structure, such as Erdős-Rényi networks, no simple power-law scaling behavior exists above the percolation threshold.
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Affiliation(s)
- Omar Malik
- Department of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic Institute, Troy, New York 12180, USA
- Network Science and Technology Center, Rensselaer Polytechnic Institute, Troy, New York 12180, USA
| | - Melinda Varga
- Department of Physics and Astronomy, University of Notre Dame, Notre Dame, Indiana 46556, USA
| | - Alaa Moussawi
- Department of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic Institute, Troy, New York 12180, USA
- Network Science and Technology Center, Rensselaer Polytechnic Institute, Troy, New York 12180, USA
| | - David Hunt
- Department of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic Institute, Troy, New York 12180, USA
- Network Science and Technology Center, Rensselaer Polytechnic Institute, Troy, New York 12180, USA
| | - Boleslaw K Szymanski
- Department of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic Institute, Troy, New York 12180, USA
- Network Science and Technology Center, Rensselaer Polytechnic Institute, Troy, New York 12180, USA
- Department of Computer Science, Rensselaer Polytechnic Institute, Troy, New York 12180, USA
| | - Zoltan Toroczkai
- Department of Physics and Astronomy, University of Notre Dame, Notre Dame, Indiana 46556, USA
| | - Gyorgy Korniss
- Department of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic Institute, Troy, New York 12180, USA
- Network Science and Technology Center, Rensselaer Polytechnic Institute, Troy, New York 12180, USA
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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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Radice M, Onofri M, Artuso R, Pozzoli G. Statistics of occupation times and connection to local properties of nonhomogeneous random walks. Phys Rev E 2020; 101:042103. [PMID: 32422811 DOI: 10.1103/physreve.101.042103] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/30/2020] [Accepted: 03/16/2020] [Indexed: 11/07/2022]
Abstract
We consider the statistics of occupation times, the number of visits at the origin, and the survival probability for a wide class of stochastic processes, which can be classified as renewal processes. We show that the distribution of these observables can be characterized by a single exponent, that is connected to a local property of the probability density function of the process, viz., the probability of occupying the origin at time t, P(t). We test our results for two different models of lattice random walks with spatially inhomogeneous transition probabilities, one of which of non-Markovian nature, and find good agreement with theory. We also show that the distributions depend only on the occupation probability of the origin by comparing them for the two systems: When P(t) shows the same long-time behavior, each observable follows indeed the same distribution.
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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, I-22100 Como, Italy and I.N.F.N. Sezione di Milano, Via Celoria 16, I-20133 Milano, Italy
| | - Manuele Onofri
- Dipartimento di Scienza e Alta Tecnologia and Center for Nonlinear and Complex Systems, Università degli studi dell'Insubria, Via Valleggio 11, I-22100 Como, Italy and I.N.F.N. Sezione di Milano, Via Celoria 16, I-20133 Milano, Italy
| | - Roberto Artuso
- Dipartimento di Scienza e Alta Tecnologia and Center for Nonlinear and Complex Systems, Università degli studi dell'Insubria, Via Valleggio 11, I-22100 Como, Italy and I.N.F.N. Sezione di Milano, Via Celoria 16, I-20133 Milano, Italy
| | - Gaia Pozzoli
- Dipartimento di Scienza e Alta Tecnologia and Center for Nonlinear and Complex Systems, Università degli studi dell'Insubria, Via Valleggio 11, I-22100 Como, Italy and I.N.F.N. Sezione di Milano, Via Celoria 16, I-20133 Milano, Italy
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5
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Ghosh A, Chakraborty D. Persistence in Brownian motion of an ellipsoidal particle in two dimensions. J Chem Phys 2020; 152:174901. [PMID: 32384838 DOI: 10.1063/5.0004134] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We investigate the persistence probability p(t) of the position of a Brownian particle with shape asymmetry in two dimensions. The persistence probability is defined as the probability that a stochastic variable has not changed its sign in the given time interval. We explicitly consider two cases-diffusion of a free particle and that of a harmonically trapped particle. The latter is particularly relevant in experiments that use trapping and tracking techniques to measure the displacements. We provide analytical expressions of p(t) for both the scenarios and show that in the absence of the shape asymmetry, the results reduce to the case of an isotropic particle. The analytical expressions of p(t) are further validated against numerical simulation of the underlying overdamped dynamics. We also illustrate that p(t) can be a measure to determine the shape asymmetry of a colloid and the translational and rotational diffusivities can be estimated from the measured persistence probability. The advantage of this method is that it does not require the tracking of the orientation of the particle.
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Affiliation(s)
- Anirban Ghosh
- Indian Institute of Science Education and Research Mohali, Sec. 81, S.A.S. Nagar, Knowledge City, Manauli, Punjab 140306, India
| | - Dipanjan Chakraborty
- Indian Institute of Science Education and Research Mohali, Sec. 81, S.A.S. Nagar, Knowledge City, Manauli, Punjab 140306, India
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Chanphana R, Chatraphorn P, Dasgupta C. Effects of initial height on the steady-state persistence probability of linear growth models. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:062402. [PMID: 24483456 DOI: 10.1103/physreve.88.062402] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/06/2013] [Revised: 11/04/2013] [Indexed: 06/03/2023]
Abstract
The effects of the initial height on the temporal persistence probability of steady-state height fluctuations in up-down symmetric linear models of surface growth are investigated. We study the (1+1)-dimensional Family model and the (1+1)- and (2+1)-dimensional larger curvature (LC) model. Both the Family and LC models have up-down symmetry, so the positive and negative persistence probabilities in the steady state, averaged over all values of the initial height h(0), are equal to each other. However, these two probabilities are not equal if one considers a fixed nonzero value of h(0). Plots of the positive persistence probability for negative initial height versus time exhibit power-law behavior if the magnitude of the initial height is larger than the interface width at saturation. By symmetry, the negative persistence probability for positive initial height also exhibits the same behavior. The persistence exponent that describes this power-law decay decreases as the magnitude of the initial height is increased. The dependence of the persistence probability on the initial height, the system size, and the discrete sampling time is found to exhibit scaling behavior.
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Affiliation(s)
- R Chanphana
- Department of Physics, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand and Research Center in Thin Film Physics, Thailand Center of Excellence in Physics, CHE, Bangkok 10400, Thailand
| | - P Chatraphorn
- Department of Physics, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand and Research Center in Thin Film Physics, Thailand Center of Excellence in Physics, CHE, Bangkok 10400, Thailand
| | - C Dasgupta
- Department of Physics, Indian Institute of Science, Bangalore 560012, India
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7
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Takikawa Y, Orihara H. Persistence of Brownian motion in a shear flow. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:062111. [PMID: 24483390 DOI: 10.1103/physreve.88.062111] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/17/2013] [Indexed: 06/03/2023]
Abstract
The persistence of a Brownian particle in a shear flow is investigated. The persistence probability P(t), which is the probability that the particle does not return to its initial position up to time t, is known to obey a power law P(t)∝t(-θ). Since the displacement of a particle along the flow direction due to convection is much larger than that due to Brownian motion, we define an alternative displacement in which the convection effect is removed. We derive theoretically the two-time correlation function and the persistence exponent θ of this displacement. The exponent has different values at short and long times. The theoretical results are compared with experiment and a good agreement is found.
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Affiliation(s)
- Yoshinori Takikawa
- Division of Applied Physics, Faculty of Engineering, Hokkaido University, Sapporo 060-8628, Japan
| | - Hiroshi Orihara
- Division of Applied Physics, Faculty of Engineering, Hokkaido University, Sapporo 060-8628, Japan
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8
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Chakraborty D. Persistence of a Brownian particle in a time-dependent potential. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:051101. [PMID: 23004697 DOI: 10.1103/physreve.85.051101] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/06/2011] [Revised: 02/27/2012] [Indexed: 06/01/2023]
Abstract
We investigate the persistence probability of a Brownian particle in a harmonic potential, which decays to zero at long times, leading to an unbounded motion of the Brownian particle. We consider two functional forms for the decay of the confinement, an exponential decay and an algebraic decay. Analytical calculations and numerical simulations show that for the case of the exponential relaxation, the dynamics of Brownian particle at short and long times are independent of the parameters of the relaxation. On the contrary, for the algebraic decay of the confinement, the dynamics at long times is determined by the exponent of the decay. Finally, using the two-time correlation function for the position of the Brownian particle, we construct the persistence probability for the Brownian walker in such a scenario.
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Affiliation(s)
- D Chakraborty
- Institute for Theoretical Physics, University of Leipzig, Vor dem Hospitaltore 1, 04103 Leipzig, Germany.
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9
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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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10
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Constantin M, Das Sarma S. Mapping spatial persistent large deviations of nonequilibrium surface growth processes onto the temporal persistent large deviations of stochastic random walk processes. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:041602. [PMID: 15600416 DOI: 10.1103/physreve.70.041602] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/29/2004] [Revised: 06/18/2004] [Indexed: 05/24/2023]
Abstract
Spatial persistent large deviations probability of surface growth processes governed by the Edwards-Wilkinson dynamics, Px(x,s), with -1< or =s< or =1 is mapped isomorphically onto the temporal persistent large deviations probability Pt(t,s) associated with the stochastic Markovian random walk problem. We show using numerical simulations that the infinite family of spatial persistent large deviations exponents thetax(s) characterizing the power-law decay of Px(x,s ) agrees, as predicted on theoretical grounds by Phys. Rev. Lett. 86, 3700 (2001)], with the numerical measurements of thetat(s), the continuous family of exponents characterizing the long-time power law behavior of Pt(t,s). We also discuss the simulations of the spatial persistence probability corresponding to a discrete model in the Mullins-Herring universality class, where our discrete simulations do not agree well with the theoretical predictions perhaps because of the severe finite-size corrections which are known to strongly inhibit the manifestation of the asymptotic continuum behavior in discrete models involving large values of the dynamical exponent and the associated extremely slow convergence to the asymptotic regime.
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Affiliation(s)
- M Constantin
- Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA
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11
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Constantin M, Dasgupta C, Chatraphorn PP, Majumdar SN, Sarma SD. Persistence in nonequilibrium surface growth. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:061608. [PMID: 15244586 DOI: 10.1103/physreve.69.061608] [Citation(s) in RCA: 23] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/22/2004] [Revised: 03/29/2004] [Indexed: 05/24/2023]
Abstract
Persistence probabilities of the interface height in ( 1+1 ) - and ( 2+1 ) -dimensional atomistic, solid-on-solid, stochastic models of surface growth are studied using kinetic Monte Carlo simulations, with emphasis on models that belong to the molecular beam epitaxy (MBE) universality class. Both the initial transient and the long-time steady-state regimes are investigated. We show that for growth models in the MBE universality class, the nonlinearity of the underlying dynamical equation is clearly reflected in the difference between the measured values of the positive and negative persistence exponents in both transient and steady-state regimes. For the MBE universality class, the positive and negative persistence exponents in the steady-state are found to be theta(S)(+) =0.66+/-0.02 and theta(S)(-) =0.78+/-0.02, respectively, in ( 1+1 ) dimensions, and theta(S)(+) =0.76+/-0.02 and theta(S)(-) =0.85+/-0.02, respectively, in ( 2+1 ) dimensions. The noise reduction technique is applied on some of the ( 1+1 ) -dimensional models in order to obtain accurate values of the persistence exponents. We show analytically that a relation between the steady-state persistence exponent and the dynamic growth exponent, found earlier to be valid for linear models, should be satisfied by the smaller of the two steady-state persistence exponents in the nonlinear models. Our numerical results for the persistence exponents are consistent with this prediction. We also find that the steady-state persistence exponents can be obtained from simulations over times that are much shorter than that required for the interface to reach the steady state. The dependence of the persistence probability on the system size and the sampling time is shown to be described by a simple scaling form.
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Affiliation(s)
- M Constantin
- Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA
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12
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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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Constantin M, Das Sarma S, Dasgupta C, Bondarchuk O, Dougherty DB, Williams ED. Infinite family of persistence exponents for interface fluctuations. PHYSICAL REVIEW LETTERS 2003; 91:086103. [PMID: 14525262 DOI: 10.1103/physrevlett.91.086103] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/11/2003] [Revised: 05/28/2003] [Indexed: 05/24/2023]
Abstract
We show experimentally and theoretically that the persistence of large deviations in equilibrium step fluctuations is characterized by an infinite family of independent exponents. These exponents are obtained by carefully analyzing dynamical experimental images of Al/Si(111) and Ag(111) equilibrium steps fluctuating at high (970 K) and low (320 K) temperatures, respectively, and by quantitatively interpreting our observations on the basis of the corresponding coarse-grained discrete and continuum theoretical models for thermal surface step fluctuations under attachment/detachment ("high-temperature") and edge-diffusion limited kinetics ("low-temperature"), respectively.
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Affiliation(s)
- M Constantin
- Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, MD 20742-4111, USA
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14
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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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15
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Dougherty DB, Lyubinetsky I, Williams ED, Constantin M, Dasgupta C, Das Sarma S. Experimental persistence probability for fluctuating steps. PHYSICAL REVIEW LETTERS 2002; 89:136102. [PMID: 12225042 DOI: 10.1103/physrevlett.89.136102] [Citation(s) in RCA: 38] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/26/2001] [Indexed: 05/23/2023]
Abstract
The persistence behavior for fluctuating steps on the Si(111)-(sqrt[3]xsqrt[3])R30 degrees -Al surface was determined by analyzing time-dependent STM images for temperatures between 770 and 970 K. Using the standard persistence definition, the measured persistence probability displays power-law decay with an exponent of theta=0.77+/-0.03. This is consistent with the value of theta=3/4 predicted for attachment-detachment limited step kinetics. If the persistence analysis is carried out in terms of return to a fixed-reference position, the measured probability decays exponentially. Numerical studies of the Langevin equation used to model step motion corroborate the experimental observations.
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Affiliation(s)
- D B Dougherty
- Materials Research Science and Engineering Center, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA
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Hellén EKO, Alava MJ. Persistence in cluster-cluster aggregation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:026120. [PMID: 12241250 DOI: 10.1103/physreve.66.026120] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/20/2001] [Revised: 04/25/2002] [Indexed: 11/07/2022]
Abstract
Persistence is considered in one-dimensional diffusion-limited cluster-cluster aggregation when the diffusion coefficient of a cluster depends on its size s as D(s) approximately s(gamma). The probabilities that a site has been either empty or covered by a cluster all the time define the empty and filled site persistences. The cluster persistence gives the probability of a cluster remaining intact. The empty site and cluster persistences are universal whereas the filled site depends on the initial concentration. For gamma>0 the universal persistences decay algebraically with the exponent 2/(2-gamma). For the empty site case the exponent remains the same for gamma<0 but the cluster persistence shows a stretched exponential behavior as it is related to the small s behavior of the cluster size distribution. The scaling of the intervals between persistent regions demonstrates the presence of two length scales: the one related to the distances between clusters and that between the persistent regions.
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Affiliation(s)
- E K O Hellén
- Laboratory of Physics, Helsinki University of Technology, P.O. Box 1100, FIN-02015 HUT, Finland
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17
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Majumdar SN, Comtet A. Local and occupation time of a particle diffusing in a random medium. PHYSICAL REVIEW LETTERS 2002; 89:060601. [PMID: 12190571 DOI: 10.1103/physrevlett.89.060601] [Citation(s) in RCA: 33] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/04/2002] [Indexed: 05/23/2023]
Abstract
We consider a particle moving in a one-dimensional potential which has a symmetric deterministic part and a quenched random part. We study analytically the probability distributions of the local time (spent by the particle around its mean value) and the occupation time (spent above its mean value) within an observation time window of size t. In the absence of quenched randomness, these distributions have three typical asymptotic behaviors depending on whether the deterministic potential is unstable, stable, or flat. These asymptotic behaviors are shown to get drastically modified when the random part of the potential is switched on, leading to the loss of self-averaging and wide sample to sample fluctuations.
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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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Majumdar SN, Bray AJ. Large-deviation functions for nonlinear functionals of a Gaussian stationary Markov process. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:051112. [PMID: 12059534 DOI: 10.1103/physreve.65.051112] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/08/2002] [Indexed: 05/23/2023]
Abstract
We introduce a general method, based on a mapping onto quantum mechanics, for investigating the large-T limit of the distribution P(r,T) of the nonlinear functional r[V]=(1/T)integral(T)(0)dT' V[X(T')], where V(X) is an arbitrary function of the stationary Gaussian Markov process X(T). For T-->infinity at fixed r we obtain P(r,T) approximately exp[-theta(r)T], where theta(r) is a large-deviation function. We present explicit results for a number of special cases including V(X)=XH(X) [where H(X) is the Heaviside function], which is related to the cooling and the heating degree days relevant to weather derivatives.
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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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Ehrhardt GCMA, Bray AJ, Majumdar SN. Persistence of a continuous stochastic process with discrete-time sampling: non-Markov processes. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:041102. [PMID: 12005801 DOI: 10.1103/physreve.65.041102] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/11/2001] [Indexed: 05/23/2023]
Abstract
We consider the problem of "discrete-time persistence," which deals with the zero crossings of a continuous stochastic process X(T) measured at discrete times T=nDeltaT. For a Gaussian stationary process the persistence (no crossing) probability decays as exp(-theta(D)T)=[rho(a)](n) for large n, where a=exp(-DeltaT/2) and the discrete persistence exponent theta(D) is given by theta(D)=(ln rho)/(2 ln a). Using the "independent interval approximation," we show how theta(D) varies with DeltaT for small DeltaT and conclude that experimental measurements of persistence for smooth processes, such as diffusion, are less sensitive to the effects of discrete sampling than measurements of a randomly accelerated particle or random walker. We extend the matrix method developed by us previously [Phys. Rev. E 64, 015101(R) (2001)] to determine rho(a) for a two-dimensional random walk and the one-dimensional random-acceleration problem. We also consider "alternating persistence," which corresponds to a<0, and calculate rho(a) for this case.
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Affiliation(s)
- George C M A Ehrhardt
- Department of Physics and Astronomy, University of Manchester, Manchester, M13 9PL, United Kingdom
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Newman TJ, Loinaz W. Critical dimensions of the diffusion equation. PHYSICAL REVIEW LETTERS 2001; 86:2712-2715. [PMID: 11290021 DOI: 10.1103/physrevlett.86.2712] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/27/2000] [Indexed: 05/23/2023]
Abstract
We study the evolution of a random initial field under pure diffusion in various space dimensions. From numerical calculations we find that the persistence properties of the system show sharp transitions at critical dimensions d(1) approximately 26 and d(2) approximately 46. We also give refined measurements of the persistence exponents for low dimensions.
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Affiliation(s)
- T J Newman
- Department of Physics, University of Virginia, Charlottesville, VA 22904, USA.
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Smedt GD, Godrèche C, Luck JM. Statistics of the occupation time for a class of Gaussian Markov processes. ACTA ACUST UNITED AC 2001. [DOI: 10.1088/0305-4470/34/7/303] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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