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Zamorategui AL, Lecomte V, Kolton AB. Statistics of zero crossings in rough interfaces with fractional elasticity. Phys Rev E 2018; 97:042129. [PMID: 29758659 DOI: 10.1103/physreve.97.042129] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/29/2017] [Indexed: 06/08/2023]
Abstract
We study numerically the distribution of zero crossings in one-dimensional elastic interfaces described by an overdamped Langevin dynamics with periodic boundary conditions. We model the elastic forces with a Riesz-Feller fractional Laplacian of order z=1+2ζ, such that the interfaces spontaneously relax, with a dynamical exponent z, to a self-affine geometry with roughness exponent ζ. By continuously increasing from ζ=-1/2 (macroscopically flat interface described by independent Ornstein-Uhlenbeck processes [Phys. Rev. 36, 823 (1930)PHRVAO0031-899X10.1103/PhysRev.36.823]) to ζ=3/2 (super-rough Mullins-Herring interface), three different regimes are identified: (I) -1/2<ζ<0, (II) 0<ζ<1, and (III) 1<ζ<3/2. Starting from a flat initial condition, the mean number of zeros of the discretized interface (I) decays exponentially in time and reaches an extensive value in the system size, or decays as a power-law towards (II) a subextensive or (III) an intensive value. In the steady state, the distribution of intervals between zeros changes from an exponential decay in (I) to a power-law decay P(ℓ)∼ℓ^{-γ} in (II) and (III). While in (II) γ=1-θ with θ=1-ζ the steady-state persistence exponent, in (III) we obtain γ=3-2ζ, different from the exponent γ=1 expected from the prediction θ=0 for infinite super-rough interfaces with ζ>1. The effect on P(ℓ) of short-scale smoothening is also analyzed numerically and analytically. A tight relation between the mean interval, the mean width of the interface, and the density of zeros is also reported. The results drawn from our analysis of rough interfaces subject to particular boundary conditions or constraints, along with discretization effects, are relevant for the practical analysis of zeros in interface imaging experiments or in numerical analysis.
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Affiliation(s)
- Arturo L Zamorategui
- Laboratoire de Probabilités, Statistique et Modélisation (LPSM, UMR 8001), Université Pierre et Marie Curie and Université Paris Diderot, 75013 Paris, France
| | - Vivien Lecomte
- Université Grenoble Alpes, CNRS, LIPhy, 38000 Grenoble, France
| | - Alejandro B Kolton
- CONICET and Instituto Balseiro (UNCu), Centro Atómico Bariloche, 8400 S.C. de Bariloche, Argentina
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Nyberg M, Lizana L. Persistence of non-Markovian Gaussian stationary processes in discrete time. Phys Rev E 2018; 97:040101. [PMID: 29758684 DOI: 10.1103/physreve.97.040101] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/26/2018] [Indexed: 06/08/2023]
Abstract
The persistence of a stochastic variable is the probability that it does not cross a given level during a fixed time interval. Although persistence is a simple concept to understand, it is in general hard to calculate. Here we consider zero mean Gaussian stationary processes in discrete time n. Few results are known for the persistence P_{0}(n) in discrete time, except the large time behavior which is characterized by the nontrivial constant θ through P_{0}(n)∼θ^{n}. Using a modified version of the independent interval approximation (IIA) that we developed before, we are able to calculate P_{0}(n) analytically in z-transform space in terms of the autocorrelation function A(n). If A(n)→0 as n→∞, we extract θ numerically, while if A(n)=0, for finite n>N, we find θ exactly (within the IIA). We apply our results to three special cases: the nearest-neighbor-correlated "first order moving average process", where A(n)=0 for n>1, the double exponential-correlated "second order autoregressive process", where A(n)=c_{1}λ_{1}^{n}+c_{2}λ_{2}^{n}, and power-law-correlated variables, where A(n)∼n^{-μ}. Apart from the power-law case when μ<5, we find excellent agreement with simulations.
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Affiliation(s)
- Markus Nyberg
- Integrated Science Lab, Department of Physics, Umeå University, SE-901 87 Umeå, Sweden
| | - Ludvig Lizana
- Integrated Science Lab, Department of Physics, Umeå University, SE-901 87 Umeå, Sweden
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Mercadier C. Numerical bounds for the distributions of the maxima of some one- and two-parameter Gaussian processes. ADV APPL PROBAB 2016. [DOI: 10.1239/aap/1143936145] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
We consider the class of real-valued stochastic processes indexed on a compact subset of R or R2 with almost surely absolutely continuous sample paths. We obtain an implicit formula for the distributions of their maxima. The main result is the derivation of numerical bounds that turn out to be very accurate, in the Gaussian case, for levels that are not large. We also present the first explicit upper bound for the distribution tail of the maximum in the two-dimensional Gaussian framework. Numerical comparisons are performed with known tools such as the Rice upper bound and expansions based on the Euler characteristic. We deal numerically with the determination of the persistence exponent.
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Sen P, Ray P. A+A→∅ model with a bias towards nearest neighbor. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:012109. [PMID: 26274127 DOI: 10.1103/physreve.92.012109] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/19/2015] [Indexed: 06/04/2023]
Abstract
We have studied the A+A→∅ reaction-diffusion model on a ring, with a bias ε(0≤ε≤0.5) of the random walkers A to hop towards their nearest neighbor. Though the bias is local in space and time, we show that it alters the universality class of the problem. The z exponent, which describes the growth of average spacings between the walkers with time, changes from the value 2 at ε=0 to the mean-field value of unity for any nonzero ε. We study the problem analytically using independent interval approximation and compare the scaling results with those obtained from simulation. The distribution P(k,t) (per site) of the spacing between two walkers is given by t(-2/z)f(k/t(1/z)) and is obtained both analytically and numerically. We also obtain the result that εt becomes the new time scale for ε≠0.
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Affiliation(s)
- Parongama Sen
- Department of Physics, University of Calcutta, 92 Acharya Prafulla Chandra Road, Kolkata 700009, India
| | - Purusattam Ray
- The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, 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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Majumdar SN, Dasgupta C. Spatial survival probability for one-dimensional fluctuating interfaces in the steady state. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:011602. [PMID: 16486156 DOI: 10.1103/physreve.73.011602] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/31/2005] [Revised: 10/24/2005] [Indexed: 05/06/2023]
Abstract
We report numerical and analytic results for the spatial survival probability for fluctuating one-dimensional interfaces with Edwards-Wilkinson or Kardar-Parisi-Zhang dynamics in the steady state. Our numerical results are obtained from analysis of steady-state profiles generated by integrating a spatially discretized form of the Edwards-Wilkinson equation to long times. We show that the survival probability exhibits scaling behavior in its dependence on the system size and the "sampling interval" used in the measurement for both "steady-state" and "finite" initial conditions. Analytic results for the scaling functions are obtained from a path-integral treatment of a formulation of the problem in terms of one-dimensional Brownian motion. A "deterministic approximation" is used to obtain closed-form expressions for survival probabilities from the formally exact analytic treatment. The resulting approximate analytic results provide a fairly good description of the numerical data.
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Affiliation(s)
- Satya N Majumdar
- Laboratoire de Physique Theorique et Modeles Statistiques, Universite Paris-Sud, Bat. 100, 91405 ORSAY cedex, France
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Dougherty DB, Tao C, Bondarchuk O, Cullen WG, Williams ED, Constantin M, Dasgupta C, Das Sarma S. Sampling-time effects for persistence and survival in step structural fluctuations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:021602. [PMID: 15783332 DOI: 10.1103/physreve.71.021602] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/25/2004] [Indexed: 05/24/2023]
Abstract
The effects of sampling rate and total measurement time have been determined for single-point measurements of step fluctuations within the context of first-passage properties. Time dependent scanning tunneling microscopy has been used to evaluate step fluctuations on Ag(111) films grown on mica as a function of temperature (300-410 K) , on screw dislocations on the facets of Pb crystallites at 320 K , and on Al-terminated Si(111) over the temperature range 770-970 K . Although the fundamental time constant for step fluctuations on Ag and Al/Si varies by orders of magnitude over the temperature ranges of measurement, no dependence of the persistence amplitude on temperature is observed. Instead, the persistence probability is found to scale directly with t/delta t where delta t is the time interval used for sampling. Survival probabilities show a more complex scaling dependence, which includes both the sampling interval and the total measurement time t(m) . Scaling with t/delta t occurs only when delta t/ t(m) is a constant. We show that this observation is equivalent to theoretical predictions that the survival probability will scale as delta t/ L(z) , where L is the effective length of a step. This implies that the survival probability for large systems, when measured with fixed values of t(m) or delta t , should also show little or no temperature dependence.
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Affiliation(s)
- D B Dougherty
- Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA
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Constantin M, Sarma SD. Generalized survival in equilibrium step fluctuations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:052601. [PMID: 15244864 DOI: 10.1103/physreve.69.052601] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/23/2003] [Indexed: 05/24/2023]
Abstract
We investigate the dynamics of a generalized survival probability S(t,R) defined with respect to an arbitrary reference level R (rather than the average) in equilibrium step fluctuations. The exponential decay at large time scales of the generalized survival probability is numerically analyzed. S(t,R) is shown to exhibit simple scaling behavior as a function of system size L, sampling time deltat, and the reference level R. The generalized survival time scale tau(s) (R) associated with S(t,R) is shown to decay exponentially as a function of R.
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Affiliation(s)
- M Constantin
- Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA
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Dasgupta C, Constantin M, Das Sarma S, Majumdar SN. Survival in equilibrium step fluctuations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:022101. [PMID: 14995501 DOI: 10.1103/physreve.69.022101] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/03/2003] [Indexed: 05/24/2023]
Abstract
We report the results of analytic and numerical investigations of the time scale of survival or non-zero-crossing probability S(t) in equilibrium step fluctuations described by Langevin equations appropriate for attachment/detachment and edge-diffusion limited kinetics. An exact relation between long-time behaviors of the survival probability and the autocorrelation function is established and numerically verified. S(t) is shown to exhibit a simple scaling behavior as a function of system size and sampling time. Our theoretical results are in agreement with those obtained from an analysis of experimental dynamical scanning tunneling microscopy data on step fluctuations on Al/Si(111) and Ag(111) surfaces.
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Affiliation(s)
- C Dasgupta
- Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA
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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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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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