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De Bruyne B, Majumdar SN, Schehr G. Generating discrete-time constrained random walks and Lévy flights. Phys Rev E 2021; 104:024117. [PMID: 34525540 DOI: 10.1103/physreve.104.024117] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/19/2021] [Accepted: 07/27/2021] [Indexed: 11/07/2022]
Abstract
We introduce a method to exactly generate bridge trajectories for discrete-time random walks, with arbitrary jump distributions, that are constrained to initially start at the origin and return to the origin after a fixed time. The method is based on an effective jump distribution that implicitly accounts for the bridge constraint. It is illustrated on various jump distributions and is shown to be very efficient in practice. In addition, we show how to generalize the method to other types of constrained random walks such as generalized bridges, excursions, and meanders.
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Affiliation(s)
- Benjamin De Bruyne
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Satya N Majumdar
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Grégory Schehr
- Sorbonne Université, Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589, 4 Place Jussieu, 75252 Paris Cedex 05, France
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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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Belardinelli D, Sbragaglia M, Gross M, Andreotti B. Thermal fluctuations of an interface near a contact line. Phys Rev E 2016; 94:052803. [PMID: 27967049 DOI: 10.1103/physreve.94.052803] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/03/2016] [Indexed: 11/07/2022]
Abstract
The effect of thermal fluctuations near a contact line of a liquid interface partially wetting an impenetrable substrate is studied analytically and numerically. Promoting both the interface profile and the contact line position to random variables, we explore the equilibrium properties of the corresponding fluctuating contact line problem based on an interfacial Hamiltonian involving a "contact" binding potential. To facilitate an analytical treatment, we consider the case of a one-dimensional interface. The effective boundary condition at the contact line is determined by a dimensionless parameter that encodes the relative importance of thermal energy and substrate energy at the microscopic scale. We find that this parameter controls the transition from a partial wetting to a pseudopartial wetting state, the latter being characterized by a thin prewetting film of fixed thickness. In the partial wetting regime, instead, the profile typically approaches the substrate via an exponentially thinning prewetting film. We show that, independently of the physics at the microscopic scale, Young's angle is recovered sufficiently far from the substrate. The fluctuations of the interface and of the contact line give rise to an effective disjoining pressure, exponentially decreasing with height. Fluctuations therefore provide a regularization of the singular contact forces occurring in the corresponding deterministic problem.
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Affiliation(s)
- D Belardinelli
- Department of Physics & INFN, University of Rome "Tor Vergata", Via della Ricerca Scientifica 1, 00133, Rome, Italy
| | - M Sbragaglia
- Department of Physics & INFN, University of Rome "Tor Vergata", Via della Ricerca Scientifica 1, 00133, Rome, Italy
| | - M Gross
- Max-Planck-Institut für Intelligente Systeme, Heisenbergstr. 3, 70569 Stuttgart, Germany and IV. Institut für Theoretische Physik, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
| | - B Andreotti
- Physique et Mécanique des Milieux Hétérogènes, UMR 7636 ESPCI-CNRS, Université Paris-Diderot, 10 rue Vauquelin, 75005, Paris, France
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Zamorategui AL, Lecomte V, Kolton AB. Distribution of zeros in the rough geometry of fluctuating interfaces. Phys Rev E 2016; 93:042118. [PMID: 27176265 DOI: 10.1103/physreve.93.042118] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/11/2015] [Indexed: 06/05/2023]
Abstract
We study numerically the correlations and the distribution of intervals between successive zeros in the fluctuating geometry of stochastic interfaces, described by the Edwards-Wilkinson equation. For equilibrium states we find that the distribution of interval lengths satisfies a truncated Sparre-Andersen theorem. We show that boundary-dependent finite-size effects induce nontrivial correlations, implying that the independent interval property is not exactly satisfied in finite systems. For out-of-equilibrium nonstationary states we derive the scaling law describing the temporal evolution of the density of zeros starting from an uncorrelated initial condition. As a by-product we derive a general criterion of the von Neumann's type to understand how discretization affects the stability of the numerical integration of stochastic interfaces. We consider both diffusive and spatially fractional dynamics. Our results provide an alternative experimental method for extracting universal information of fluctuating interfaces such as domain walls in thin ferromagnets or ferroelectrics, based exclusively on the detection of crossing points.
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Affiliation(s)
- Arturo L Zamorategui
- Laboratoire Probabilités et Modèles Aléatoires (UMR CNRS 7599), Université Pierre et Marie Curie and Université Paris Diderot, 75013 Paris, France
| | - Vivien Lecomte
- Laboratoire Probabilités et Modèles Aléatoires (UMR CNRS 7599), Université Pierre et Marie Curie and Université Paris Diderot, 75013 Paris, France
| | - Alejandro B Kolton
- CONICET-Centro Atómico Bariloche and Instituto Balseiro (UNCu), 8400 S.C. de Bariloche, Argentina
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Sire C. Crossing intervals of non-Markovian Gaussian processes. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:011121. [PMID: 18763933 DOI: 10.1103/physreve.78.011121] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/13/2008] [Indexed: 05/26/2023]
Abstract
We review the properties of time intervals between the crossings at a level M of a smooth stationary Gaussian temporal signal. The distribution of these intervals and the persistence are derived within the independent interval approximation (IIA). These results grant access to the distribution of extrema of a general Gaussian process. Exact results are obtained for the persistence exponents and the crossing interval distributions, in the limit of large |M|. In addition, the small-time behavior of the interval distributions and the persistence is calculated analytically, for any M. The IIA is found to reproduce most of these exact results, and its accuracy is also illustrated by extensive numerical simulations applied to non-Markovian Gaussian processes appearing in various physical contexts.
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Affiliation(s)
- Clément Sire
- Laboratoire de Physique Théorique--IRSAMC, CNRS, Université Paul Sabatier, 31062 Toulouse, France.
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Tao CG, Cullen WG, Williams ED, Dasgupta C. Generalized survival in step fluctuations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:021601. [PMID: 17930045 DOI: 10.1103/physreve.76.021601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/11/2006] [Indexed: 05/25/2023]
Abstract
The properties of the generalized survival probability, that is, the probability of not crossing an arbitrary location R during relaxation, have been investigated experimentally (via scanning tunneling microscope observations) and numerically. The results confirm that the generalized survival probability decays exponentially with a time constant tau(s) (R). The distance dependence of the time constant is shown to be tau(s) (R) = tau(s0) exp[-R/w (T)], where w2 (T) is the material-dependent mean-squared width of the step fluctuations. The result reveals the dependence on the physical parameters of the system inherent in the prior prediction of the time constant scaling with R/L(alpha), with L the system size and alpha the roughness exponent. The survival behavior is also analyzed using a contrasting concept, the generalized inside survival S(in) (t,R), which involves fluctuations to an arbitrary location R further from the average. Numerical simulations of the inside survival probability also show an exponential time dependence, and the extracted time constant empirically shows (R/w)(lambda) behavior, with lambda varying over 0.6 to 0.8 as the sampling conditions are changed. The experimental data show similar behavior, and can be well fit with lambda = 1.0 for T = 300 K, and 0.5 < lambda < 1 for T = 460 K. Over this temperature range, the ratio of the fixed sampling time to the underlying physical time constant, and thus the true correlation time, increases by a factor of approximately 10(3). Preliminary analysis indicates that the scaling effect due to the true correlation time is relevant in the parameter space of the experimental observations.
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Affiliation(s)
- C G Tao
- Department of Physics and Materials Research Science and Engineering Center, University of Maryland, College Park, Maryland 20742-4111, USA
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Mukherjee B, Maiti PK, Dasgupta C, Sood AK. Strong correlations and Fickian water diffusion in narrow carbon nanotubes. J Chem Phys 2007; 126:124704. [PMID: 17411149 DOI: 10.1063/1.2565806] [Citation(s) in RCA: 73] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/27/2023] Open
Abstract
The authors have used atomistic molecular dynamics (MD) simulations to study the structure and dynamics of water molecules inside an open ended carbon nanotube placed in a bath of water molecules. The size of the nanotube allows only a single file of water molecules inside the nanotube. The water molecules inside the nanotube show solidlike ordering at room temperature, which they quantify by calculating the pair correlation function. It is shown that even for the longest observation times, the mode of diffusion of the water molecules inside the nanotube is Fickian and not subdiffusive. They also propose a one-dimensional random walk model for the diffusion of the water molecules inside the nanotube. They find good agreement between the mean-square displacements calculated from the random walk model and from MD simulations, thereby confirming that the water molecules undergo normal mode diffusion inside the nanotube. They attribute this behavior to strong positional correlations that cause all the water molecules inside the nanotube to move collectively as a single object. The average residence time of the water molecules inside the nanotube is shown to scale quadratically with the nanotube length.
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Affiliation(s)
- Biswaroop Mukherjee
- Centre for Condensed Matter Theory, Department of Physics, Indian Institute of Science, Bangalore, India.
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Conrad BR, Cullen WG, Dougherty DB, Lyubinetsky I, Williams ED. Spatial first-passage statistics of Al/Si(111)-(square root 3 x square root 3) step fluctuations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:021603. [PMID: 17358349 DOI: 10.1103/physreve.75.021603] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/12/2006] [Revised: 12/11/2006] [Indexed: 05/14/2023]
Abstract
Spatial step edge fluctuations on a multicomponent surface of Al/Si(111)-(square root 3 x square root 3) were measured via scanning tunneling microscopy over a temperature range of 720-1070 K, for step lengths of L=65-160 nm. Even though the time scale of fluctuations of steps on this surface varies by orders of magnitude over the indicated temperature range, measured first-passage spatial persistence and survival probabilities are temperature independent. The power law functional form for spatial persistence probabilities is confirmed and the symmetric spatial persistence exponent is measured to be theta=0.498+/-0.062 in agreement with the theoretical prediction theta=1/2. The survival probability is found to scale directly with y/L, where y is the distance along the step edge. The form of the survival probabilities agrees quantitatively with the theoretical prediction, which yields exponential decay in the limit of small y/L. The decay constant is found experimentally to be y(s)/L=0.076+/-0.033 for y/L<or=0.2.
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Affiliation(s)
- B R Conrad
- Department of Physics and MRSEC, University of Maryland, College Park, Maryland 20742-4111, USA
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Sire C. Probability distribution of the maximum of a smooth temporal signal. PHYSICAL REVIEW LETTERS 2007; 98:020601. [PMID: 17358591 DOI: 10.1103/physrevlett.98.020601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/06/2006] [Indexed: 05/14/2023]
Abstract
We present an approximate calculation for the distribution of the maximum of a smooth stationary temporal signal X(t). As an application, we compute the persistence exponent associated with the probability that the process remains below a nonzero level M. When X(t) is a Gaussian process, our results are expressed explicitly in terms of the two-time correlation function, f(t)=X(0)X(t).
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Affiliation(s)
- Clément Sire
- Laboratoire de Physique Théorique, UMR 5152 du CNRS, Université Paul Sabatier, Toulouse Cedex 4, France
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Schehr G, Majumdar SN. Universal asymptotic statistics of maximal relative height in one-dimensional solid-on-solid models. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:056103. [PMID: 16802994 DOI: 10.1103/physreve.73.056103] [Citation(s) in RCA: 23] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/16/2006] [Indexed: 05/10/2023]
Abstract
We study the probability density function P(h(m), L) of the maximum relative height h(m) in a wide class of one-dimensional solid-on-solid models of finite size L. For all these lattice models, in the large-L limit, a central limit argument shows that, for periodic boundary conditions, P(h(m), L) takes a universal scaling form P(h(m), L) approximately radical(12w(L))(-1) f(h(m)radical(12w(L))(-1), with w(L) the width of the fluctuating interface f(x) and the Airy distribution function. For one instance of these models, corresponding to the extremely anisotropic Ising model in two dimensions, this result is obtained by an exact computation using the transfer matrix technique, valid for any L > 0. These arguments and exact analytical calculations are supported by numerical simulations, which show in addition that the subleading scaling function is also universal, up to a nonuniversal amplitude, and simply given by the derivative of the Airy distribution function f'(x).
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Affiliation(s)
- Grégory Schehr
- Theoretische Physik, Universität des Saarlandes, Saarbrücken, Germany
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