Distribution of zeros in the rough geometry of fluctuating interfaces

Zamorategui, Arturo L.; Lecomte, Vivien; Kolton, Alejandro B.

doi:10.1103/physreve.93.042118

Cited by 4 publications

(11 citation statements)

References 38 publications

(88 reference statements)

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“…The ICD was previously found, for example, for different models of Lévy walks [15,39]. Since dual scaling of the moments and fat tailed distributions are very common, we speculate that ICDs will describe a large class of systems, e.g., Lévy glasses [40], fluctuating surfaces [41], motion of tracer particles in the cell [42] and diffusion on lipid bi-layers [43]. To identify the ICDs in these diverse systems requires further work.…”

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confidence: 69%

Large Fluctuations for Spatial Diffusion of Cold Atoms

2017

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Large-deviations theory deals with tails of probability distributions and the rare events of random processes, for example spreading packets of particles. Mathematically, it concerns the exponential fall-of of the density of thin-tailed systems. Here we investigate the spatial density Pt(x) of laser cooled atoms, where at intermediate length scales the shape is fat-tailed. We focus on the rare events beyond this range, which dominate important statistical properties of the system. Through a novel friction mechanism induced by the laser fields, the density is explored with the recently proposed nonnormalized infinite-covariant density approach. The small and large fluctuations give rise to a bi-fractal nature of the spreading packet.PACS numbers: 05.40. Jc,46.65.+g In diffusion processes such as Brownian motion, the concentration of particles starting at the origin spreads out like a Gaussian, which is fully characterized by the mean squared-displacement. This is the result of the widely applicable Gaussian central limit theorem (CLT) [1]. Of no less importance is large-deviations theory [2], which deals with the rare fluctuations of processes such as simple coin tossing random walks (see e.g., [3]), extreme variations of the surface height in the KardarParisi-Zhang model [4] and the tails of the position distribution in single-file diffusion [5,6]. Mathematically, a prerequisite of the theory is that the cumulant generating function be "well behaved", i.e. smooth and differentiable. Large-deviations theory works when the decay of the probability of the observable of interest is exponential (see details in [2]). However many systems do not meet this requirement [2], for example Lévy fat-tailed processes [7][8][9], where the decay rate is a power-law. This is the case for a cloud of atoms undergoing Sisyphus laser-cooling [10], where both theoretically [11,12] and experimentally [13], it was shown that the central part of the spreading particle packet is described by the Lévy CLT [14]. The latter deals with the sum of independent identically-distributed random variables, but unlike the classical Gaussian CLT, here the summands' own distribution is heavy-tailed. As a result, Lévy's CLT yields an infinite mean squared-displacement for the sum [14], and consequently also the second cumulant. Largedeviations theory mainly deals with thin-tailed processes where extreme events are rare, but in Lévy processes these large fluctuations are dominant. To study the fluctuations in this system, we will show that the relevant tool is the asymptotic moment-generating function, which yields an infinite-covariant density (ICD) [12,15]. We will discuss the generality of this approach and its results below.In an experimental situation, diverging moments are unphysical. For example, although the experiment in [13] shows a nice fit of the particles' density to a symmetric Lévy distribution, clearly at finite times no particles traveling at finite velocities can ever be found infinitely far from their origin. The finiteness of all t...

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confidence: 69%

Large Fluctuations for Spatial Diffusion of Cold Atoms

2017

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“…[29] Analytical calculations for the survival probability of the EW interface have indeed shown the importance of the zero-area constraint in finite interfaces [30]. More recently, we have shown that the same model displays subtle correlations effects between intervals and long-range correlations between increments [29]. Thus, finite-size effects are expected to become even more important for fractional dynamics, specially for large values of ζ where excursions get even more constrained by the zero-area condition.…”

Section: Introductionmentioning

confidence: 78%

“…This prompts the question of why the same constraint does not equally affect the γ exponent in the rough interface for regime II. At this respect we also note that even for ζ = 1/2 the zero-area constraint is responsible for the breakdown of the independent interval approximation (a modified version of the Sparre-Andersen theorem is needed for a correct description of P ( ) as studied in [29]).…”

Section: Discussionmentioning

confidence: 94%

“…The position of a zero of the discretized interface is given by the immediate integer to the left of the crossing point at which the interface height changes its sign. This definition was shown [29] to describe correctly the firstpassage distribution of the interface in the case ζ = 1/2. In the steady-state, we will be interested in the distribution P ( ) of the length of the intervals between consecutive zeros, and also in the total number of zeros n and its distribution P (n).…”

Section: Model and Methodsmentioning

confidence: 90%

“…x /L which are global random variables of each configuration. In [29], we studied in detail the one dimensional case with ζ = 1/2 corresponding to the one-dimensional Edwards-Wilkinson equation. We showed that a truncated form of the Sparre-Andersen theorem [36], which remains valid for describing the interval distribution, is found to be P ( ) ∼ −3/2 below a size dependent cut-off.…”

Section: A Interval Distributionsmentioning

confidence: 99%

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Statistics of zero crossings in rough interfaces with fractional elasticity

2018

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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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Persistence probabilities of height fluctuation in thin film growth of the Das Sarma–Tamborenea model

Chanphana

Chatraphorn

2020

Indian J Phys

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Distribution of zeros in the rough geometry of fluctuating interfaces

Cited by 4 publications

References 38 publications

Large Fluctuations for Spatial Diffusion of Cold Atoms

Large Fluctuations for Spatial Diffusion of Cold Atoms

Statistics of zero crossings in rough interfaces with fractional elasticity

Persistence probabilities of height fluctuation in thin film growth of the Das Sarma–Tamborenea model

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