Occurrence of normal and anomalous diffusion in polygonal billiard channels

Sanders, David P.; Larralde, Hernán

doi:10.1103/physreve.73.026205

Cited by 43 publications

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“…Higher moments for the finitehorizon Lorentz gas were studied in [30]. The numerical calculation of higher moments is difficult, due to the weak effect of free flights [26]. Nonetheless, by taking means over a very large number of initial conditions, it is possible to see the effect of the different types of gaps for our 3D Lorentz gas model: as shown in fig.…”

Section: C(s)ds So That D(t) Converges To the Diffusion Coefficientmentioning

confidence: 99%

“…Normal diffusion corresponds to an asymptotically flat graph, since the logarithmic correction is absent, and the diffusion coefficient is then proportional to the asymptotic height of the graph. Weak superdiffusive t logt behavior for the mean-squared displacement, on the other hand, gives asymptotic linear growth [26].…”

Section: C(s)ds So That D(t) Converges To the Diffusion Coefficientmentioning

confidence: 99%

“…Numerical results:-To test the above hypotheses, we perform careful numerical simulations of our model to calculate the time-evolution of the mean-squared displacement r(t) 2 in each regime. We use a stringent test to distinguish normal from weakly anomalous diffusion: r(t) 2 /t is plotted as a function of logt [22,26,27]. Normal diffusion corresponds to an asymptotically flat graph, since the logarithmic correction is absent, and the diffusion coefficient is then proportional to the asymptotic height of the graph.…”

Section: C(s)ds So That D(t) Converges To the Diffusion Coefficientmentioning

confidence: 99%

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Normal diffusion in crystal structures and higher-dimensional billiard models with gaps

Sanders

2008

Phys. Rev. E

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We show, both heuristically and numerically, that three-dimensional periodic Lorentz gases-clouds of particles scattering off crystalline arrays of hard spheres-often exhibit normal diffusion, even when there are gaps through which particles can travel without ever colliding, i.e., when the system has an infinite horizon. This is the case provided that these gaps are not "too big", as measured by their dimension. The results are illustrated with simulations of a simple three-dimensional model having different types of diffusive regime, and are then extended to higher-dimensional billiard models, which include hard-sphere fluids.PACS numbers: 05.60. Cd, 05.45.Jn, 05.45.Pq, 66.10.cg The Lorentz gas is a classical model of transport processes, in which a cloud of non-interacting point particles (modelling electrons) undergo free motion between elastic collisions with fixed hard spheres (atoms) [1]. It has been much studied as a model system for which the programme of statistical physics can be carried out in detail: to relate the known microscopic dynamics to the macroscopic behavior of the system, which in this case is diffusive [2][3][4].When the scatterers are arranged in a periodic crystal structure, the dynamics of this billiard model can be reduced to a single unit cell [2]. The curved shape of the scatterers implies that nearby trajectories separate exponentially fast, so that the system is hyperbolic (chaotic) and ergodic [5].In two dimensions, it has been shown that the cloud of particles in the periodic Lorentz gas undergoes normal diffusion, provided that the geometrical finite horizon condition is satisfied: particles cannot travel arbitrarily far without colliding with a scatterer [5,6]. By normal diffusion, we mean that the distribution of particle positions behaves like solutions of the diffusion equation; in particular, the mean-squared displacement (variance) grows asymptotically linearly in time:r(t) 2 ∼ 2dDt when t → ∞, where r(t) is the displacement of a particle at time t from its initial position, d is the spatial dimension, · denotes a mean over initial conditions, and the diffusion coefficient D gives the asymptotic growth rate. When the horizon is infinite, however, particles can undergo arbitrarily long free flights along corridors in the structure. It was long argued [7][8][9] and has recently been proved [10], that there is then weak superdiffusive behavior, with r(t) 2 ∼ t logt, so that the diffusion coefficient no longer exists.For higher-dimensional periodic Lorentz gases, rigorous results on ergodic properties [11] and diffusive properties [12] have been obtained; recent progress in their analysis has been made [13,14], including in the limit of small scatterers [15]. In particular, higher-dimensional Lorentz gases are believed to exhibit normal diffusion when the horizon is finite [12].Nonetheless, the study of billiard models in higher dimensions, especially three dimensions, has received surprisingly little attention from the physics community, despite their interest as simple ...

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Section: C(s)ds So That D(t) Converges To the Diffusion Coefficientmentioning

confidence: 99%

Section: C(s)ds So That D(t) Converges To the Diffusion Coefficientmentioning

confidence: 99%

Section: C(s)ds So That D(t) Converges To the Diffusion Coefficientmentioning

confidence: 99%

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Normal diffusion in crystal structures and higher-dimensional billiard models with gaps

Sanders

2008

Phys. Rev. E

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“…Therefore, it seems a natural way to adopt CTRW to study multifractal types of random walk. This can be based on the well-known characterization of the stochastic phenomena using a spectrum of fractional moments, both temporal and spatial [28,[93][94][95][96][97][98][99][100][101]. The scaling of the fractional moments of the Lévy walk has a characteristic fractal, bifractal, and multifractal behaviour.…”

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confidence: 99%

The continuous time random walk, still trendy: fifty-year history, state of art and outlook

2017

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Abstract.In this article we demonstrate the very inspiring role of the continuous-time random walk (CTRW) formalism, the numerous modifications permitted by its flexibility, its various applications, and the promising perspectives in the various fields of knowledge. A short review of significant achievements and possibilities is given. However, this review is still far from completeness. We focused on a pivotal role of CTRWs mainly in anomalous stochastic processes discovered in physics and beyond. This article plays the role of an extended announcement of the Eur. Inspiring properties and achievementsIn their pioneering work published in year 1965 [1], physicists Eliott W. Montroll and George H. Weiss introduced the concept of continuous-time random walk (CTRW) as a way to make the interevent-time continuous and fluctuating. It is characterized by some distribution associated with a stochastic process, giving an insight into the process activity. This distribution, called pausing-or waiting-time one (WTD), permitted the description of both Debye (exponential) and, what is most significant, non-Debye (slowly-decaying) relaxations as well as normal and anomalous transport and diffusion [2,3] -thus the model involves fundamental aspects of the stochastic world -a real, complex world. Notably, ancestors of this concept are presented by Michael Shlesinger in [4].Let us incidentally comment that term "walk" in the name "continuous-time random walk" is commonly used in the generic sense comprising two concepts: namely, both the walk (associated with finite displacement velocity of the process) and flight (associated with an instantaneous displacement of the process). Thus we have to specify in a detailed way with what kind of process we are considering.The CTRW formalism was most conveniently developed by physicists Scher and Lax in terms of recursion relations [5][6][7][8][9][10]. In this context the distinction between Contribution to the Topical Issue "Continuous Time Random Walk Still Trendy: Fifty-year History, Current State and Outlook", edited by Ryszard Kutner and Jaume Masoliver. a e-mail: ryszard.kutner@fuw.edu.pl discrete and continuous times [11] and also between separable and non-separable WTDs were introduced [12]. A thorough analysis of the latter, called also the nonindependent CTRW, was performed a decade ago [13] although, in the context of the concentrated lattice gases, it was performed much earlier [14,15]. These analyses took into account dependences over many correlated consecutive particle displacements and waiting (or interevent) times. The Scher and Lax formulation of the CTRW formalism is particularly convenient to study as well anomalous transport and diffusion as the non-Debye relaxation and their anomalous scaling properties (e.g., the nonlinear time growth of the process variance). Examples are the span of the walk, the first-passage times, survival probabilities, the number of distinct sites visited and, of course, mean and mean-square displacement if they exist. It is very interesting...

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“…Muitos sistemas foram arduamente estudados com o intuito de observar o fenômeno de aprisionamento e uma possível conexão com o transporte. Os mais frequentes são mapas com preservação deárea [12], embora algo tenha sido feito considerando dissipações [13], e bilhares com plano 2 [14] ou horizonte infinito 3 [15,16]. Alguns destes sistemas conduzem ao transporte anômalo (TA) istoé, ao crescimento não linear do desvio quadrático médio 2 Como um exemplo mais conhecido temos o bilhar de Sinai extendido por todo o plano conhecido como gás de Lorentz.…”

Section: Capítulo 1 Introduçãounclassified

Propriedades de tranporte, caos e dissipação num sistema dinâmico não linear

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Occurrence of normal and anomalous diffusion in polygonal billiard channels

Cited by 43 publications

References 31 publications

Normal diffusion in crystal structures and higher-dimensional billiard models with gaps

Normal diffusion in crystal structures and higher-dimensional billiard models with gaps

The continuous time random walk, still trendy: fifty-year history, state of art and outlook

Propriedades de tranporte, caos e dissipação num sistema dinâmico não linear

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