Measuring logarithmic corrections to normal diffusion in infinite-horizon billiards

Cristadoro, Giampaolo; Gilbert, Thomas; Lenci, Marco; Sanders, David P.

doi:10.1103/physreve.90.022106

Cited by 28 publications

(40 citation statements)

References 49 publications

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“…The position of the moving particle q(t) at time t satisfies a non-classical limit theorem with a √ t log t scaling factor (this is proved in [13]; see also [38]). Regarding the convergence of the second moment, a doubling effect analogous to our Theorem 4 has been observed and studied in [22]; for further discussion and numerical evidence see also [19,23,20]. However, this model is quite different from dispersing billiards with cusps and requires a different approach.…”

Section: Belowmentioning

confidence: 61%

Convergence of moments for dispersing billiards with cusps

Bálint¹,

Chernov²,

Dolgopyat³

2017

Contemporary Mathematics

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Dispersing billiards with cusps are deterministic dynamical systems with a mild degree of chaos, exhibiting "intermittent" behavior that alternates between regular and chaotic patterns. They are characterized by decay of correlations of order 1/n and a central limit theorem with a non-classical scaling factor of √ n log n. As for the growth of the pth moments of the appropriately normalized Birkhoff sums, it follows from the results of [28] that these converge to the moments of the limit normal distribution only for p < 2 and diverge for p > 2. Here we focus on the critical case p = 2 and prove a doubling effect: the second moments converge, but their limit is twice the second moment of the limit normal distribution.

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Section: Belowmentioning

confidence: 61%

Convergence of moments for dispersing billiards with cusps

Bálint¹,

Chernov²,

Dolgopyat³

2017

Contemporary Mathematics

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“…To proceed, we substitute equation (28) into equation (33) and let t ≡ kτ B . Having dropped all terms of order 2 and higher, we obtain…”

Section: Small-parameter Expansionmentioning

confidence: 99%

Lévy walks on lattices as multi-state processes

Cristadoro¹,

Gilbert²,

Lenci³

et al. 2015

J. Stat. Mech.

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Continuous-time random walks combining diffusive scattering and ballistic propagation on lattices model a class of Lévy walks. The assumption that transitions in the scattering phase occur with exponentially-distributed waiting times leads to a description of the process in terms of multiple states, whose distributions evolve according to a set of delay differential equations, amenable to analytic treatment. We obtain an exact expression of the mean squared displacement associated with such processes and discuss the emergence of asymptotic scaling laws in regimes of diffusive and superdiffusive (subballistic) transport, emphasizing, in the latter case, the effect of initial conditions on the transport coefficients. Of particular interest is the case of rare ballistic propagation, in which case a regime of superdiffusion may lurk underneath one of normal diffusion.

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“…For example, it reproduces Hamiltonian kinetics in egg-crate potentials [8] and in infinite horizon billiards [10]. Depending on the symmetry of a potential or size of the scatterers in a billiard, the motion can be restricted to four, eight, or larger even number of basic directions [11]. The XY-model can be generalized to reproduce kinetics of these systems [12].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic densities of planar Lévy walks: a non-isotropic case

Bystrik,

Denisov

2021

Preprint

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Lévy walks are a particular type of continuous-time random walks which results in a superdiffusive spreading of an initially localized packet. The original one-dimensional model has a simple schematization that is based on starting a new unidirectional motion event either in the positive or in the negative direction. We consider two-dimensional generalization of Lévy walks in the form of the so-called XY-model. It describes a particle moving with a constant velocity along one of the four basic directions and randomly switching between them when starting a new motion event. We address the ballistic regime and derive solutions for the asymptotic density profiles. The solutions have a form of first-order integrals which can be evaluated numerically. For specific values of parameters we derive an exact expression. The analytic results are in perfect agreement with the results of finite-time numerical samplings.

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Measuring logarithmic corrections to normal diffusion in infinite-horizon billiards

Cited by 28 publications

References 49 publications

Convergence of moments for dispersing billiards with cusps

Convergence of moments for dispersing billiards with cusps

Lévy walks on lattices as multi-state processes

Asymptotic densities of planar Lévy walks: a non-isotropic case

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