Exponential Decay of Correlations in Multi-Dimensional Dispersing Billiards

Bálint, Péter; Tóth, Imre

doi:10.1007/s00023-008-0389-1

Cited by 37 publications

(91 citation statements)

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“…Our results complement classical studies in ergodic theory that characterize the stochastic properties of the periodic Lorentz gas in the limit of long times, see [6,2,9,21,22,1,25,12] for details.…”

Section: Introductionsupporting

confidence: 77%

The Boltzmann-Grad limit of the periodic Lorentz gas

2011

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Section: Introductionsupporting

confidence: 77%

The Boltzmann-Grad limit of the periodic Lorentz gas

2011

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“…Analogous results hold for the stadium billiard (Bálint and Gouëzel [2]) and billiards with cusps (Bálint et al [1]). The difficulty in extending the above findings to dimensions greater than two lies in the possibly exponential growth of the complexity of singularities (Bálint and Tóth [3,4], Chernov [13]) and, in the case of infinite horizon, the subtle geometry of channels (Dettmann [14], Nándori et al [24]). …”

Section: Introductionmentioning

confidence: 98%

Superdiffusion in the Periodic Lorentz Gas

Marklof

Tóth

2016

Commun. Math. Phys.

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We prove a superdiffusive central limit theorem for the displacement of a test particle in the periodic Lorentz gas in the limit of large times t and low scatterer densities (Boltzmann-Grad limit). The normalization factor is √ t log t, where t is measured in units of the mean collision time. This result holds in any dimension and for a general class of finite-range scattering potentials. We also establish the corresponding invariance principle, i.e., the weak convergence of the particle dynamics to Brownian motion.

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“…The case of higher dimensions is still open, even for models with finite horizon, cf. Chernov [9], and Balint and Toth [1]. Scaling limits that are intermediate between the Boltzmann-Grad and the limit of large times have been explored by Klages and Dellago [14].…”

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confidence: 99%

Kinetic transport in the two-dimensional periodic Lorentz gas

Marklof

Strömbergsson

2008

Nonlinearity

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Abstract. The periodic Lorentz gas describes an ensemble of non-interacting point particles in a periodic array of spherical scatterers. We have recently shown that, in the limit of small scatterer density (Boltzmann-Grad limit), the macroscopic dynamics converges to a stochastic process, whose kinetic transport equation is not the linear Boltzmann equation-in contrast to the Lorentz gas with a disordered scatterer configuration. The present paper focuses on the two-dimensional set-up, and reports an explicit, elementary formula for the collision kernel of the transport equation.One of the central challenges in kinetic theory is the derivation of macroscopic evolution equations-describing for example the dynamics of an electron gas-from the underlying fundamental microscopic laws of classical or quantum mechanics. An iconic mathematical model in this research area is the Lorentz gas [15], which describes an ensemble of non-interacting point particles in an infinite array of spherical scatterers. In the case of a disordered scatterer configuration, well known results by Gallavotti [12], Spohn [22], and Boldrighini, Bunimovich and Sinai [5] show that the time evolution of a macroscopic particle cloud is governed, in the limit of small scatterer density (Boltzmann-Grad limit), by the linear Boltzmann equation. We have recently proved an analogous statement for a periodic configuration of scatterers [16], [17]. In this case the linear Boltzmann equation fails (cf. also Golse [13]), and the random flight process that emerges in the Boltzmann-Grad limit is substantially more complicated.In the present paper we focus on the two-dimensional case, and derive explicit formulae for the collision kernels of the limiting random flight process. These include information not only of the velocity before and after the collision (as in the case of the linear Boltzmann equation), but also the path length until the next hit and the velocity thereafter. Our formulae thus generalize those for the limiting distributions of the free path length found by Dahlqvist [10] Our results on the Boltzmann-Grad limit complement classical studies in ergodic theory, where the scatterer size remains fixed. Bunimovich and Sinai [7] showed that the dynamics of the two-dimensional periodic Lorentz gas is diffusive in the limit of large times, and satisfies a central limit theorem. They assumed that the periodic Lorentz gas has a finite horizon, i.e., the scatterers are configured in such a way that the path length between consecutive collisions is bounded. The corresponding result for infinite horizon has recently been established by Szasz and Varju [21] following initial work by Bleher [2]. See also the recent papers by Dolgopyat, Szasz and Varju [11], and Melbourne and Nicol [19], [20] for related studies of statistical properties of the two-dimensional periodic Lorentz gas. The case of higher dimensions is still open, even for models with finite horizon, cf. Chernov [9], and Balint and Toth [1]. Scaling limits that are intermediate between the ...

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Exponential Decay of Correlations in Multi-Dimensional Dispersing Billiards

Abstract: Abstract. We prove exponential decay of correlations for a "reasonable" class of multi-dimensional dispersing billiards. The scatterers are required to be C 3 smooth, the horizon is finite, there are no corner points. In addition, we assume subexponential complexity of the singularity set.

Cited by 37 publications

References 14 publications

The Boltzmann-Grad limit of the periodic Lorentz gas

The Boltzmann-Grad limit of the periodic Lorentz gas

Superdiffusion in the Periodic Lorentz Gas

Kinetic transport in the two-dimensional periodic Lorentz gas

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