Superdiffusion in the Periodic Lorentz Gas

Marklof, Jens; Tóth, Bálint

doi:10.1007/s00220-016-2578-y

Cited by 24 publications

(63 citation statements)

References 29 publications

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“…We thus expect a classical diffusion equation with a non-classical coefficient in the asymptotic limit. For this simplified equation, this reproduces the result of Marklof & Tóth [13] who proved a superdiffusive central limit theorem directly for the particle billards underlying the periodic Lorentz gas equation. They showed that the periodic Lorentz gas is superdiffusive, but only logarithmically.…”

Section: Concluding Remarks and Future Worksupporting

confidence: 83%

Fractional diffusion limits of non-classical transport equations

Frank¹,

Sun²

2018

Kinetic &Amp; Related Models

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We establish asymptotic diffusion limits of the non-classical transport equation derived in [10]. By introducing appropriate scaling parameters, the limits will be either regular or fractional diffusion equations depending on the tail behaviour of the path-length distribution. Our analysis uses the Fourier transform combined with a moment method. We conclude with remarks on the diffusion limit of the periodic Lorentz gas equation.

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Section: Concluding Remarks and Future Worksupporting

confidence: 83%

Fractional diffusion limits of non-classical transport equations

Frank¹,

Sun²

2018

Kinetic &Amp; Related Models

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“…Theorem 3.2: the particle density is approximately linear (point particles are enlarged for better visibility).case. Case (i) has been analyzed in [6] for periodic Lorentz gas and in [3,17,28] for random Lorentz gas in the Boltzmann-Grad limit (see also [2,23] for recent results on deterministic motion in the Boltzmann-Grad limit). Case (ii) has been studied in [15] for periodic Lorentz gas and in [20] for random Lorentz gas in the Boltzmann-Grad limit.…”

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confidence: 99%

Nonequilibrium Density Profiles in Lorentz Tubes with Thermostated Boundaries

Dolgopyat

Nándori

2015

Comm Pure Appl Math

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Abstract. We consider a long Lorentz tube with absorbing boundaries. Particles are injected to the tube from the left end. We compute the equilibrium density profiles in two cases: the semi-infinite tube (in which case the density is constant) and a long finite tube (in which case the density is linear). In the latter case, we also show that convergence to equilibrium is well described by the heat equation. In order to prove these results, we obtain new results for the Lorentz particle which are of independent interest. First, we show that a particle conditioned not to hit the boundary for a long time converges to the Brownian meander. Second, we prove several local limit theorems for particles having a prescribed behavior in the past.

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“…(19) despite the differing manner in which the limit is taken [18]. Marklof and Tóth have recently proved a central limit theorem, also in arbitary dimension [113]. Their approach, which uses dynamics in the space of lattices, has also led to interesting results in other fields, such as the distribution of Frobenius numbers [114].…”

Section: The Boltzmann-grad Limitmentioning

confidence: 99%

Diffusion in the Lorentz Gas

Dettmann¹

2014

Commun. Theor. Phys.

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General rightsThis document is made available in accordance with publisher policies. Please cite only the published version using the reference above. The Lorentz gas, a point particle making mirror-like reflections from an extended collection of scatterers, has been a useful model of deterministic diffusion and related statistical properties for over a century. This survey summarises recent results, including periodic and aperiodic models, finite and infinite horizon, external fields, smooth or polygonal obstacles, and in the Boltzmann-Grad limit. New results are given for several moving particles and for obstacles with flat points. Finally, a variety of applications are presented.

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Superdiffusion in the Periodic Lorentz Gas

Cited by 24 publications

References 29 publications

Fractional diffusion limits of non-classical transport equations

Fractional diffusion limits of non-classical transport equations

Nonequilibrium Density Profiles in Lorentz Tubes with Thermostated Boundaries

Diffusion in the Lorentz Gas

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