Abstract
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 \({\sqrt{t {\rm log} t}}\), 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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Communicated by L. Erdös
The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007–2013) / ERC Grant Agreement No. 291147. J.M. is furthermore supported by a Royal Society Wolfson Research Merit Award. The research of B.T. is partially supported by the Hungarian National Science Foundation (OTKA) through Grant K100473 and by the Leverhulme Trust through International Network Grant “Laplacians, Random Walks, Quantum Spin Systems”. Both authors thank the Isaac Newton Institute, Cambridge for its support and hospitality during the programmes “Periodic and Ergodic Spectral Problems” and “Random Geometry”.
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Marklof, J., Tóth, B. Superdiffusion in the Periodic Lorentz Gas. Commun. Math. Phys. 347, 933–981 (2016). https://doi.org/10.1007/s00220-016-2578-y
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DOI: https://doi.org/10.1007/s00220-016-2578-y