Lutsko, Christopher and Tóth, Bálint (2020) Invariance Principle for the Random Lorentz Gas  Beyond the BoltzmannGrad Limit. COMMUNICATIONS IN MATHEMATICAL PHYSICS. ISSN 00103616

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Abstract
We prove the invariance principle for a \emph{random Lorentzgas} particle in 3 dimensions under the BoltzmannGrad limit and simultaneous diffusive scaling. That is, for the trajectory of a pointlike particle moving among infinitemass, hardcore, spherical scatterers of radius r, placed according to a Poisson point process of density ϱ, in the limit ϱ→∞, r→0, ϱr2→1 up to time scales of order T=o(r−2logr−2). To our knowledge this represents the first significant progress towards solving rigorously this problem in classical nonequilibrium statistical physics, since the groundbreaking work of Gallavotti (1969), Spohn (1978) and BoldrighiniBunimovichSinai (1983). The novelty is that the diffusive scaling of particle trajectory and the kinetic (BoltzmannGrad) limit are taken simultaneously. The main ingredients are a coupling of the mechanical trajectory with the Markovian random flight process, and probabilistic and geometric controls on the efficiency of this coupling. Similar results have been earlier obtained for the weak coupling limit of classical and quantum random Lorentz gas, by KomorowskiRyzhik (2006), respectively, Erd\H osSalmhoferYau (2007). However, the following are substantial differences between our work and these ones: (1) The physical setting is different: low density rather than weak coupling. (2)The method of approach is different: probabilistic coupling rather than analytic/perturbative. (3) Due to (2), the time scale of validity of our diffusive approximation  expressed in terms of the kinetic time scale  is much longer and fully explicit.
Item Type:  Article 

Subjects:  Q Science / természettudomány > QA Mathematics / matematika 
SWORD Depositor:  MTMT SWORD 
Depositing User:  MTMT SWORD 
Date Deposited:  29 Aug 2020 06:28 
Last Modified:  21 Apr 2023 10:38 
URI:  http://real.mtak.hu/id/eprint/112591 
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