Abstract
We prove the invariance principle for a random Lorentzgas particle in 3 dimensions under the Boltzmann Grad 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 $\varrho$ , in the limit $\varrho \to \infty$ , $r \to 0$ , $\varrho r^{2} \to 1$ up to time scales of order $T \ll r^{2}\abs{\log r}^{2}$ . 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 Boldrighini  Bunimovich Sinai (1983). The novelty is that the diffusive scaling of particle trajectory and the kinetic (Boltzmann Grad ) 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 Komorowski  Ryzhik (2006) , respectively, Erd \H os  Salmhofer  Yau (2007) . However, the following are substantial differences between our work and those 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.
Original language  English 

Pages (fromto)  589–632 
Number of pages  44 
Journal  Communications in Mathematical Physics 
Volume  379 
Issue number  2 
DOIs  
Publication status  Published  16 Sept 2020 
Keywords
 Lorentzgas
 invariance principle
 scaling limit
 coupling
 exploration process
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Professor Balint A Toth
 School of Mathematics  Chair in Probability
 Probability, Analysis and Dynamics
 Probability
Person: Academic , Member, Group lead