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We prove the invariance principle for a \emph{random Lorentz-gas} particle in3 dimensions under the Boltzmann-Grad limit and simultaneous diffusive scaling.That is, for the trajectory of a point-like particle moving amonginfinite-mass, hard-core, spherical scatterers of radius $r$, placed accordingto a Poisson point process of density $\varrho$, in the limit$\varrho\to\infty$, $r\to0$, $\varrho r^{2}\to1$ up to time scales of order$T=o(r^{-2}{|\log r|}^{-2})$. To our knowledge this represents the firstsignificant progress towards solving rigorously this problem in classicalnonequilibrium statistical physics, since the groundbreaking work of Gallavotti(1969), Spohn (1978) and Boldrighini-Bunimovich-Sinai (1983). The novelty isthat the diffusive scaling of particle trajectory and the kinetic(Boltzmann-Grad) limit are taken simultaneously. The main ingredients are acoupling 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 ofclassical and quantum random Lorentz gas, by Komorowski-Ryzhik (2006),respectively, Erd\H os-Salmhofer-Yau (2007). However, the following aresubstantial differences between our work and these ones: (1) The physicalsetting is different: low density rather than weak coupling. (2)The method ofapproach is different: probabilistic coupling rather thananalytic/perturbative. (3) Due to (2), the time scale of validity of ourdiffusive approximation -- expressed in terms of the kinetic time scale -- ismuch longer and fully explicit.

Invariance Principle for the Random Lorentz Gas -- Beyond the Boltzmann-Grad Limit