  We prove the invariance principle for a \emph{random Lorentz-gas} particle in
3 dimensions under the Boltzmann-Grad limit and simultaneous diffusive scaling.
That is, for the trajectory of a point-like particle moving among
infinite-mass, hard-core, spherical scatterers of radius $r$, placed according
to 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 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 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.
