We prove an invariance principle for a 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 @math , placed according to a Poisson point process of density @math , in the limit @math , @math , @math up to time scales of order @math . To our knowledge this represents the first significant progress towards solving this problem in classical nonequilibrium statistical physics, since the groundbreaking work of Gallavotti (1970), 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 simulataneously. 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.
In @cite_12 and @cite_14 it is proved that in the Boltzmann-Grad limit the trajectory of the Lorentz particle in any compact time interval @math with @math fixed, converges weakly to a non-Markovian flight process which has, however, a complete description in terms of a Markov chain of the successive collision impact parameters and, conditionally on this random sequence, independent flight lengths. (For a full description in these terms see @cite_21 .) As a second limit, an invariance principle is proved in @cite_21 for this non-Markovian random flight process, with superdiffusive scaling @math . Note that in this case the second limit doesn't just drop out from Donsker's theorem as it did in the random scatterer setting. The results of @cite_12 are valid in @math while those of @cite_14 and @cite_21 in arbitrary dimension.
@cite_14 @cite_21 @cite_12
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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 (BoltzmannGrad limit). The normalization factor is ( t 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. We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math . Abstract The periodic Lorentz gas is the dynamical system corresponding to the free motion of a point particle in a periodic system of fixed spherical obstacles of radius r centered at the integer points of the Euclidian plane, assuming all collisions of the particle with the obstacles to be elastic. In this Note, we study this motion on time intervals of order 1 r as r  0 + . To cite this article: E. Caglioti, F. Golse, C. R. Acad. Sci. Paris, Ser. I 346 (2008).