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 the case of infinite horizon (e.g. the plain @math arrangement of the spherical scatterers of diameter less than the lattice spacing) the free flight distribution of a particle flying in a uniformly sampled random direction has a heavy tail which causes a different type of long time behaviour of the particle displacement. The arguments of @cite_20 indicated that in the two-dimensional case super-diffusive scaling of order @math is expected. A central limit theorem with this anomalous scaling was proved with full rigour in @cite_7 , for the Lorentz-particle displacement in the @math -dimensional periodic case with infinite horizon. The periodic infinite horizon case in dimensions @math remains open.
@cite_7 @cite_20
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We study the asymptotic statistical behavior of the 2-dimensional periodic Lorentz gas with an infinite horizon. We consider a particle moving freely in the plane with elastic reflections from a periodic set of fixed convex scatterers. We assume that the initial position of the particle in the phase space is random with uniform distribution with respect to the Liouville measure of the periodic problem. We are interested in the asymptotic statistical behavior of the particle displacement in the plane as the timet goes to infinity. We assume that the particle horizon is infinite, which means that the length of free motion of the particle is unbounded. Then we show that under some natural assumptions on the free motion vector autocorrelation function, the limit distribution of the particle displacement in the plane is Gaussian, but the normalization factor is (t logt)1 2 and nott1 2 as in the classical case. We find the covariance matrix of the limit distribution. As Bleher (J. Stat. Phys. 66(1):315–373, 1992) observed the free flight vector of the planar, infinite horizon, periodic Lorentz process S n ∣n=0,1,2,… belongs to the non-standard domain of attraction of the Gaussian law—actually with the ( n n ) scaling. Our first aim is to establish his conjecture that, indeed, ( S_ n n n ) converges in distribution to the Gaussian law (a Global Limit Theorem). Here the recent method of Balint and Gouezel (Commun. Math. Phys. 263:461–512, 2006), helped us to essentially simplify the ideas of our earlier sketchy proof (Szasz, D., Varju, T. in Modern dynamical systems and applications, pp. 433–445, 2004). Moreover, we can also derive (a) the local version of the Global Limit Theorem, (b) the recurrence of the planar, infinite horizon, periodic Lorentz process, and finally (c) the ergodicity of its infinite invariant measure. The stability of the inner Reissner-Nordstroem geometry is studied with test massless integer-spin fields. In contrast to previous mathematical treatments we present physical arguments for the processes involved and show that ray tracing and simple first-order scattering suffice to elucidate most of the results. Monochromatic waves which are of small amplitude and ingoing near the outer horizon develop infinite energy densities near the inner Cauchy horizon (as measured by a freely falling observer). Previous work has shown that certain derivatives of the field in a general (nonmonochromatic) disturbance must fall off exponentially near the inner (Cauchy) horizon (r = r sub - ) if energy densities are to remain finite. Thus the solution is unstable to physically reasonable perturbations which arise outside the black hole because such perturbations, if localized near past null infinity (I sup - ), cannot be localized near r sub + , the outer horizon. The mass-energy of an infalling disturbance would generate multipole moments on the black hole. Price, Sibgatullin, and Alekseev have shown that such moments are radiated away as ''tails'' which travel outward and are rescattered inward yielding a wave field with a time dependence t sup -p , p > 0. This decay in time is sufficiently slow that themore » tails yield infinite energy densities on the Cauchy horizon. (The amplification of the low-frequency tails upon interacting with the time-dependent potential between the horizons is an important feature guaranteeing the infinite energy density.) The interior structure of the analytically extended solution is thus disrupted by finite external disturbances. have further shown that even perturbations which are localized as they cross the outer horizon produce singularities at the inner horizon. It is shown that this singularity arises when the incoming radiation is first scattered just inside the outer horizon« less 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 (Boltzmann–Grad 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.