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The Boltzmann-Grad limit of the periodic Lorentz gas. (English) Zbl 1237.37014

Consider the \(d\)-dimensional Lorentz gas defined by infinitely many spherical scatterers of radius \(\rho\), whose centers are placed in the points of a lattice \(\mathcal{L} \subset \mathbb{R}^d\). For this configuration space, the authors study the statistical properties of the billiard dynamics (point particle traveling freely between scatterers and experiencing Fresnel reflections at each collision with a scatterer), in the Boltzmann-Grad limit. The Boltzmann-Grad limit is defined as the limit \(\rho \to 0^+\), together with a rescaling of the spatial coordinates by a factor \(\rho^{d-1}\), so that the average cross-section and the mean free path between collisions remain constant.
Building on the very strong results the same authors had achieved in [“The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems”, Ann. Math. (2) 172, No. 3, 1949–2033 (2010; Zbl 1211.82011)], they prove various forms of the following limit theorem, which we state in a non-rigorous way for the sake of clarity.
Denoting by \((q(t), v(t))\) a trajectory in the Lorentz gas, which depends on the initial conditions \((q_0, v_0)\) and on the choice of \(\rho\), define the stochastic process \(\{ Q(T), V(T) \}_{T \geq 0}\) via the relation \((Q(T), V(T)) = (\rho^{d-1} q(\rho^{-(d-1)}T), v(\rho^{-(d-1)}T))\) and initial conditions randomly chosen according to a certain probability distribution. This process depends on \(\rho\). As \(\rho \to 0^+,\) the process tends (in the sense of the finite-dimensional distributions for finite sets of times) to a persistent Markov process. The discrete-time version of this process, using the collision times, is a Markov chain with memory two.

MSC:

37A60 Dynamical aspects of statistical mechanics
37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
60J25 Continuous-time Markov processes on general state spaces

Citations:

Zbl 1211.82011
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References:

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