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.


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


Zbl 1211.82011
Full Text: DOI arXiv


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