# On the Boltzmann-Grad limit for the two dimensional periodic Lorentz gas - Mathematics > Analysis of PDEs

On the Boltzmann-Grad limit for the two dimensional periodic Lorentz gas - Mathematics > Analysis of PDEs - Download this document for free, or read online. Document in PDF available to download.

Abstract: The two-dimensional, periodic Lorentz gas, is the dynamical systemcorresponding with the free motion of a point particle in a planar system offixed circular obstacles centered at the vertices of a square lattice in theEuclidian plane. Assuming elastic collisions between the particle and theobstacles, this dynamical system is studied in the Boltzmann-Grad limit,assuming that the obstacle radius $r$ and the reciprocal mean free path areasymptotically equivalent small quantities, and that the particle-sdistribution function is slowly varying in the space variable. In this limit,the periodic Lorentz gas cannot be described by a linear Boltzmann equationsee F. Golse, Ann. Fac. Sci. Toulouse 17 2008, 735-749, but involves anintegro-differential equation conjectured in E. Caglioti, F. Golse, C.R. Acad.Sci. S\-er. I Math. 346 2008 477-482 and proved in J. Marklof, A.Str\-ombergsson, preprint arXiv:0801.0612, set on a phase-space larger thanthe usual single-particle phase-space. The main purpose of the present paper isto study the dynamical properties of this integro-differential equation:identifying its equilibrium states, proving a H Theorem and discussing thespeed of approach to equilibrium in the long time limit. In the first part ofthe paper, we derive the explicit formula for a transition probabilityappearing in that equation following the method sketched in E. Caglioti, F.Golse, loc. cit

Author: ** Emanuele Caglioti, François Golse**

Source: https://arxiv.org/