The Boltzmann-Grad Limit of the Lorentz Gas in a Union of Lattices

Abstract

The Lorentz gas describes an ensemble of noninteracting point particles in an infinite array of spherical scatterers. In the present paper we consider the case when the scatterer configuration P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {P}}}$$\\end{document} is a fixed union of (translated) lattices in Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^d$$\\end{document}, and prove that in the limit of low scatterer density, the particle dynamics converges to a random flight process. In the special case when the lattices in P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {P}}}$$\\end{document} are pairwise incommensurable, this settles a conjecture from Marklof and Strömbergsson (J Stat Phys 155:1072–1086, 2014). The proof is carried out by applying a framework developed in recent work by Marklof and Strömbergsson (Mem AMS 294, 2024), and central parts of our proof are the construction of an admissible marking of the point set P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {P}}}$$\\end{document}, and the verification of the uniform spherical equidistribution condition required in Marklof and Strömbergsson (Mem AMS 294, 2024). Regarding the random flight process obtained in the low density limit of the Lorentz gas, we prove that it can be reconstructed from the corresponding limiting flight processes arising from the individual commensurability classes of lattices in P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {P}}}$$\\end{document}. We furthermore prove that the free path lengths of the limit flight process have a distribution with a power law tail, whose exponent depends on the number of commensurability classes in P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {P}}}$$\\end{document}.