On the existence of the Boltzmann-Grad limit for a system of hard smooth spheres

Abstract

Despite the progress achieved by kinetic theory, its rigorous theoretical foundations still remain unsolved to date. This concerns in particular the search of possible exact kinetic equations and, specifically, the conjecture proposed by Grad (Grad, 1972) and developed in a seminal work by Lanford (Lanford, 1974) that kinetic equations - such as the Boltzmann equation for a gas of classical hard spheres - might result exact in an appropriate asymptotic limit, usually denoted as Boltzmann-Grad limit. The Lanford conjecture has actually had a profound influence on the scientific community, giving rise to a whole line of original research in kinetic theory and mathematical physics. Nevertheless, several aspects of the theory remain to be addressed and clarified. In fact, its validity has been proven for the Boltzmann equation only at most in a weak sense, i.e., if the Boltzmann-Grad limit is defined according to the weak * convergence. While it is doubtful whether the result applies for arbitrary times and for general situations (and in particular more generally for classical systems of particles interacting via binary forces), it remains completely unsolved the issue whether the conjecture might be valid also in a stronger sense (\textit{strong Lanford conjecture}). This paper will point out a physical model providing a counter-example to the strong Lanford conjecture, representing a straightforward generalization of the classical model based on a gas of hard-smooth spheres. In particular we claim that that the one-particle limit function, defined in the sense of the strong Boltzmann-Grad limit, does not generally satisfy the BBGKY (or Boltzmann) hierarchy. The result is important for the theoretical foundations of kinetic theory.