On the Existence of the Boltzmann-Grad Limit for a System of HardSmooth Spheres

A fundamental aspect of kinetic theory concerns 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. Several aspects of the related theory remain to be addressed and clarified. In fact, up to now the validity of the Lanford conjecture 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 meaning (strong Lanford conjecture). This problem arises,in fact, when the Boltzmann-Grad limit is intended in the sense of the uniform convergence in phase-space of the relevant joint-particle probability densities. The answer to this question seems doubtful not just for the Boltzmann equation but also for the related BBGKY hierarchy from which it is derived. In this paper we intend to 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.