Approaches to Derivation of the Boltzmann Equation with Hard Sphere Collisions

In the paper the possible approaches to the rigorous derivation of the Boltzmann kinetic equation with hard sphere collisions from underlying dynamics are considered. In particular, a formalism for the description of the evolution of infinitely many hard spheres within the framework of marginal observables in the Boltzmann--Grad scaling limit is developed. Also we give consideration to one more approach of the description of the kinetic evolution of hard spheres in terms of a one-particle distribution function governed by the non-Markovian generalization of the Enskog equation and the Boltzmann--Grad asymptotic behavior of its non-perturbative solution is established.