  The communication deals with the problem of a rigorous description of the
evolution of the observables of many colliding particles. The method of cluster
expansions of the groups of operators of the Liouville equations is used to
establish the generating operators of expansions, which represent a solution of
the Cauchy problem of the dual BBGKY hierarchy. The structure of series
expansions, which represent a non-perturbative solution of the Cauchy problem
of the BBGKY hierarchy describing the state evolution of many hard spheres, is
given in the appendix to the article.
