Microscopic statistical description of classical matter around black holes

The problem of statistical description of classical matter, represented by a N-body system of relativistic point particles falling into a black hole, is investigated, adopting a classical framework. A covariant microscopic statistical description is developed for the classical particle system by introducing the definition of observer-related N-particle microscopic distribution function and by suitably extending to general relativity the concept of Gibbs microscopic entropy. The corresponding entropy production rate is calculated taking into account the presence of an event horizon and proven to obey, for N>>1, an exact microscopic H-theorem which holds for arbitrary geometries.