BILLIARDS, INVARIANT MEASURES, AND EQUILIBRIUM THERMODYNAMICS. II

The kinetics of collisionless continuous medium is studied in a bounded region on a curved manifold. We have assumed that in statistical equilibrium, the probability distribution density depends only on the total energy. It is shown that in this case, all the fundamental relations for a multi-dimensional ideal gas in thermal equilibrium hold true. According to Gibbs, the basic object of statistical mechanics is an ensemble of identical Hamilto-nian systems. The systems do not interact with each other and the assembly of them makes essentially a collisionless continuous medium. From the viewpoint of kinetics, the Hamiltonian systems with elas-tic impacts, i. e. billiards, are especially interesting. These are systems where particles move inertially inside a bounded region and bounce elastically against the boundaries of the region. As it is shown in Refs. [1, 2], in a billiards, the probability distribution density as a function of time t (this function satis es the classic Liouville equation) necessarily has the weak limit as t! 1. This result justies the Zeroth Law of Thermodynamics in the Gibbs theory. The weak limit is a rst integral of the Hamilton equations and depends, in the ergodic case, only on the system's energy. It is noted in Ref. [3] that this is very often justied even without any ergodic hypothesis: it is important here to keep in mind the function class, to which the probability distribution density function belongs. In Ref. [3], we developed the thermodynamics of billiards in the Euclidean space. It turns out t