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We introduce the notion of an ensemble of self-adjoint operators and formulate theorems relating the occupation numbers to the number of eigenvalues of the ensemble. We formulate a theorem for the Gibbs distribution in classical mechanics. [ABSTRACT FROM AUTHOR]

A rigorous derivation of the Gibbs and Bose-Einstein distributions on a finite set of energies is stated, in essence, as a theorem in number theory. If there are not very many particles, then the discussion of this question reduces to an exact theorem. [ABSTRACT FROM AUTHOR]

We show that the BBGKY hierarchy in thermodynamic equilibrium degenerates into an equilibrium hierarchy. In turn, the latter can be used to obtain both the fundamental system of equations describing the state of matter inside a correlation sphere of the radius R ≈ 10-7 cm and the canonical Gibbs distribution, which, according to the current concepts, describes the state of the entire macroscopic system. We show that the Gibbs distribution is indeed also local, i.e., describes the state of matter only inside the correlation sphere; in the thermostat surrounding this sphere, the Gibbs distribution degenerates into a constant. We state that these two approaches are equivalent to each other. [ABSTRACT FROM AUTHOR]