Method of $m$-particle density matrices of the canonical ensemble in the description of states of quantum systems

A rigorous mathematical description is given of equilibrium states of quantum systems on the basis of the theory of the canonical ensemble. For the $m$-particle density matrices of the canonical ensemble in finite volume relations are obtained which go over into the Kirkwood–Salzburg integral equations in the case of infinite Systems. Existence and uniqueness is proved for the limit $m$-particle density matrices of the canonical ensemble; their analytic dependence on the density is investigated; and it is shown that the canonical and the grand canonical ensemble are equivalent in the thermodynamic limit.