Mathematical description of the states of bose and fermi systems by the method of partial density matrices of the canonical ensemble

In the framework of the theory of the canonical ensemble, a rigorous mathematical description is given of equilibrium states of quantum systems that satisfy Bose or Fermi statistics at low densities. Study of the properties of the solutions of the corresponding Kirkwood–Salsburg equations leads to proof of the existence and uniqueness of limit partial density matrices of the canonical ensemble as analytic functions of the density; the equivalence of the canonical and the grand canonical ensembles in the thermodynamic limit is proved.