Journal of Statistical Physics, Vol. 28, No. 4, 1982. p. 685-710.
The cluster expansion for the classical and the quantum canonical partition function are related to the Bell polynomials. This observation is exploited in derivation of a set of recursion relations that render tractable numerical evaluation of quantities such as mean cluster size distributions and pressure isotherms. The exact volume dependences of properties of an ideal Bose gas are calculated under periodic boundary conditions. Numerical calculations with volume-independent cluster integrals show bimodal distributions in the mean cluster
weight for two- and three-dimensional ideal Bose gases at sufficiently low temperature and high density. The variation of the size at which the liquid (condensate) peak appears indicates that the liquid clusters are macroscopic in macroscopic systems. The similarity between the Bose-Einstein condensation and the sol->gel transition in nonlinear chemically polymerizing systems is discussed. When the exact volume dependence of the cluster integrals is taken into account, the mean cluster weight distribution becomes "chair shaped" rather than bimodal and displays no diagonal long-range order in the canonical ensemble. The "Kac density" for an ideal Bose gas implies that in the canonical ensemble the Ursell function satisfies a cluster property in the limit in which the coordinates of the particles are widely separated.