The Yang-Lee circle theorem is extended to an ideal pseudospin-1/2 Bose gas in an external magnetic field. It is found that the zeros of the canonical partition function are located on the unit circle in the complex activity plane if the temperature is above the critical temperature of ideal Bose-Einstein condensation. No zeros exist if the temperature is below the critical temperature.

A much simpler method is proposed for the evaluation of the exact finite-temperature particle and energy densities for noninteracting Bose and Fermi gases in d-dimensional anisotropic harmonic traps. For the Bose gas in the whole range of temperature, the exact results are obtained. For the Fermi gas with chemical potential less than the single-particle ground-state energy, the exact results are obtained.

We observe that for finite N and V, the canonical partition function QN of a fluid system of N particles is a polynomial of degree N in variable V/Nl3, which has N zeros that depend only on the cluster integrals b2 (V,T),..., bN (V,T). In the thermodynamic limit, if the zero distribution approaches the positive real axis, a phase transition arises. The behavior of phase transition is determined solely by the zero distribution near the positive real axis. Below the critical temperature, the Maxwell' s equal-area rule must be used to obtain the gas-liquid coexistence regime. Several examples are given.

Using the Huang-Yang-Luttinger and Lee-Yang virial expansions of a dilute Bose gas as well as our theory of phase transition, we show that in the dilute limit, the critical temperature of Bose-Einstein condensation of a Bose gas is given by [Tc-T(0)c ] = T(0)c = γn1 = 3a, = 4 [ζ(3/2)]5/3/[3 ζ(5/2)] = 4. 92506..., which is in good agreement with the experimental result of Reppy et al.,γ= 5.1 ± 0.9.

Using some observations and some mean-field approximations, we develop a mean-field cage theory for the freezing of hard-sphere fluids with υf≥ad and obtain the freezing densities as functions of the closest-packing densities and the spatial densities, which are in good agreement with the experimental and simulation results.