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 Yang-Lee theory of phase transition and our extension, it is found that for a real fluid, both the singularity of canonical partition function and the critical point of the gas-liquid phase transition occur precisely at the temperature when all the cluster integrals become positive. The critical temperature is determined by the limit of the first zeros of the cluster integrals.

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.

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 use the real-space renormalization-group approach to calculate the fractal dimension of diffusion-limited aggregation for a alttice with an arbitrary Euclidean dimension d. The results obtained are in good agreement with numerical numerical results. We prove that D → d-1 as d → ∞