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.

Based on Meakin' s simulation results [Phys. Rev. A 34, 710 (1986); 35, 2234 (1987)], we use the bino-mial distribution to discuss the growth probability for the screened-growth model and develop a formula of the generalized dimensions. The results from this model are in good agreement with those from the simulations.

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 → ∞

We show that the singularity of the free energy of Ising models in the absence of a magnetic field on the triangular, square, and honeycomb lattices is related to zeros of the pseudopartition function on an elementary cycle. Using the Griffiths' smoothness postulate, we extend these results to the case in a magnetic field and derive a formula of the critical line of an Ising antiferromagnet, which is in good agreement with the numerical results.

Using the approach that we developed recently, we find the critical line of an anisotropic Ising antiferromagnet on two-dimensional square and honeycomb lattices. We extend our previous lemma and conjecture to be useful in the antiferromagnetic system. We find two interesting behaviors: (1) An antiferromagnet is not necessarily most inert at the absolute zero. (2) The field-driven antiferromagnetic phase transition is possible in the honeycomb lattice.

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.