We have studied quadric type energy (frequency) recursion relations resulting from fractal lattices. We found that the allowed energy intervals in successive levels form two similar two-scale Cantor sets. If WE imagine the iterating procedure as a dynamical process, the iterating results in different levels gen-erate a"time"sequence, and we can introduce an equal probability measure Pu on a Cantor set and con-struct，following Helsey et a1. [Phys. Rev. A 33, 1141 (1986)], a partition function F (q,T) =∑ipq/17; finally, we obtain Dq-q and f (a)-a curves. A number of exactly soluable examples are investigated.

pin-1 model onto the spin-1/2 Ising model and have employed the exact results of the latter to find the exactly solvable cases of the spin-1 model. Our transformations are not one-to-one correspon-dences between Sf and Or and they are applicable to all lattices, depending on the lattice structure only through the coordination number. On square, triangle, and honeycomb lattices the exact critical points are shown in certain subspaces of parameter space (K, J, L, H, and△).

In this paper, the single-spin transition dynamics is used to investigate the kinetic Gaussian model in a periodic external field. We first derive the fundamental dynamic equations, and then treat an isotropic d-dimensional hypercubic lattice Gaussian spin system with Fourier's transformation method. We obtain exactly the local magnetization and the equal-time pair-correlation function. The critical characteristics of the dynamical relaxation tq, the complex susceptibility x(v,q), and the dynamical response are discussed. The results show that the time evolution of the dynamical quantities and the dynamical responses of the system strongly depend on the frequency and the wave vector of the external field.

A catalytic reaction model, the Ziff-Gulari-Barshad model, is studied on fractal lattices, and the influence of the order of ramification of the lattice on the dynamic behavior of the model is investigated. According to the Monte Carlo simulation results, the order of ramification of the lattice is not crucial to the existence of the continuous transition. This is different from the equilibrium phase transitions in discrete-symmetry spin models such as the Ising model!. Our results indicate that the criterion of the existence of the reactive phase may be complicated.

We study the aggregation-annihilation processes involving three or more kinds of distinct monomers species based on mean-field theory. We propose several solvable models in which irreversible bonding occurs only between similar species and annihilation occurs only between dissimilar species. Under constant aggregation and annihilation reaction rates, the exact solutions of these models are obtained. These analytical solutions show that the kinetic evolution behaviors of each species are quite different. They are scaling or nonscaling depending conclusively on all reaction rates and initial mass distribution of each of the reactants.