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.

With a special Sierpfnski carpet (SC)，the Ising model is exactly solved by a combinatorial approach and graph technique. The rigorous partition function and free energy are obtained and the phase transi-tion is investigated. We argue that the existence of a phase transition strongly depends on the order of ramification of the SC. Our method is extended to deal with other lattices.

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.

We use the position-space renormalisation group and exact enumeration methods to study configurational properties of directed lattice animals. These are clusters made up of directed bonds in which the'tail'of a new bond must be added to the 'head' of already existing bonds. Furthermore, the directed bonds have a net orientational order with respect to some preferred anisotropy axis. In the limit that the number of bonds N+CQ, directed animals become extremely anisotropic and two independent correlation lengths, one parallel and one perpendicular to the preferred axis, are required to describe their shape. Mean-field theory suggests that these lengths diverge with exponents of Y I I=f and vl=$ respectively. Below seven dimensions mean-field theory breaks down, and we analyse our enumeration data to estimate the location of the critical point, the exponents YII and v l, and the exponent 0 characterising the singularity of the directed animal generating function. In two dimensions, we estimate that YH=0.8, vl=0.5 and 0=0.5, and our estimates interpolate smoothly with dimension to their respective mean-field limits. In two dimensions, we also apply a one-parameter position-space renormalisation group using small cells which gives reasonable estimates for the location of the critical point and q. In addition, we formulate a two-parameter renormalisation which allows us to study animals with a variable fraction of directed bonds, and thereby describe the crossover between directed and isotropic animals. We find that any anisotropy ensures that the critical behaviour belongs to the universality class of directed animals.

In this paper the phase transition of the Gaussian model on m-sheet fractals (mSG)l and (mDH)l is investigated by the real-space renormalization-group method, i.e., decimation following a spin rescaling. The latter is introduced to keep the parameter b constant. Fixed points of the renormalization-group transformation are found and discussed. Our results show the existence of different properties of phase transition between the Gaussian model and the Ising model on fractals. In addition, we find that the critical point k*=b/4 in a regular Sierpinski gasket is identified, with result of k*=b/d d is the coordination number! in Euclidean space. This indicates that the critical point of the Gaussian model may be uniquely determined by the coordination number whether on homogeneous fractals or translationally invariant lattices.