Critical dynamics of the kinetic Ising model on regular fractals

Critical slowing down of the kinetic Gaussian model on hierarchical lattices

The critical slowing down of the kinetic Gaussian model on hierarchical lattices is studied by means of a real-space time-dependent renormalization-group transformation. The dynamic critical exponent z and the exponent Delta are calculated. For hierarchical lattices with reducible generators, both the dynamic critical exponent z and the exponent Delta are independent of the fractal dimension Df of the lattice, the number of branches m, and the number of bonds per branch b of the generator--i.e., z = 2 and Delta = 1. For hierarchical lattices with irreducible generators, the exponent Delta is the same--i.e., Delta = 1; however, the dynamic critical exponent z is dependent on the concrete geometrical structure of these lattices. In addition, it was found that the lattice dependence of the correlation-length critical exponent nu is the same as that of the dynamic critical exponent z. Finally we give a brief discussion about universality for critical dynamics.

Critical dynamics of the kinetic Glauber-Ising model on hierarchical lattices

The critical dynamics of the kinetic Glauber-Ising model is studied on a family of the diamond-type hierarchical lattices with various branches. By carrying out the time-dependent real-space renormalization-group transformation to the master equation of the systems considered, the dynamic exponent is calculated. We find that the dynamic exponent depends on fractal dimension d(f) or the branch number m in a generator, and that it increases with the increase of d(f) or m. We notice that for the case of m=1 (one-dimensional spin chain, d(f)=1) our result z=2 is the same as the exact result obtained by Glauber, and for the case of m=2 (the simplest one in the diamond-type hierarchical lattices, d(f)=2) the exponent z=2.626 is higher than those of the two-dimensional regular lattice and the triangular lattice.

Solvable kinetic gaussian model in an external field

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 tau(q), the complex susceptibility chi(omega,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.