A systematic method is developed for studying the time-dependent Ising model proposed by Glauber and extended by others. The graphical technique developed for the equilibrium statistical mechanics has been found to be useful in this nonequilibrium problem as well with some modifications. A hierarchy of equations linking various spin cluster functions is obtained. If we choose as the expansion parameter the reciprocal range of the spin interaction which is taken to be of Kac's type, the hierarchy is terminated, and closed sets of nonlinear kinetic equations satisfied by spin cluster functions follow. The dynamical susceptibility is studied using the hierachy, and the polydispersive nature of the spin relaxation is found to be closely connected to the deviation of the spin pair correlation from the Ornstein-Zernike form via multi-spin cluster functions. A sum rule on the distribution of relaxation times is found, which provides a precise expression for the critical slowing-down at the Curie point. The closed set of kinetic equations involving at most three spin cluster functions is used to evaluate explicitly the approximate dynamical susceptibility above the Curie point, and the characteristic time-scale of the polydispersive relaxation is found to increase indefinitely near the Curie point.
We study the time evolution of the one-dimensional random-bond transverse Ising model with four-spin interactions. We calculate the time-dependent correlation function as well as the longitudinal relaxation function of the infinite chain. We analyze how the presence of disorder affect the dynamical behavior of the system in comparison with the pure model. We find that the main effect of disorder is to produce a crossover from a central mode to a collective-mode type of dynamics, as the concentration of weaker bonds is enhanced. Such crossover is also present in the case of an increase in bond dilution.
We investigate the dynamics of the one-dimensional S=(1/2) isotropic XY model and transverse Ising model in the high-temperature limit by using the method of recurrence relations. We obtain the relaxation functions as well as some Brownian analogs of the generalized Langevin equation for a tagged spin
Sjx in these models, namely, the memory functions and the random forces. We find that the realized dynamical Hilbert spaces of the two models have the same structure, which leads to similar dynamical behavior apart from a time scale.
The correlation functions ${\ensuremath{\rho}}_{\mathrm{xx}}(R,t)$ and ${\ensuremath{\rho}}_{\mathrm{yy}}(R,t)$ are calculated for the one-dimensional Ising model with a transverse field
h at
T=0. This model corresponds to the
XY model with $\ensuremath{\gamma}=1$, and is equivalent to the two-dimensional anisotropic (${J}_{1}\ensuremath{\rightarrow}\ensuremath{\infty}$, ${J}_{2}\ensuremath{\rightarrow}0$) Ising model. The additional dimension in the classical model is the imaginary time of the quantum model. For all values of
h, ${\ensuremath{\rho}}_{\mathrm{yy}}(R,t)=\ensuremath{-}(\frac{1}{{h}^{2}})(\frac{{\ensuremath{\partial}}^{2}}{\ensuremath{\partial}{t}^{2}}){\ensuremath{\rho}}_{\mathrm{xx}}(R,t)$. At the critical field
h=hc=1, ${\ensuremath{\rho}}_{\mathrm{xx}}(R,t)\ensuremath{\sim}{({R}^{2}\ensuremath{-}{t}^{2})}^{\frac{\ensuremath{-}1}{8}}$. For
h<1,h>1 the results already obtained for the
XY model are recovered. We give some consequences from this equivalence in higher dimension, concerning the behavior of the correlation function at the critical field at
T=0.