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arXiv:cond-mat/0207006 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Glauber critical dynamics: Exact solution of the kinetic Gaussian model

Jian-Yang Zhu, Zhao‐Di Yang

Published Feb 1, 1999• 18 citations
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Abstract

In this paper, we have exactly solved Glauber's critical dynamics of the Gaussian model in three dimensions. Of course, it is much easier to apply in the low-dimensional case. The key steps are that we generalize the spin change mechanism from Glauber's single-spin flipping to single-spin transition and give a normalized version of the transition probability. We have also investigated the dynamical critical exponent and found surprisingly that the dynamical critical exponent is highly universal; that is, for one, two, and three dimensions they have the same value independent of spatial dimensionality in contrast to static (equilibrium) critical exponents.

DOI
10.1103/physreve.59.1551

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In this paper, we have exactly solved Glauber's critical dynamics of the Gaussian model in three dimensions. Of course, it is much easier to apply in the low-dimensional case. The key steps are that we generalize the spin change mechanism from Glauber's single-spin flipping to single-spin transition and give a normalized version of the transition probability. We have also investigated the dynamical critical exponent and found surprisingly that the dynamical critical exponent is highly universal; that is, for one, two, and three dimensions they have the same value independent of spatial dimensionality in contrast to static (equilibrium) critical exponents.
In this paper, we present a multispin transition mechanism, which is an extension of the Glauber one, to investigate critical dynamics. By exactly solving the master equation, the influence of the multispin transition mechanism on the dynamic critical behavior is studied for the Gaussian model with nearest-neighbor interactions on d-dimensional lattices (d=1, 2, and 3). The time evolution of magnetization is exactly calculated, and the exact results of relaxation time and dynamic critical exponent are obtained. Our models are divided into two kinds: one is the spin-cluster transition and the other is the arbitrary multispin transition. It is found that there are different relaxation times, but the same dynamical critical exponent for different kinds of multispin transitions. The results show that the dynamical critical exponents are independent of spatial dimensions and configurations of transitional spins, and that the dynamical critical exponent is the same as that of the Glauber dynamics, and thus give a strong support to the simple single-spin-transition dynamics. Finally, we give a brief discussion on the results.
We generalize Kawasaki's dynamics, spin-pair exchange mechanism, to a spin-pair redistribution mechanism, and we present a normalized redistribution probability. As an application, we treat the one-dimensional kinetic Gaussian model and obtain the exact diffusion equation and the temperature-dependent diffusion coefficient. We find that the diffusion process can slow down infinitely near the critical point and obtain the critical dynamic exponent z = 2 that is independent of the assumed mechanism, either Glauber-type or Kawasaki-type.
A single-spin transitional critical dynamics based on the Glauber theory is used to investigate the d-dimensional kinetic S 4 model. We derive the fundamental dynamical equations by a cumulant expansion, and then, a one-dirnensional model is treated. We obtain approximately the local magnetization and equal-time spin-pair correlation function, in which the terms of k 3 and the higher orders are ignored (corresponding to high-temperature cutoff). These are finally extended to d-dimensional case. In addition, we obtain the critical point k c of d-dimensional S 4 model, and give a relation satisfied by the dynamic critical exponent z and the correlation length critical exponent ν .
In this work we study the Glauber dynamics of the one-dimensional Ising model with nearest-neighbor and next-nearest-neighbor interactions, for which an approximate solution of the magnetization per site is obtained. When the dynamical critical exponent z is investigated following the treatment of Cordery, Sarker, and Tobochnik [Phys. Rev. B 24, 5402 (1981)], our observation shows that its upper-bound value is the same as the known value, thus implying that z is independent of the range of the interaction. We also suggest a high-temperature expansion approximation which is then used to solve the two-dimensional Glauber dynamics governed by a master equation; this solution is compared with that of the decoupling method. The time-delayed correlation function is also calculated.
We make use of the calculation of Achiam and Kosterlitz and investigate the generalization of the Migdal-type renormalization-group calculation to critical dynamics, by looking at the one- and two-dimensional kinetic Ising model with no conserved magnetization. In the two-dimensional case the dynamical critical index ZM(magneticperturbation)=2.064{Z}_{M}(\mathrm{magnetic}\mathrm{perturbation})=2.064ZM​(magneticperturbation)=2.064 and ZE(energylikeperturbation)=1.819{Z}_{E}(\mathrm{energylike}\mathrm{perturbation})=1.819ZE​(energylikeperturbation)=1.819 for scale factor $\ensuremath{\lambda}=1$. ZM{Z}_{M}ZM​ involves the static $\frac{\ensuremath{\beta}}{\ensuremath{\nu}}$ exponent, while ZE{Z}_{E}ZE​ involves $\frac{1}{\ensuremath{\nu}}$, and therefore, it is not surprising that ZM{Z}_{M}ZM​ is closer to the high-temperature expansion results, since $\frac{\ensuremath{\beta}}{\ensuremath{\nu}}$ in the Migdal approximation for statics is much closer to the exact value than $\frac{1}{\ensuremath{\nu}}$. In one dimension we obtain ZM=ZE=2{Z}_{M}={Z}_{E}=2ZM​=ZE​=2, which is the exact result of Glauber.
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.