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Physical Review A

Droplet model for autocorrelation functions in an Ising ferromagnet

Chao Tang, Hiizu Nakanishi, James S. Langer

Published Jul 1, 1989• 29 citations
Publisher

Abstract

The autocorrelation function, C(t)=〈${S}_{i}$(0)${S}_{i}$(t)〉-〈${S}_{i}^{2}$ (0)〉, of Ising spins in an ordered phase (T${T}_{c}$) is studied via a droplet model. Only noninteracting spherical droplets are considered. The Langevin equation for droplet fluctuations is studied in detail. The relaxation-rate spectra for the corresponding Fokker-Planck equation are found to be (1) continuous from zero for dimension d=2, (2) continuous with a finite gap for d=3, and (3) discrete for d\ensuremath{\ge}4. These spectra are different from the gapless form assumed by Takano, Nakanishi, and Miyashita for the kinetic Ising model. The asymptotic form of C(t) is found to be exponential for d\ensuremath{\ge}3 and stretched exponential with the exponent \ensuremath{\beta}=1/2 for d=2. Our results for C(t) are consistent with the scaling argument of Huse and Fisher, but not with Ogielski's Monte Carlo simulations.

DOI
10.1103/physreva.40.995

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A semiphenomenological cluster theory is developed for dynamic critical properties, which is not limited to small deviations from equilibrium. Explicit numerical expressions are derived for linear and nonlinear response functions of the kinetic Ising model, which are compatible with dynamic scaling and recent calculations of dynamic critical exponents. An uniaxial magnet (or Ising system) and its critical fluctuations are approximated by a "droplet" or "cluster" model, the critical part of the free energy being the sum of the contributions of clusters each containing lll spins. These clusters are assumed to grow and shrink with a phenomenological rate $\ensuremath{\propto}{l}^{r}$, $r\ensuremath{\approx}1$. These rates are compatible with direct Monte Carlo simulations in two and three dimensions. The resulting cluster reaction and diffusion equation is an approximation to the master equation of the Glauber kinetic Ising model (in which the magnetization is not conserved), and has the same structure as used in nucleation theories. In linear response to space- and time-dependent fields the relaxation times of energy and magnetization are expressed as triple integrals. All relaxation times diverge as ${|1\ensuremath{-}\frac{T}{{T}_{c}}|}^{\ensuremath{-}(2\ensuremath{-}r)\ensuremath{\beta}\ensuremath{\delta}}$. This treatment is consistent with dynamic scaling and universality, and is more general than the conventional (Van Hove) theory of critical slowing down. Also the (smaller) exponents found for the response to localized variations ("autocorrelation functions") agree with dynamic scaling. The wave-vector dependence of the relaxation times arises from the static correlation function only, and not from the cluster diffusion term. If the equilibrium cluster distribution is assumed to be that of the Fisher droplet model, and for special values of rrr, the (discrete) eigenvalue spectrum and eigenfunctions (generalized Laguerre polynomials) of the cluster reaction equation are found explicitly. Relaxation functions and frequency-dependent susceptibilities are expressed by hypergeometric series, even in the case of nonlinear response. At the critical point, the spectrum becomes continuous and the exponential decay for large times is replaced by power-law behavior. Similarly, it is found that dynamic scaling applies also to the nucleation rate for this model, and the scaling exponent of the nucleation rate is $2\ensuremath{-}\ensuremath{\alpha}+(2\ensuremath{-}r)\ensuremath{\beta}\ensuremath{\delta}$.
The dynamics of a random Ising ferromagnet is considered, for a general distribution P(J)P(J)P(J) of exchange interactions, in the Griffiths phase. The autocorrelation function C(t)C(t)C(t) is found to have the previously derived asymptotic form, $C(t)\ensuremath{\sim}\mathrm{exp}[\ensuremath{-}A{(\mathrm{ln}t)}^{\frac{d}{(d\ensuremath{-}1)}}]$, where ddd is the spatial dimension, only if $\ensuremath{-}\mathrm{ln}[P(J)]$ grows faster than ${J}^{\frac{d}{(d\ensuremath{-}1)}}$ as $J\ensuremath{\rightarrow}\ensuremath{\infty}$. Otherwise, the asymptotic form of C(t)C(t)C(t) depends explicitly on the form of P(J)P(J)P(J) at large JJJ.
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