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Physical Review

Diffusion Constants near the Critical Point for Time-Dependent Ising Models. II

Kyozi Kawasaki

Published Aug 5, 1966• 132 citations
Publisher

Abstract

The nature of the local-equilibrium approximation employed in our previous work is examined with the use of the correlation-function expression for the general frequency-dependent transport coefficient for systems described by master equations. The transport coefficient is found to be a nondecreasing function of frequency which in the high-frequency limit reduces to the transport coefficient calculated in the local-equilibrium approximation. Thus the local-equilibrium approximation provides an upper bound to the zero-frequency transport coefficient. For the case of spin diffusion, combined with the result of our previous work, this limit implies that the diffusion constant must vanish at least as fast as the inverse of the magnetic susceptibility near the Curie point, which establishes the existence of the critical slowing down for our model of spin diffusion.

DOI
10.1103/physrev.148.375

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The nature of the local-equilibrium approximation employed in our previous work is examined with the use of the correlation-function expression for the general frequency-dependent transport coefficient for systems described by master equations. The transport coefficient is found to be a nondecreasing function of frequency which in the high-frequency limit reduces to the transport coefficient calculated in the local-equilibrium approximation. Thus the local-equilibrium approximation provides an upper bound to the zero-frequency transport coefficient. For the case of spin diffusion, combined with the result of our previous work, this limit implies that the diffusion constant must vanish at least as fast as the inverse of the magnetic susceptibility near the Curie point, which establishes the existence of the critical slowing down for our model of spin diffusion.
The diffusion constant of spins in ferromagnets or of molecules in binary mixtures near the critical point is discussed employing a time-dependent Ising model in which spin interactions are replaced by certain temperature-dependent transition probabilities of spin exchange. The spin diffusion constant is calculated with the single approximation of replacing a reduced spin distribution function by its value for local equilibrium with a given inhomogeneous spin density. The behavior of the diffusion constant near the critical point is dominated by a factor ${\ensuremath{\chi}}^{\ensuremath{-}1}$, where $\ensuremath{\chi}$ is the magnetic susceptibility. This problem is also studied with the use of the Bethe lattice. The effects of surrounding spins on the transition probability for spin exchange are found to be essential for obtaining the critical slowing-down near the critical point. In view of this, Koci\ifmmode \acute{n}\else \'{n}\fi{}ski's calculation of the spin diffusion constant is critically discussed.
Employing a time-dependent Ising model of the kind used in the preceding works of this series, we have studied the self-diffusion constant. In the local-equilibrium approximation, the self-diffusion constant is expressed in terms of certain equal-time spin correlation functions. At the critical point the self-diffusion constant is shown to have a finite value, and it is conjectured that it may have an infinite slope as a function of temperature at the critical point. The two relevant experiments are discussed.
A statistical theory of the localized model of ferromagnetism is developed by considering the effective density matrices for a single spin, and two nearest-neighbor spins, respectively. The Ising Model and the Heisenberg model, and even the case of the anisotropic exchange interaction, are treated within a unified mathematical scheme. The result is reduced to that of the Bethe approximation in the case of the Ising model, and the constant coupling approximation in the case of the Heisenberg model. The method is generalized to calculate the static spin-correlation function. It is obtained in Ornstein-Zernike type. In the nonequilibrium state, the expectation value of the spin decays by the homogeneous damping as well as the spin-diffusion process. This relaxation equation is different from that by Van Hove because of the existence of the homogeneous damping. The diffusion constant does not vanish even at the Curie point TC, contrary to Van Hove's result, although the damping constant vanishes at TC. The latter fact causes the critical slowing down of the regression of fluctuation. The solution of this relaxation equation gives the time-dependent spin-correlation function. The method can be applied to the antiferromagnetic case, too.
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 $2\ensuremath{-}\ensuremath{\alpha}+(2\ensuremath{-}r)\ensuremath{\beta}\ensuremath{\delta}$.