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Physical review. B, Solid state

Theory for the dynamics of clusters near the critical point. I. Relaxation of the Glauber kinetic Ising model

Kurt Binder, Dietrich Stauffer, Heiner Muller-Krumbhaar

Published Dec 1, 1975• 98 citations
Publisher

Abstract

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 $l$ 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 $r$, 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 $2\ensuremath{-}\ensuremath{\alpha}+(2\ensuremath{-}r)\ensuremath{\beta}\ensuremath{\delta}$.

DOI
10.1103/physrevb.12.5261

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Critical fluctuations in kinetic Ising models are interpreted in terms of cluster reactions. The basic assumption, that clusters with $l$ spins grow at a rate $\ensuremath{\propto}{l}^{r}$, is tested by Monte Carlo computations in the single-spin-flip case. The dynamic susceptibilities associated with order parameter and energy are then calculated also for nonzero magnetic field, and are shown to fulfill dynamic scaling. The exponent $(2\ensuremath{-}r)\ensuremath{\beta}\ensuremath{\delta}$ of the relaxation times can be different from the susceptibility exponent $\ensuremath{\gamma}$.

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Extending the Monte Carlo method to dynamic critical phenomena we investigated the time-dependent correlation functions in the two-dimensional one-spin-flip Ising model and the critical behavior of the associated relaxation times. These relaxation times are the following: ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}}^{\ensuremath{\Delta}T}$, characterizing the approach of the order parameter to equilibrium after a change of temperature $\ensuremath{\Delta}T$ of the system; ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}$ and ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}^{A}$ characterizing the slowing down of the order-parameter correlation and autocorrelation functions, respectively; ${\ensuremath{\tau}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}$ and ${\ensuremath{\tau}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}^{A}$, characterizing the slowing down of the energy correlation and autocorrelation functions; and finally ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\mathcal{H}}$, characterizing the cross-correlation function. We give estimates for the associated exponents ${\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}}^{\ensuremath{\Delta}T}\ensuremath{\approx}{\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}\ensuremath{\approx}{\ensuremath{\Delta}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}\ensuremath{\approx}{\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\mathcal{H}}\ensuremath{\approx}1.90\ifmmode\pm\else\textpm\fi{}0.10$, and ${\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}^{A}\ensuremath{\approx}1.60\ifmmode\pm\else\textpm\fi{}0.10$, ${\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\mathcal{H}}^{A}\ensuremath{\approx}0.95\ifmmode\pm\else\textpm\fi{}0.10$, ${\ensuremath{\Delta}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}^{A}\ensuremath{\approx}0$, which are consistent with the dynamic scaling hypothesis and with exact inequalities. A detailed comparison with recent high-temperature-expansion estimates is performed, and the reliability of the Monte Carlo results is carefully analyzed.

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Critical dynamical behavior of the Ising model

Zihua Liu, Erol Vatansever, G. T. Barkema +1 more

We investigate the dynamical critical behavior of the two- and three-dimensional Ising models with Glauber dynamics in equilibrium. In contrast to the usual standing, we focus on the mean-squared deviation of the magnetization $M, {\mathrm{MSD}}_{M}$, as a function of time, as well as on the autocorrelation function of $M$. These two functions are distinct but closely related. We find that ${\mathrm{MSD}}_{M}$ features a first crossover at time ${\ensuremath{\tau}}_{1}\ensuremath{\sim}{L}^{{z}_{1}}$, from ordinary diffusion with ${\mathrm{MSD}}_{M}\ensuremath{\sim}t$, to anomalous diffusion with ${\mathrm{MSD}}_{M}\ensuremath{\sim}{t}^{\ensuremath{\alpha}}$. Purely on numerical grounds, we obtain the values ${z}_{1}=0.45(5)$ and $\ensuremath{\alpha}=0.752(5)$ for the two-dimensional Ising ferromagnet. Related to this, the magnetization autocorrelation function crosses over from an exponential decay to a stretched-exponential decay. At later times, we find a second crossover at time ${\ensuremath{\tau}}_{2}\ensuremath{\sim}{L}^{{z}_{2}}$. Here, ${\mathrm{MSD}}_{M}$ saturates to its late-time value $\ensuremath{\sim}{L}^{2+\ensuremath{\gamma}/\ensuremath{\nu}}$, while the autocorrelation function crosses over from stretched-exponential decay to simple exponential one. We also confirm numerically the value ${z}_{2}=2.1665(12)$, earlier reported as the single dynamic exponent. Continuity of ${\mathrm{MSD}}_{M}$ requires that $\ensuremath{\alpha}({z}_{2}\ensuremath{-}{z}_{1})=\ensuremath{\gamma}/\ensuremath{\nu}\ensuremath{-}{z}_{1}$. We speculate that ${z}_{1}=1/2$ and $\ensuremath{\alpha}=3/4$, values that indeed lead to the expected ${z}_{2}=13/6$ result. A complementary analysis for the three-dimensional Ising model provides the estimates ${z}_{1}=1.35(2), \ensuremath{\alpha}=0.90(2)$, and ${z}_{2}=2.032(3)$. While ${z}_{2}$ has attracted significant attention in the literature, we argue that for all practical purposes ${z}_{1}$ is more important, as it determines the number of statistically independent measurements during a long simulation.

arXiv:2307.01837•Physical review. E•Sep 21, 2023•14 citations
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∝lr\ensuremath{\propto}{l}^{r}\ensuremath∝lr, r\ensuremath≈1r\ensuremath{\approx}1r\ensuremath≈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−TTc∣\ensuremath−(2\ensuremath−r)\ensuremathβ\ensuremathδ{|1\ensuremath{-}\frac{T}{{T}_{c}}|}^{\ensuremath{-}(2\ensuremath{-}r)\ensuremath{\beta}\ensuremath{\delta}}∣1\ensuremath−Tc​T​∣\ensuremath−(2\ensuremath−r)\ensuremathβ\ensuremathδ. 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−\ensuremathα+(2\ensuremath−r)\ensuremathβ\ensuremathδ2\ensuremath{-}\ensuremath{\alpha}+(2\ensuremath{-}r)\ensuremath{\beta}\ensuremath{\delta}2\ensuremath−\ensuremathα+(2\ensuremath−r)\ensuremathβ\ensuremathδ.
Critical fluctuations in kinetic Ising models are interpreted in terms of cluster reactions. The basic assumption, that clusters with lll spins grow at a rate $\ensuremath{\propto}{l}^{r}$, is tested by Monte Carlo computations in the single-spin-flip case. The dynamic susceptibilities associated with order parameter and energy are then calculated also for nonzero magnetic field, and are shown to fulfill dynamic scaling. The exponent $(2\ensuremath{-}r)\ensuremath{\beta}\ensuremath{\delta}$ of the relaxation times can be different from the susceptibility exponent $\ensuremath{\gamma}$.
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.
Extending the Monte Carlo method to dynamic critical phenomena we investigated the time-dependent correlation functions in the two-dimensional one-spin-flip Ising model and the critical behavior of the associated relaxation times. These relaxation times are the following: ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}}^{\ensuremath{\Delta}T}$, characterizing the approach of the order parameter to equilibrium after a change of temperature $\ensuremath{\Delta}T$ of the system; ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}$ and ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}^{A}$ characterizing the slowing down of the order-parameter correlation and autocorrelation functions, respectively; ${\ensuremath{\tau}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}$ and ${\ensuremath{\tau}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}^{A}$, characterizing the slowing down of the energy correlation and autocorrelation functions; and finally ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\mathcal{H}}$, characterizing the cross-correlation function. We give estimates for the associated exponents ${\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}}^{\ensuremath{\Delta}T}\ensuremath{\approx}{\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}\ensuremath{\approx}{\ensuremath{\Delta}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}\ensuremath{\approx}{\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\mathcal{H}}\ensuremath{\approx}1.90\ifmmode\pm\else\textpm\fi{}0.10$, and ${\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}^{A}\ensuremath{\approx}1.60\ifmmode\pm\else\textpm\fi{}0.10$, ${\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\mathcal{H}}^{A}\ensuremath{\approx}0.95\ifmmode\pm\else\textpm\fi{}0.10$, ${\ensuremath{\Delta}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}^{A}\ensuremath{\approx}0$, which are consistent with the dynamic scaling hypothesis and with exact inequalities. A detailed comparison with recent high-temperature-expansion estimates is performed, and the reliability of the Monte Carlo results is carefully analyzed.
In the preceding chapter, we have introduced several levels for the mathematical description of stochastic dynamics. We now use the kinetic Ising models introduced in Section 2.3.3 to formulate the dynamic scaling hypothesis which appropriately generalizes the homogeneity property of the static correlation function in the vicinity of a critical point, as established in Chapter 1. The dynamic critical exponent z is defined to characterize both the critical dispersion and the basic phenomenon of critical slowing-down. As a next step, and building on the results of Section 2.4, a continuum effective theory for the mesoscopic order parameter density, basically the dynamical analog to the Ginzburg–Landau approach, is constructed in terms of a non-linear Langevin equation. The distinction between dissipative and diffusive dynamics for the purely relaxational kinetics of either a non-conserved or conserved order parameter field, respectively, defines the universality classes A and B. Following the analysis of these models in the Gaussian approximation, they also serve to outline the construction of a dynamical perturbation theory for non-linear stochastic differential equations through direct iteration. In general, however, the order parameter alone does not suffice to fully capture the critical dynamics near a second-order phase transition. Additional hydrodynamic modes originating from conservation laws need to be accounted for as well. The simplest such situation is entailed in the relaxational models C and D, which encompass the static coupling of the order parameter to the energy density. Further scenarios emerge through reversible non-linear mode couplings in the Langevin equations of motion.
The Master equation of the kinetic Ising model with spin-flip and spin-exchange dynamics is written in a second quantized in terms of Pauli operators. The introduction of weighted transition rates allows a general formulation in accordance with the principle of detailed balance. Static and dynamic correlation functions are calculated. The structure factor reveals three different relaxation times already in an extended mean-field approximation. Additionally, we derive a nonlinear evolution equation for the magnetization under the influence of nonthermal external noise. Already in lowest order, the correlation function offers a stretched-exponential behavior with a universal exponent \ensuremath{\sigma}=4(d+2) below the critical dimensionality dc{\mathit{d}}_{\mathit{c}}dc​=2. A one-loop renormalization group approach is used to calculate the critical exponents for the growth of the magnetization. The value of the growth exponent \ensuremath{\beta}=2(2-d)(8+d) is supported by simulations in one dimension. \textcopyright{} 1996 The American Physical Society.
We investigate the dynamical critical behavior of the two- and three-dimensional Ising models with Glauber dynamics in equilibrium. In contrast to the usual standing, we focus on the mean-squared deviation of the magnetization M,MSDMM, {\mathrm{MSD}}_{M}M,MSDM​, as a function of time, as well as on the autocorrelation function of MMM. These two functions are distinct but closely related. We find that MSDM{\mathrm{MSD}}_{M}MSDM​ features a first crossover at time ${\ensuremath{\tau}}_{1}\ensuremath{\sim}{L}^{{z}_{1}}$, from ordinary diffusion with ${\mathrm{MSD}}_{M}\ensuremath{\sim}t$, to anomalous diffusion with ${\mathrm{MSD}}_{M}\ensuremath{\sim}{t}^{\ensuremath{\alpha}}$. Purely on numerical grounds, we obtain the values z1=0.45(5){z}_{1}=0.45(5)z1​=0.45(5) and $\ensuremath{\alpha}=0.752(5)$ for the two-dimensional Ising ferromagnet. Related to this, the magnetization autocorrelation function crosses over from an exponential decay to a stretched-exponential decay. At later times, we find a second crossover at time ${\ensuremath{\tau}}_{2}\ensuremath{\sim}{L}^{{z}_{2}}$. Here, MSDM{\mathrm{MSD}}_{M}MSDM​ saturates to its late-time value $\ensuremath{\sim}{L}^{2+\ensuremath{\gamma}/\ensuremath{\nu}}$, while the autocorrelation function crosses over from stretched-exponential decay to simple exponential one. We also confirm numerically the value z2=2.1665(12){z}_{2}=2.1665(12)z2​=2.1665(12), earlier reported as the single dynamic exponent. Continuity of MSDM{\mathrm{MSD}}_{M}MSDM​ requires that $\ensuremath{\alpha}({z}_{2}\ensuremath{-}{z}_{1})=\ensuremath{\gamma}/\ensuremath{\nu}\ensuremath{-}{z}_{1}$. We speculate that z1=1/2{z}_{1}=1/2z1​=1/2 and $\ensuremath{\alpha}=3/4$, values that indeed lead to the expected z2=13/6{z}_{2}=13/6z2​=13/6 result. A complementary analysis for the three-dimensional Ising model provides the estimates ${z}_{1}=1.35(2), \ensuremath{\alpha}=0.90(2)$, and z2=2.032(3){z}_{2}=2.032(3)z2​=2.032(3). While z2{z}_{2}z2​ has attracted significant attention in the literature, we argue that for all practical purposes z1{z}_{1}z1​ is more important, as it determines the number of statistically independent measurements during a long simulation.