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arXiv:hep-lat/9411074 · Physical review. B, Condensed matter

Dynamical scaling from multiscale measurements

N. Persky, Radel Ben-Av, S. Solomon

Published Mar 1, 1995• 1 citation
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Abstract

We present a measure of the dynamical critical behavior: the ``multiscale dynamical exponent (MDE),'' ${\mathit{z}}^{\mathit{m}\mathit{d}}$. Using dynamical renormalization-group concepts we study the relaxation times, ${\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\Lambda}}}$, of a family of space scales defined by \ensuremath{\Lambda}=1/k. Assuming dynamical universality we argue, and verify numerically, that ${\mathit{z}}^{\mathit{m}\mathit{d}}$ has the same value as the usually defined z. We measure ${\mathit{z}}^{\mathit{m}\mathit{d}}$ in the two-dimensional Ising model with the Metropolis and cluster dynamics and find ${\mathit{z}}_{\mathrm{met}}^{\mathit{m}\mathit{d}}$=2.1\ifmmode\pm\else\textpm\fi{}0.1 and ${\mathit{z}}_{\mathrm{wolff}}^{\mathit{m}\mathit{d}}$=0\ifmmode\pm\else\textpm\fi{}0.15, respectively. We note that in our approach ${\mathit{z}}^{\mathit{m}\mathit{d}}$ is measured using a single temperature and a single lattice size. In addition, in the Metropolis case we present a method which helps to overcome critical slowing down in the dynamical measurements themselves.

DOI
10.1103/physrevb.51.6100

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We present a measure of the dynamical critical behavior: the ``multiscale dynamical exponent (MDE),'' zmd{\mathit{z}}^{\mathit{m}\mathit{d}}zmd. Using dynamical renormalization-group concepts we study the relaxation times, \ensuremathτ\ensuremathΛ{\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\Lambda}}}\ensuremathτ\ensuremathΛ​, of a family of space scales defined by \ensuremath{\Lambda}=1/k. Assuming dynamical universality we argue, and verify numerically, that zmd{\mathit{z}}^{\mathit{m}\mathit{d}}zmd has the same value as the usually defined z. We measure zmd{\mathit{z}}^{\mathit{m}\mathit{d}}zmd in the two-dimensional Ising model with the Metropolis and cluster dynamics and find zmetmd{\mathit{z}}_{\mathrm{met}}^{\mathit{m}\mathit{d}}zmetmd​=2.1\ifmmode\pm\else\textpm\fi{}0.1 and zwolffmd{\mathit{z}}_{\mathrm{wolff}}^{\mathit{m}\mathit{d}}zwolffmd​=0\ifmmode\pm\else\textpm\fi{}0.15, respectively. We note that in our approach zmd{\mathit{z}}^{\mathit{m}\mathit{d}}zmd is measured using a single temperature and a single lattice size. In addition, in the Metropolis case we present a method which helps to overcome critical slowing down in the dynamical measurements themselves.
Values of dynamic critical exponents are numerically estimated for various models with the nonequilibrium relaxation method to test the dynamic universality hypothesis. The dynamics used here are single-spin update with Metropolis-type transition probabities. The estimated values of nonequilibrium relaxation exponent of magnetization λ m (=β/ z ν) of Ising models on bcc and fcc lattices are estimated to be 0.251(3) and 0.252(3), respectively, which are consistent with the value of the model on simple-cubic lattice, 0.250(2). The dynamic critical exponents of three-states Potts models on square, honeycomb and triangular lattices are also estimated to be 2.193(5), 2.198(4), and 2.199(3), respectively. They are consistent within the error bars. It is also confirmed that Ising models with regularly modulated coupling constants on square lattice have the same dynamic critical exponents with the uniformly ferromagnetic Ising model.
The random-field Ising model shows an extreme critical slowdown that has been described by activated dynamic scaling: The characteristic time of relaxation toward equilibrium diverges exponentially with the correlation length $ln\ensuremath{\tau}\ensuremath{\sim}{\ensuremath{\xi}}^{\ensuremath{\psi}}/T$, with $\ensuremath{\psi}$ an a priori unknown barrier exponent. Through a nonperturbative functional renormalization group, we show that for spatial dimensions ddd less than a critical value ${d}_{\text{DR}}\ensuremath{\simeq}5.1$, also associated with dimensional-reduction breakdown, $\ensuremath{\psi}=\ensuremath{\theta}$ with $\ensuremath{\theta}$ the temperature exponent near the zero-temperature fixed point that controls the critical behavior. For d>dDRd>{d}_{\text{DR}}d>dDR​, on the other hand, $\ensuremath{\psi}=\ensuremath{\theta}\ensuremath{-}2\ensuremath{\lambda}$, where $\ensuremath{\theta}=2$ and $\ensuremath{\lambda}>0$ an additional exponent. At the upper critical dimension $d=6,\ensuremath{\lambda}=1$, so that $\ensuremath{\psi}=0$, and activated scaling gives way to conventional scaling. We give a physical interpretation of the results in terms of collective events in real space, avalanches, and droplets. We also propose a way to check the two regimes by computer simulations of long-range one-dimensional systems.
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We extend the Monte Carlo renormalization-group technique to investigate the scaling behavior in the vicinity of the critical point of the two-dimensional Ising model in the presence of linearly changing temperatures, instead of some static temperatures. We find that it is possible in such a dynamically driven, nonequilibrium case to measure both the dynamic and static critical exponents. Two additional critical exponents associated with the temperature sweep rate are presented, and scaling laws relating them to other known exponents are obtained. Also a scaling form of the order parameter is extracted.
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 $2\ensuremath{-}\ensuremath{\alpha}+(2\ensuremath{-}r)\ensuremath{\beta}\ensuremath{\delta}$.