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Journal of the Physical Society of Japan

Statistical Dependence Time and Its Application to Dynamical Critical Exponent

Macoto Kikuchi, Nobuyasu Ito

Published Sep 15, 1993• 29 citations
Publisher

Abstract

A new method for studying dynamical correlation of temporal sequence is proposed. We introduce a new quantity named statistical dependence time τ dep as an estimator for the equilibrium relaxation time. The calculation of τ dep does not require calculation of any time-displaced correlation function nor numerical fitting in extracting the relaxation time; as a result, the estimation of τ dep and statistical analysis such as the error estimation are quite straightforward and simple. We apply this method to the critical dynamics of 3D Ising model, and estimate the dynamical critical exponent z as 2.030±0.004, provided K c =0.221654 (if possible error in K c is taken into account, the error in z becomes 0.01).

DOI
10.1143/jpsj.62.3052

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A new method for studying dynamical correlation of temporal sequence is proposed. We introduce a new quantity named statistical dependence time τ dep as an estimator for the equilibrium relaxation time. The calculation of τ dep does not require calculation of any time-displaced correlation function nor numerical fitting in extracting the relaxation time; as a result, the estimation of τ dep and statistical analysis such as the error estimation are quite straightforward and simple. We apply this method to the critical dynamics of 3D Ising model, and estimate the dynamical critical exponent z as 2.030±0.004, provided K c =0.221654 (if possible error in K c is taken into account, the error in z becomes 0.01).
We review our recent studies on the dynamical correlations in MC simulations from the view point of the statistical dependence. Attentions are paid to the reduction of the statistical degrees of freedom for correlated data. Possible biases on several cumulants, such as the susceptibility and the Binder number due to finite MC length are discussed. A new method for calculating the equilibrium relaxation time from the analysis of the statistical dependence is presented. We apply it to the critical dynamics of the Ising model to estimate the dynamical critical exponent accurately.
We review our recent studies on the dynamical correlations in MC simulations from the view point of the statistical dependence. Attentions are paid to the reduction of the statistical degrees of freedom for correlated data. Possible biases on several cumulants, such as the susceptibility and the Binder number due to finite MC length are discussed. A new method for calculating the equilibrium relaxation time from the analysis of the statistical dependence is presented. We apply it to the critical dynamics of the Ising model to estimate the dynamical critical exponent accurately.
We derive twenty nontrivial terms of the high-temperature series expansion for the linear relaxation time τ of the time-displaced correlation function C(t) = ⟨ m (0) m(t) ⟩ of the magnetization m(t) in the two-dimensional nearest-neighbour ferromagnetic Ising model on the square lattice. We study the dynamics introduced by Glauber and compute the longest (characteristic) relaxation time of C(t) . We analyse the series by using unbiased and biased methods, such as the ratio method, Padé approximants and generalized differential approximants. It is reassuring that all the methods yield compatible results providing the estimate for the dynamical critical exponent: z = 2.183 ± 0.005.
From the non-equilibrium critical relaxation study of the two-dimensional Ising model, the dynamical critical exponent z is estimated to be 2.165 ± 0.010 for this model. The relaxation in the ordered phase of this model is consistent with exp (−√t/τ) behavior. The interface energy of the three-dimensional Ising model is studied and the critical exponent of the correlation length ν and the critical amplitude of the surface tension σ0 are estimated to be 0.6250 ± 0.0025 and 1.42 ± 0.04, respectively. A dynamic Monte Carlo renormalization group method is successfully applied to the equilibrium properties of the three-dimensional Ising model.
We review the current state of knowledge of dynamic critical phenomena in simple magnetic models and present high statistics Monte Carlo data for critical slowing down in Ising and Potts models in two and three dimensions. By fitting the relaxation functions to multi-exponential decays at the critical temperature and applying finite size scaling we are able to extract very accurate estimates for the dynamical critical exponent z. Our results are compared with previous numerical estimates and with the predictions of renormalization group theory.
The dynamic critical exponent z describes the divergence of correlation times near a continuous phase transition. Obtaining accurate estimates of z from noisy time series is complex, especially when computational resources are limited. We compare machine learning approaches for estimating z in the three-dimensional Ising model. Data are generated from Monte Carlo simulations on a standard laptop computer, requiring 15 to 20 min of runtime. Twelve time-series features are extracted from the simulation output. We compare linear regression, simple scaling, and ensemble methods. Linear regression achieves a root mean square error of 0.0164 on the validation set with a coefficient of determination R2=0.974. The integrated autocorrelation time logτ shows a near-perfect correlation with z (the correlation value r=0.992), consistent with the system linear size scaling relation τ∼Lz. Autocorrelation features at three lags considered show correlations with z between 0.72 and 0.87. Standardised coefficient analysis indicates that the standard deviation of successive differences and the standard deviation of magnetisation are the dominant predictors in the standardised model, with coefficients of −0.246 and 0.239, respectively. Accurate exponent estimation is possible with minimal computational resources, and fluctuation-based features capture critical slowing down effectively.