We report the results of a Monte Carlo investigation of the (equilibrium) time-displaced correlation functions for the magnetization and energy of a simple cubic Ising model as a function of time, temperature, and lattice size. The simulations were carried out on a CDC CYBER 205 supercomputer employing a high-speed, vectorized multispin coding program and using a total of 5\ifmmode\times\else\texttimes\fi{}
1012 Monte Carlo spin-flip trials. We used L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L lattices with periodic boundary conditions and L as large as 96. The short-time and long-time behaviors of the correlation functions are analyzed by fits to a sum of exponential decays, and the critical exponent z for the largest relaxation time is extracted using a finite-size-scaling analysis. Our estimate z=2.04\ifmmode\pm\else\textpm\fi{}0.03 resolves an intriguing contradiction in the literature; it satisfies the theoretical lower bound and is in agreement with the prediction obtained by \ensuremath{\epsilon} expansion. We also consider various small systematic errors that typically occur in the analysis of relaxation functions and show how they can lead to spurious results if sufficient care is not exercised.
We study purely dissipative relaxational dynamics in the three-dimensional Ising universality class. To this end, we simulate the improved Blume-Capel model on the simple cubic lattice by using local algorithms. We perform a finite size scaling analysis of the integrated autocorrelation time of the magnetic susceptibility in equilibrium at the critical point. We obtain z=2.0245(15) for the dynamic critical exponent. As a complement, fully magnetized configurations are suddenly quenched to the critical temperature, giving consistent results for the dynamic critical exponent. Furthermore, our estimate of z is fully consistent with recent field theoretic results.
In this work we performed numerical simulations for the Ising model on three dimensional lattices with coordination number equal 5. With Monte Carlo simulations in the static case we evaluated the critical temperature and the static critical exponents
ν,
γ and
β. Once that we have the critical temperature value we investigated the dynamical critical behavior with Glauber dynamics, starting from disordered states. From our simulations we obtained the values for the dynamic exponent
z=2.037(8), the exponent for the autocorrelation
λ/z=1.364(5), the exponent of the critical initial increase
θ′=0.109(8) and the asymptotic value of the fluctuation-dissipation ratio
X∞=0.433(1). All of these results are in good agreement to the previous values reported for the 3D Ising model universality class.
The critical dynamics of `model A'' of Hohenberg and Halperin has been studied by the Monte Carlo method. Simulations have been carried out in the three-dimensional (3d) simple cubic Ising model for lattices of sizes
L=16 to
L=512. Using Wolff's cluster algorithm, the critical temperature is precisely found as ${\ensuremath{\beta}}_{c}=0.221\phantom{\rule{0.16em}{0ex}}654\phantom{\rule{0.16em}{0ex}}68(5)$. By Fourier transform of the lattice configurations, the critical scattering intensities $I(\stackrel{P\vec}{q}\phantom{\rule{0.16em}{0ex}})$ can be obtained. After circular averaging, the static simulations with
L=256 and
L=512 provide an estimate of the critical exponent $\ensuremath{\gamma}/\ensuremath{\nu}=2\ensuremath{-}\ensuremath{\eta}=1.9640(5)$. The $|\stackrel{P\vec}{q}\phantom{\rule{0.16em}{0ex}}|$-dependent distribution of $I(\stackrel{P\vec}{q}\phantom{\rule{0.16em}{0ex}})$ showed an exponential distribution, corresponding to a Gaussian distribution of the scattering amplitudes for a large
q domain. The time-dependent intensities were then used for the study of the critical dynamics of 3d lattices at the critical point. To simulate results of an x-ray photon correlation spectroscopy experiment, the time-dependent correlation function of the intensities was studied for each $|\stackrel{P\vec}{q}\phantom{\rule{0.16em}{0ex}}|$-value. In the
q region where $I(\stackrel{P\vec}{q}\phantom{\rule{0.16em}{0ex}})$ had an exponential distribution, the time correlations can be fit to a stretched exponential, where the exponent $\ensuremath{\mu}=\ensuremath{\gamma}/\ensuremath{\nu}z\ensuremath{\simeq}0.975$ provides an estimate of the dynamic exponent
z. This corresponds to
z=2.0145, in agreement with the observed variations of the characteristic fluctuation time of the intensity: $\ensuremath{\tau}(q)\ensuremath{\propto}{q}^{\ensuremath{-}z}$, which gives
z=2.015. These results agree with the $\ensuremath{\epsilon}$ expansion of field-theoretical methods (2.017). In this paper, the need to take account of the anomalous time behavior ($\ensuremath{\mu}<1$) in the dynamics is exemplified. This dynamics reflects a nonlinear time behavior of model A, and its large time extension is discussed in detail.
Comprehensive Monte Carlo simulations of the short-time dynamic behaviour are reported for the three-dimensional Ising model at criticality. Besides the exponent of the critical initial increase and the dynamic exponent z , the static critical exponents and as well as the critical temperature are determined from the power-law scaling behaviour of observables at the beginning of the time evolution. States of very high temperature as well as of zero temperature are used as initial states for the simulations.