Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

International Journal of Modern Physics C

LARGE SCALE SIMULATIONS OF THE KINETIC ISING MODEL

Christian Münkel

Published Dec 1, 1993• 4 citations
Publisher

Abstract

We present Monte Carlo simulation results for the dynamical critical exponent z of the two- and three-dimensional kinetic Ising model. The z-values were calculated from the magnetization relaxation from an ordered state into the equilibrium state at Tc for very large systems with up to (169984)2 and (3072)3 spins. To our knowledge, these are the largest Ising-systems simulated todate. We also report the successful simulation of very large lattices on a massively parallel MIMD computer with high speedups of approximately 1000 and an efficiency of about 0.93.

DOI
10.1142/s0129183193000896

Related papers

HIGH SPEED ALGORITHMS AND SIMULATIONS OF THE CRITICAL DYNAMICS OF THE ISING MODEL

Hans-Otto Heuer

We review the development of Monte Carlo algorithms for the simulation of the Ising model with local dynamics. The multi lattice coding algorithm for the pure and for the site-disordered Ising model is described in detail. Application of these algorithms on fast vector computers have led to new progress concerning the critical dynamics of the pure Ising model leading to the critical exponent z = 2:05. The importance of disorder for the critical dynamics of the Ising model has been investigated recently. We review the numerical results and compare with theoretical predictions. A crossover scenario is found as for the static behavior with an effective dynamical exponent z = 1:75 fl. 1 Introduction The Ising model is a standard model in statistical physics for the description of a ferromagnet in an extremely simplified manner. It is assumed that quantum spin operators may be replaced by classical spins S i = \\Sigma 1 which are oriented parallel or antiparallel to a magnetic filed H: H...

WORLD SCIENTIFIC eBooks•May 1, 1996•1 citation

Vectorized Monte Carlo simulation of large Ising models near the critical point

Masatoshi Mori, Yoshinori Tsuda

The relaxation of the two-dimensional Ising model above ${T}_{c}$ for lattices with size up to 10 240\ifmmode\times\else\texttimes\fi{}10 240 is studied by the Monte Carlo simulation in vectorized super-spin-coding on the HITAC S-810/20 supercomputer. The present estimation of the dynamic critical exponent z gives z=2.076\ifmmode\pm\else\textpm\fi{}0.005, which is equal to the critical exponent ${\ensuremath{\Delta}}^{(l)}$ of the linear relaxation time in a two-dimensional system. Also, the critical exponent, ${\ensuremath{\Delta}}^{(\mathrm{nl})}$ of the nonlinear relaxation time is obtained: ${\ensuremath{\Delta}}^{(\mathrm{nl})=1.932\ifmmode\pm\else\textpm\fi{}0.018}$. These results support the Racz scaling law ${\ensuremath{\Delta}}^{(l)}$=${\ensuremath{\Delta}}^{(\mathrm{nl})+\mathrm{\ensuremath{\beta}}}$ (\ensuremath{\beta} is the critical exponent of the order parameter).

Physical review. B, Condensed matter•Apr 1, 1988•32 citations

Monte Carlo investigation of critical dynamics in the three-dimensional Ising model

Stephan Wansleben, D. P. Landau

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{}${10}^{12}$ 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.

Physical review. B, Condensed matter•Mar 1, 1991•129 citations

Linear relaxation in large two-dimensional Ising models

Yue Lin, Fuming Wang

Critical dynamics in two-dimension Ising lattices up to 2048×2048 is simulated on field-programmable-gate-array- based computing devices. Linear relaxation times are measured from extremely long Monte Carlo simulations. The longest simulation has 7.1×10(16) spin updates, which would take over 37 years to simulate on a general purpose computer. The linear relaxation time of the Ising lattices is found to follow the dynamic scaling law for correlation lengths as long as 2048. The dynamic exponent z of the system is found to be 2.179(12), which is consistent with previous studies of Ising lattices with shorter correlation lengths. It is also found that Monte Carlo simulations of critical dynamics in Ising lattices larger than 512×512 are very sensitive to the statistical correlations between pseudorandom numbers, making it even more difficult to study such large systems.

Physical review. E•Feb 8, 2016•20 citations

Nonequilibrium relaxation of the two-dimensional Ising model: Series-expansion and Monte Carlo studies

Jian‐Sheng Wang, Chee Kwan Gan

We study the critical relaxation of the two-dimensional Ising model from a fully ordered configuration by series expansion in time $t$ and by Monte Carlo simulation. Both the magnetization $(m)$ and energy series are obtained up to 12th order. An accurate estimate from series analysis for the dynamical critical exponent $z$ is difficult but compatible with 2.2. We also use Monte Carlo simulation to determine an effective exponent, ${z}_{\mathrm{eff}}(t)=\ensuremath{-}\frac{1}{8}d\mathrm{ln}t/d\mathrm{ln}m$, directly from a ratio of three-spin correlation to $m$. Extrapolation to $\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty}$ leads to an estimate $z=2.169\ifmmode\pm\else\textpm\fi{}0.003$.

arXiv:cond-mat/9712198•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Jun 1, 1998•20 citations

Accurate Estimation of the Dynamical Critical Exponent of the Two-Dimensional Kinetic Ising Model Based on the Ising Machinem-TIS

Nobuyasu Ito, Makoto Taiji, Masuo Suzuki

The dynamical critical exponent, z , of the two-dimensional kinetic Ising model is estimated by the Monte Carlo method using finite-size scaling. The Monte Carlo simulalion is performed on a special-purpose computer system, m -TIS, designed and constructed by the present authors. The obtained value of z is 2.132±0.008. This result supports the conjecture by Racz and Collins that z is equal to 17/8 in the two-dimensional kinetic Ising model (Racz and Collins; Rhys. Rev. B 13 (1976) 3074).

Journal of the Physical Society of Japan•Dec 15, 1987•24 citations
We present Monte Carlo simulation results for the dynamical critical exponent z of the two- and three-dimensional kinetic Ising model. The z-values were calculated from the magnetization relaxation from an ordered state into the equilibrium state at Tc for very large systems with up to (169984)2 and (3072)3 spins. To our knowledge, these are the largest Ising-systems simulated todate. We also report the successful simulation of very large lattices on a massively parallel MIMD computer with high speedups of approximately 1000 and an efficiency of about 0.93.
We review the development of Monte Carlo algorithms for the simulation of the Ising model with local dynamics. The multi lattice coding algorithm for the pure and for the site-disordered Ising model is described in detail. Application of these algorithms on fast vector computers have led to new progress concerning the critical dynamics of the pure Ising model leading to the critical exponent z = 2:05. The importance of disorder for the critical dynamics of the Ising model has been investigated recently. We review the numerical results and compare with theoretical predictions. A crossover scenario is found as for the static behavior with an effective dynamical exponent z = 1:75 fl. 1 Introduction The Ising model is a standard model in statistical physics for the description of a ferromagnet in an extremely simplified manner. It is assumed that quantum spin operators may be replaced by classical spins S i = \\Sigma 1 which are oriented parallel or antiparallel to a magnetic filed H: H...
The relaxation of the two-dimensional Ising model above Tc{T}_{c}Tc​ for lattices with size up to 10 240\ifmmode\times\else\texttimes\fi{}10 240 is studied by the Monte Carlo simulation in vectorized super-spin-coding on the HITAC S-810/20 supercomputer. The present estimation of the dynamic critical exponent z gives z=2.076\ifmmode\pm\else\textpm\fi{}0.005, which is equal to the critical exponent ${\ensuremath{\Delta}}^{(l)}$ of the linear relaxation time in a two-dimensional system. Also, the critical exponent, ${\ensuremath{\Delta}}^{(\mathrm{nl})}$ of the nonlinear relaxation time is obtained: ${\ensuremath{\Delta}}^{(\mathrm{nl})=1.932\ifmmode\pm\else\textpm\fi{}0.018}$. These results support the Racz scaling law ${\ensuremath{\Delta}}^{(l)}$=${\ensuremath{\Delta}}^{(\mathrm{nl})+\mathrm{\ensuremath{\beta}}}$ (\ensuremath{\beta} is the critical exponent of the order parameter).
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{10}^{12}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.
Critical dynamics in two-dimension Ising lattices up to 2048×2048 is simulated on field-programmable-gate-array- based computing devices. Linear relaxation times are measured from extremely long Monte Carlo simulations. The longest simulation has 7.1×10(16) spin updates, which would take over 37 years to simulate on a general purpose computer. The linear relaxation time of the Ising lattices is found to follow the dynamic scaling law for correlation lengths as long as 2048. The dynamic exponent z of the system is found to be 2.179(12), which is consistent with previous studies of Ising lattices with shorter correlation lengths. It is also found that Monte Carlo simulations of critical dynamics in Ising lattices larger than 512×512 are very sensitive to the statistical correlations between pseudorandom numbers, making it even more difficult to study such large systems.
We study the critical relaxation of the two-dimensional Ising model from a fully ordered configuration by series expansion in time ttt and by Monte Carlo simulation. Both the magnetization (m)(m)(m) and energy series are obtained up to 12th order. An accurate estimate from series analysis for the dynamical critical exponent zzz is difficult but compatible with 2.2. We also use Monte Carlo simulation to determine an effective exponent, ${z}_{\mathrm{eff}}(t)=\ensuremath{-}\frac{1}{8}d\mathrm{ln}t/d\mathrm{ln}m$, directly from a ratio of three-spin correlation to mmm. Extrapolation to $\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty}$ leads to an estimate $z=2.169\ifmmode\pm\else\textpm\fi{}0.003$.