Physical Review Letters

Order-Parameter Relaxation Times of Finite Three-Dimensional Ising-like Systems

Wulf Koch, V. Dohm, Dietrich Stauffer

Published Aug 26, 1996• 27 citations

Abstract

We present field-theoretic and numerical studies of finite-size effects on the exponential relaxation times ${\ensuremath{\tau}}_{1}$ and ${\ensuremath{\tau}}_{2}$ of the order parameter and the square of the order parameter near the critical point of three-dimensional Ising-like systems. Finite-size scaling functions of ${\ensuremath{\tau}}_{1}$ and ${\ensuremath{\tau}}_{2}$ are calculated for the relaxational dynamics of the ${\ensuremath{\phi}}^{4}$ model in cubic geometry with periodic boundary conditions. At ${T}_{c}$ we predict the universal ratio $\frac{{\ensuremath{\tau}}_{1}}{{\ensuremath{\tau}}_{2}}=5.4$. Below ${T}_{c}$, a maximum of ${\ensuremath{\tau}}_{2}$ is predicted. New Monte Carlo data for the Ising model are presented which are in good agreement with the predictions.

DOI: 10.1103/physrevlett.77.1789

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Physical Review Letters

Order-Parameter Relaxation Times of Finite Three-Dimensional Ising-like Systems

Wulf Koch, V. Dohm, Dietrich Stauffer

Published Aug 26, 1996• 27 citations

Publisher

Abstract

We present field-theoretic and numerical studies of finite-size effects on the exponential relaxation times

{\ensuremath{\tau}}_{1}

and

{\ensuremath{\tau}}_{2}

of the order parameter and the square of the order parameter near the critical point of three-dimensional Ising-like systems. Finite-size scaling functions of

{\ensuremath{\tau}}_{1}

and

{\ensuremath{\tau}}_{2}

are calculated for the relaxational dynamics of the

{\ensuremath{\phi}}^{4}

model in cubic geometry with periodic boundary conditions. At

{T}_{c}

we predict the universal ratio

\frac{{\ensuremath{\tau}}_{1}}{{\ensuremath{\tau}}_{2}}=5.4

. Below

{T}_{c}

, a maximum of

{\ensuremath{\tau}}_{2}

is predicted. New Monte Carlo data for the Ising model are presented which are in good agreement with the predictions.

DOI: 10.1103/physrevlett.77.1789

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M. M. Tsypin, Henk W. J. Blöte

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Rudolf Hilfer A, Nigel B. Wilding

Critical finite-size scaling functions for the order-parameter distribution of the two- and three-dimensional Ising model are investigated. Within a recently introduced classification theory of phase transitions the universal part of critical finite-size scaling functions has been derived by employing a scaling limit which differs from the traditional finite-size scaling limit. In this paper the analytical predictions are compared with Monte Carlo simulation results. We find good agreement between the analytical expression and the simulation results. The agreement is consistent with the possibility that the functional form of the critical finite-size scaling function for the order-parameter distribution is determined uniquely by only a few universal parameters, most notably the equation of state exponent.Critical finite-size scaling functions for the order-parameter distribution of the two- and three-dimensional Ising model are investigated. Within a recently introduced classification theory of phase transitions the universal part of critical finite-size scaling functions has been derived by employing a scaling limit which differs from the traditional finite-size scaling limit. In this paper the analytical predictions are compared with Monte Carlo simulation results. We find good agreement between the analytical expression and the simulation results. The agreement is consistent with the possibility that the functional form of the critical finite-size scaling function for the order-parameter distribution is determined uniquely by only a few universal parameters, most notably the equation of state exponent.

arXiv:cond-mat/9503157Journal of Physics A Mathematical and GeneralMay 21, 199562 citations

We present field-theoretic and numerical studies of finite-size effects on the exponential relaxation times

{\ensuremath{\tau}}_{1}

and

{\ensuremath{\tau}}_{2}

of the order parameter and the square of the order parameter near the critical point of three-dimensional Ising-like systems. Finite-size scaling functions of

{\ensuremath{\tau}}_{1}

and

{\ensuremath{\tau}}_{2}

are calculated for the relaxational dynamics of the

{\ensuremath{\phi}}^{4}

model in cubic geometry with periodic boundary conditions. At

{T}_{c}

we predict the universal ratio

\frac{{\ensuremath{\tau}}_{1}}{{\ensuremath{\tau}}_{2}}=5.4

. Below

{T}_{c}

, a maximum of

{\ensuremath{\tau}}_{2}

is predicted. New Monte Carlo data for the Ising model are presented which are in good agreement with the predictions.

We present an analytic study of finite-size effects on critical diffusion above and below

{T}_{c}

of three-dimensional Ising-like systems whose order parameter is coupled to a conserved density. We also calculate the finite-size relaxation time that governs the critical order-parameter relaxation towards a metastable equilibrium state below

{T}_{c}.

Two universal dynamic amplitude ratios at

{T}_{c}

are predicted and quantitative predictions of dynamic finite-size scaling functions are given that can be tested by Monte Carlo simulations.

{\mathit{T}}_{\mathit{c}}

=(0.6929${)}^{\mathrm{\ensuremath{-}}1}$, \ensuremath{\tau} grows like \ensuremath{\tau}\ensuremath{\simeq}

{\mathit{L}}^{\mathit{z}}

with z=1.96(6). In the paramagnetic region we show that the relaxation time is represented by a universal scaling function \ensuremath{\tau}${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{z}}$ vs (T/

{\mathit{T}}_{\mathit{c}}

-1)${\mathit{L}}^{1/\ensuremath{\nu}}$ with z=1.96(6). This estimate is consistent with theoretical predictions and results of experimental studies concerning the dynamical critical behavior of Heisenberg magnets perturbed by order-parameter nonconserving dipolar interactions. We also describe how a simple over-relaxation algorithm of Creutz and of Brown and Woch can be incorporated into the Metropolis sampling method for the classical Heisenberg model, thereby significantly reducing the critical slowing down; the resultant correlation time varies as \ensuremath{\tau}\ensuremath{\sim}(T/

{\mathit{T}}_{\mathit{c}}

-1${)}^{\mathrm{\ensuremath{-}}0.77}$, which implies z\ensuremath{\simeq}1.10.

{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.