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arXiv:cond-mat/9802206 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Finite-size effects on critical diffusion and relaxation towards metastable equilibrium

Wulf Koch, V. Dohm

Published Aug 1, 1998• 21 citations
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Abstract

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.

DOI
10.1103/physreve.58.r1179

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We present an analytic study of finite-size effects on critical diffusion above and below Tc{T}_{c}Tc​ 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 Tc.{T}_{c}.Tc​. Two universal dynamic amplitude ratios at Tc{T}_{c}Tc​ are predicted and quantitative predictions of dynamic finite-size scaling functions are given that can be tested by Monte Carlo simulations.
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 Tc{T}_{c}Tc​ we predict the universal ratio $\frac{{\ensuremath{\tau}}_{1}}{{\ensuremath{\tau}}_{2}}=5.4$. Below Tc{T}_{c}Tc​, 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.
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.
We report a Monte Carlo study on the critical relaxation of the three-dimensional kinetic Ising model from a nonequilibrium initial state. It is shown that the details of the method, namely, a choice of the sampling procedure and a functional form of the transition probability do not affect an asymptotic power-law relaxation. The finite-size effects on the critical relaxation is also studied. Cross-over from the power-law relaxation to the exponential one is found, which is discussed in view of the dynamical finite-size scaling theory.
The finite-size-scaling method in the equilibrium Monte Carlo (MC) simulations and the finite-time-scaling method in the nonequilibrium relaxation simulations are compromised. MC time data of various physical quantities are scaled by the MC time data of the dynamic correlation length. It corresponds to changing the system size in the finite-size-scaling method. This scaling method is tested in the three-dimensional ferromagnetic Ising model and in the three-dimensional $\ifmmode\pm\else\textpm\fi{}J$ Ising spin-glass model. The transition temperature and the critical exponents are obtained. We also comment on the definition of the dynamic correlation length in the nonequilibrium relaxation process. The Ornstein-Zernike formula is not always appropriate.
We have carried out extensive Monte Carlo simulations of the dynamic critical behavior of L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L simple-cubic classical Heisenberg models with periodic boundary conditions. The equilibrium relaxation time \ensuremath{\tau} is determined from the long-time exponential decay of the time-displaced correlation function of the magnetization. Using finite-size scaling and corrections to scaling we find that at the critical point Tc{\mathit{T}}_{\mathit{c}}Tc​=(0.6929${)}^{\mathrm{\ensuremath{-}}1}$, \ensuremath{\tau} grows like \ensuremath{\tau}\ensuremath{\simeq}Lz{\mathit{L}}^{\mathit{z}}Lz 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/Tc{\mathit{T}}_{\mathit{c}}Tc​-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/Tc{\mathit{T}}_{\mathit{c}}Tc​-1${)}^{\mathrm{\ensuremath{-}}0.77}$, which implies z\ensuremath{\simeq}1.10.
We investigate the large-time scaling regimes arising from a variety of metastable structures in a chain of Ising spins with both first- and second-neighbor couplings while subject to Kawasaki dynamics. Depending on the ratio and sign of these former, different dynamic exponents are suggested by finite-size scaling analyses of relaxation times. At low but nonzero temperatures these are calculated via exact diagonalizations of the evolution operator in finite chains under several activation barriers. In the absence of metastability the dynamics is always diffusive.