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Journal of Zhejiang University(Science Edition)

Study for aging phenomena of two-dimensional Ising and XY model

Lei Xiao-wei

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

With dynamic Monte Carlo simulations,aging phenomena of two-dimensional Ising model and dynamic XY model were studied numerically.Systems were quenched to their critical point from a disordered initial state.The autocorrelation function was calculated by heat-bath algorithm.Critical exponents were also calculated.It was confirmed that for the 2D XY model,there is a logarithmic correction to scaling.The universality proposal and their scale relation in aging phenomena were predicted.The scale relation agreed with renormalization group calculations.

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With dynamic Monte Carlo simulations,aging phenomena of two-dimensional Ising model and dynamic XY model were studied numerically.Systems were quenched to their critical point from a disordered initial state.The autocorrelation function was calculated by heat-bath algorithm.Critical exponents were also calculated.It was confirmed that for the 2D XY model,there is a logarithmic correction to scaling.The universality proposal and their scale relation in aging phenomena were predicted.The scale relation agreed with renormalization group calculations.
With a large-scale Monte Carlo simulation, the aging of the two-dimensional dynamic Ising and XY models is numerically studied. Starting from a disordered state, the systems are quenched to critical temperature Tc and then evolve according to dynamics of model A. By calculating the autocorrelation function A(t,t′) we observe the aging phenomenon for both spin systems. Then, relatively accurate values of the dynamic critical exponent λc and z are estimated. Especially, we confirm that for the two-dimensional XY model, there is a strong logarithmic correction to scaling in the dynamic relaxation starting from a completely disordered.
With Monte Carlo simulation method,the aging phenomenon of the 2D XY model is numerically studied.Starting from an ordered initial state,the system is suddenly quenched to critical temperature Tc.The two-time autocorrelation function A(t,t') is measured.For a dynamic process with an ordered initial state,the difficult problem is that accurate analysis is lagging behind the Monte Carlo simulation study.So many activities have been devoted to dynamic scaling form.According to the numerical experimental data of 2D XY non-equilibrium system,the relation between A(t,t') and ratio of spatial correlation length is detected.The aging phenomenon exhibits a dynamic scaling behavior induced by the time correlation.The aging phenomenon is characterized by the dynamic scale invariance.The aging phenomenon of 2D XY model is confirmed.Especially,the correction of scaling behavior is numerically verified,the form of dynamic scaling of A(t,t') is more creditable than existing literatures.For a dynamic process starting from an ordered initial state,the scaling form assumption fits the numerical data well.The scaling form holds not only at lower temperatures but also around Tc.This is because of the simulation temperature is close to the critical temperature.Based on the dynamic scaling,the experimental data indicate that more accurate values of the critical exponents can be obtained.In addition,the data collapse phenomenon is observed.
With Monte Carlo methods, we simulate dynamic relaxation processes of the two-dimensional XY model at the Berezinskii-Kosterlitz-Thouless phase transition temperature and below, starting from both ordered and disordered initial states. The two-time correlation function A(t',t) is measured, and aging phenomena are investigated. The power-law correction in the spatial correlation length xi(t) for relaxation with an ordered initial state and the logarithmic correction for relaxation with a disordered initial state are carefully analyzed. The scaling functions of A(t',t) are then extracted.
A Monte Carlo numerical simulation of the specific features of nonequilibrium critical behavior is carried out for the two-dimensional structurally disordered XY model during its evolution from a low-temperature initial state. On the basis of the analysis of the two-time dependence of autocorrelation functions and dynamic susceptibility for systems with spin concentrations of p = 1.0, 0.9, and 0.6, aging phenomena characterized by a slowing down of the relaxation system with increasing waiting time and the violation of the fluctuation–dissipation theorem (FDT) are revealed. The values of the universal limiting fluctuation–dissipation ratio (FDR) are obtained for the systems considered. As a result of the analysis of the two-time scaling dependence for spin–spin and connected spin autocorrelation functions, it is found that structural defects lead to subaging phenomena in the behavior of the spin–spin autocorrelation function and superaging phenomena in the behavior of the connected spin autocorrelation function.
We employ Monte Carlo simulations to study the nonequilibrium relaxation of driven Ising lattice gases in two dimensions. Whereas the temporal scaling of the density autocorrelation function in the nonequilibrium steady state does not allow a precise measurement of the critical exponents, these can be accurately determined from the aging scaling of the two-time autocorrelations and the order parameter evolution following a quench to the critical point. We obtain excellent agreement with renormalization group predictions based on the standard Langevin representation of driven Ising lattice gases.
We investigate the non-equilibrium critical evolution of statistical systems and describe of its some features, such as aging and violation of the fluctuation-dissipation theorem. We consider some theoretical results of computations for universal quantities that have been obtained in recent years, such as the exponents determining the scaling behavior of dynamic response and correlation functions and the fluctuation-dissipation ratio, associated with the non-equilibrium critical dynamics. We derive our original Monte Carlo simulation results for 3D pure and diluted Ising models with Glauber and Metropolis dynamics. For these models, we analyze the influence of critical fluctuations, different non-equilibrium initial states, and the presence of nonmagnetic impurities on the two-time dependence of correlation and response functions on characteristic time variables, such as waiting time |twt_{\rm w}tw​| and time of observation |t−twt - t_{\rm w}t−tw​| with |t>twt>t_{\rm w}t>tw​|⁠. We discuss the obtained values of the non-equilibrium exponents for the autocorrelation and response functions and values of the universal long-time limit of the fluctuation-dissipation ratio |X∞X^{\infty }X∞|⁠. Our simulation results demonstrate that the insertion of disorder leads to new values of |X∞X^{\infty }X∞| with |Xdiluted∞>Xpure∞X_{\rm diluted}^{\infty }>X_{\rm pure}^{\infty }Xdiluted∞​>Xpure∞​|⁠.