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Journal of Physics Conference Series

Ageing properties of three-dimensional pure and site-diluted Ising ferromagnets

Vladimir V. Prudnikov, Pavel Vladimirovich Prudnikov, Evgenii A. Pospelov

Published May 15, 2014• 3 citations
Publisher

Abstract

Ageing properties of the three-dimensional pure and diluted Ising models are investigated for case with quenching from a high temperature initial state to their critical points. The two-times autocorrelation and response functions for the systems with spin concentrations p = 1.0, 0.8, and 0.6 are computed and analysed. The fluctuation-dissipation theorem violation for considered 3D Ising models are shown with the asymptotic values of the fluctuation-dissipation ratio X ∞ ( p ) < 1/2. The asymptotic values of universal fluctuation-dissipation ratio X ∞ were measured, and it was shown that X ∞ disorder ( p < 1) > X ∞ pure ( p = 1).

DOI
10.1088/1742-6596/510/1/012015

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Ageing properties of the three-dimensional pure and diluted Ising models are investigated for case with quenching from a high temperature initial state to their critical points. The two-times autocorrelation and response functions for the systems with spin concentrations p = 1.0, 0.8, and 0.6 are computed and analysed. The fluctuation-dissipation theorem violation for considered 3D Ising models are shown with the asymptotic values of the fluctuation-dissipation ratio X ∞ ( p ) < 1/2. The asymptotic values of universal fluctuation-dissipation ratio X ∞ were measured, and it was shown that X ∞ disorder ( p < 1) > X ∞ pure ( p = 1).
Abstract We have performed a numerical investigation of the influence of disorder on the dynamical non-equilibrium evolution of a 3D site-diluted Ising model from a low-temperature initial state with magnetization m 0 = 1. It is shown that two-time dependences of the autocorrelation and integrated response functions for systems with spin concentrations p = 1.0, 0.95, 0.8, 0.6 and 0.5 demonstrate ageing properties with anomalous slowing-down relaxation and violation of the fluctuation-dissipation ratio. It was revealed that during non-equilibrium critical dynamics in the long-time regime t − t w ≫ t w ≫ 1 the autocorrelation functions for diluted systems are extremely slow due to the pinning of domain walls on impurity sites. We have found that the fluctuation-dissipation ratio X ∞ = 0 for diluted systems with spin concentration p < 1 while the pure system is characterized by X ∞ = 0.784 ( 7 ) . The autocorrelation function power-law delay becomes the same as for the time dependence of the magnetization in the critical point and is characterized by exponent − β / z ν . Also, for diluted systems we reveal memory effects for critical evolution in the ageing regime with realization of cyclic temperature change and quenching at T < T c .
It is considered the non-equilibrium critical evolution of statistical systems which displays some features, such as ageing and violation of the fluctuation-dissipation theorem. We review here some theoretical results of computations that have been obtained in recent years for universal quantities, such as the exponents determining the scaling behaviour of dynamic response and correlation functions and the fluctuation-dissipation ratio, associated with the non-equilibrium critical dynamics, with particular focus on the 3D pure and diluted Ising models with Glauber dynamics. We analyse an influence of critical fluctuations, different non-equilibrium initial states and presence of nonmagnetic impurities in spin systems on two-time dependence of correlation and response functions on characteristic time variables as waiting time t w and time of observation t – t w with t > t w . We discuss the obtained values of non-equilibrium exponents for autocorrelation and response functions and values of the universal long-time limit of the fluctuation-dissipation ratio X ∞ . Analysis of simulation results show that the insertion of disorder leads to new values of X ∞ with X ∞ diluted > X ∞ pure .
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∞​|⁠.
We study the dynamics of ferromagnetic spin systems quenched from infinite temperature to their critical point. We perform an exact analysis of the spherical model in any dimension D >2 and numerical simulations on the two-dimensional Ising model. These systems are shown to be ageing in the long-time regime, i.e. their two-time autocorrelation and response functions, and associated fluctuation-dissipation ratio, are non-trivial scaling functions of both time variables. We show in particular that, for 1<< s (waiting time) << t (observation time), the fluctuation-dissipation ratio possesses a non-trivial limit value X ∞ , which appears as a dimensionless amplitude ratio, and is therefore a novel universal characteristic of non-equilibrium critical dynamics. For the spherical model, we obtain X ∞ = 1-2/ D for 2< D <4 and X ∞ = ½ for D >4 (mean-field regime). For the two-dimensional Ising model we measure X ∞ ≈0.26±0.01.
We study the purely relaxational dynamics (model A) at criticality in three-dimensional disordered Ising systems whose static critical behaviour belongs to the randomly diluted Ising universality class. We consider the site-diluted and bond-diluted Ising models, and the ± J Ising model along the paramagnetic–ferromagnetic transition line. We perform Monte Carlo simulations at the critical point using the Metropolis algorithm and study the dynamic behaviour in equilibrium at various values of the disorder parameter. The results provide a robust evidence of the existence of a unique model-A dynamic universality class which describes the relaxational critical dynamics in all considered models. In particular, the analysis of the size dependence of suitably defined autocorrelation times at the critical point provides the estimate z = 2.35(2) for the universal dynamic critical exponent. We also study the off-equilibrium relaxational dynamics following a quench from to T c . In agreement with the field-theory scenario, the analysis of the off-equilibrium dynamic critical behaviour gives an estimate of z that is perfectly consistent with the equilibrium estimate z = 2.35(2).
We investigate, analytically near the dimension d(uc) =4 and numerically in d=3 , the nonequilibrium relaxational dynamics of the randomly diluted Ising model at criticality. Using the exact renormalization-group method to one loop, we compute the two times t, t(w) correlation function and fluctuation dissipation ratio (FDR) for any Fourier mode of the order parameter, of finite wave vector q . In the large time separation limit, the FDR is found to reach a nontrivial value X(infinity) independently of (small) q and coincide with the FDR associated to the total magnetization obtained previously. Explicit calculations in real space show that the FDR associated to the local magnetization converges, in the asymptotic limit, to this same value X(infinity). Through a Monte Carlo simulation, we compute the autocorrelation function in three dimensions, for different values of the dilution fraction p at T(c) (p) . Taking properly into account the corrections to scaling, we find, according to the renormalization-group predictions, that the autocorrelation exponent lambda(c) is independent of p . The analysis is complemented by a study of the nonequilibrium critical dynamics following a quench from a completely ordered state.