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Journal of Statistical Mechanics Theory and Experiment

Influence of disorder on ageing and memory effects in non-equilibrium critical dynamics of 3D Ising model relaxing from an ordered state

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

Published Apr 1, 2016• 12 citations
Publisher

Abstract

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 .

DOI
10.1088/1742-5468/2016/04/043303

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•Jan 1, 2005•153 citations
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 .
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∞​|⁠.
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 .
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).
The Monte Carlo study of the nonequilibrium critical evolution of structurally disordered anisotropic magnetic materials from the low-temperature initial state with the reduced magnetization m 0 = 1 is performed within the broad range of spin densities, p = 1.0, 0.95, 0.8, 0.6, and 0.5. It is shown that, in such systems, the pinning of domain walls by structural defects occurring when the evolution starts from the low-temperature state leads to significant changes in the nonequilibrium “aging” and “memory” effects in comparison to those characteristic of the “pure” system. As a result, in the long-term regime at times t–t w » t w » 1, an anomalously strong slowing down in the correlation effects is revealed. It is shown that a decrease in the autocorrelation function with time occurs according to a power law typical of the critical relaxation of the magnetization in contrast to a usual scaling dependence. Eventually, the limiting value of the fluctuation–dissipation ratio X∞ for the structurally disordered systems with p < 1 vanishes, whereas for the pure system, we have X∞ = 0.784(5). The nonequilibrium critical “superaging” stage is found. This stage is characterized by the critical exponent µ = 2.30(6) for weakly disordered systems and by µ = 2.80(7) for the systems with strong disorder.
The effect of various initial magnetizations m0 and structural defects the nonequilibrium critical behavior of the three-dimensional Ising model is numerically studied. Based on an analysis of the time dependence of the magnetization and the two-time dependence of the autocorrelation function and dynamic susceptibility, the significant effect of initial states on relaxation magnetizations and aging effects characterized by anomalous relaxation inhibition and correlation in the system with increasing waiting time was revealed. The fluctuation–dissipation theorem violation was studied, and the values of the limit fluctuation–dissipation ratio (FDR) are calculated. It is shown that two universality subclasses can be distinguished in the nonequilibrium critical behavior of the three-dimensional Ising model with random initial magnetization m0 These subclasses correspond to the system evolution from the high-temperature (m0 = 0) and low-temperature (m0 = 1) initial states with limit FDRs characteristic of these states.
In the past few years systems with slow dynamics have attracted considerable theoretical and experimental interest. Ageing phenomena are observed during this ever-lasting non-equilibrium evolution. A simple instance of such a behaviour is provided by the dynamics that takes place when a system is quenched from its high-temperature phase to the critical point. The aim of this review is to summarize the various numerical and analytical results that have been recently obtained for this case. Particular emphasis is put to the field-theoretical methods that can be used to provide analytical predictions for the relevant dynamical quantities. Fluctuation-dissipation relations are discussed and in particular the concept of fluctuation-dissipation ratio (FDR) is reviewed, emphasizing its connection with the definition of a possible effective temperature. The Renormalization-Group approach to critical dynamics is summarized and the scaling forms of the time-dependent non-equilibrium correlation and response functions of a generic observable are discussed. From them the universality of the associated FDR follows as an amplitude ratio. It is then possible to provide predictions for ageing quantities in a variety of different models. In particular the results for Model A, B, and C dynamics of the O(N) Ginzburg-Landau Hamiltonian, and Model A dynamics of the weakly dilute Ising magnet and of a \phi^3 theory, are reviewed and compared with the available numerical results and exact solutions. The effect of a planar surface on the ageing behaviour of Model A dynamics is also addressed within the mean-field approximation.