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arXiv:cond-mat/0412447 · Physical Review E

Universal aging properties at a disordered critical point

Grégory Schehr, Raja Paul

Published Jul 6, 2005• 36 citations
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Abstract

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.

DOI
10.1103/physreve.72.016105

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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.
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.
In the past few years, systems with slow dynamics have attracted considerable theoretical and experimental interest. Ageing phenomena are observed during this everlasting 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 on 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 models A, B and C dynamics of the O ( N ) Ginzburg–Landau Hamiltonian, and model A dynamics of the weakly dilute Ising magnet and of the φ 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.
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∞​|⁠.
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.
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 global persistence properties for the non-equilibrium critical dynamics of the randomly diluted Ising model. The disorder-averaged persistence probability c ( t ) of the global magnetization is found to decay algebraically with an exponent θ c that we compute analytically in a dimensional expansion in d = 4 − . Corrections to Markov process are found to occur already at one loop order and θ c is thus a novel exponent characterizing this disordered critical point. Our result is thoroughly compared with Monte Carlo simulations in d = 3, which also include a measurement of the initial slip exponent. Taking carefully into account corrections to scaling, θ c is found to be universal, independent of the dilution factor p along the critical line at T c ( p ), and in good agreement with our one-loop calculation.