Abstract We discuss how to calculate non-equilibrium universal amplitude ratios in the functional renormalization group approach, extending its applicability. In particular, we focus on the critical relaxation of the Ising model with non-conserved dynamics (model A) and calculate the universal amplitude ratio associated with the fluctuation–dissipation ratio of the order parameter, considering a critical quench from a high-temperature initial condition. Our predictions turn out to be in good agreement with previous perturbative renormalization-group calculations and Monte Carlo simulations.
Abstract We discuss how to calculate non-equilibrium universal amplitude ratios in the functional renormalization group approach, extending its applicability. In particular, we focus on the critical relaxation of the Ising model with non-conserved dynamics (model A) and calculate the universal amplitude ratio associated with the fluctuation–dissipation ratio of the order parameter, considering a critical quench from a high-temperature initial condition. Our predictions turn out to be in good agreement with previous perturbative renormalization-group calculations and Monte Carlo simulations.
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 now begin our exposition of universal dynamic scaling behavior that emerges under non-equilibrium conditions. We are mainly concerned with the relaxational models A and B here, but also touch on other dynamic universality classes. First, we consider non-equilibrium critical relaxation from a disordered initial state, and compute the associated universal scaling exponents within the dynamic renormalization group framework. Related phenomena in the early-time regime, prior to reaching temporally translationally invariant asymptotics, are ‘aging’ and a non-equilibrium fluctuation-dissipation ratio, both of which have enjoyed prominence in the literature on ‘glassy’ kinetics. In this context, we derive the critical initial slip exponent for the order parameter, and briefly discuss interesting persistence properties. Second, we explore the long-time scaling laws in phase ordering kinetics and coarsening for the relaxational models, following a fast temperature ‘quench’ into the ordered phase, in systems with either non-conserved or conserved order parameters. Aside from a few explicit computations in the spherical model limit (wherein the number of order parameter components n → ∞), we largely employ phenomenological considerations and scaling theory, stressing the importance of topological defects for energy dissipation during coarsening. Next we address the question how violations of Einstein's relation that links relaxation coefficients and noise correlations might affect the asymptotic dynamic critical scaling behavior. Whereas in systems with non-conserved order parameter typically the equilibrium scaling laws are recovered in the vicinity of the critical point, in contrast genuinely novel universality classes may emerge in the case of a conserved parameter, provided it is driven out of equilibrium in a spatially anisotropic fashion.
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.
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.
We present a method to calculate short-time nonequilibrium universal exponents within the functional-renormalization-group scheme. As an example, we consider the classical critical dynamics of the relaxational model A after a quench of the temperature of the system and calculate the initial-slip exponent which characterizes the nonequilibrium universal short-time behavior of both the order parameter and correlation functions. The value of this exponent is found to be consistent with the result of a perturbative dimensional expansion and of Monte Carlo simulations in three spatial dimensions.
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.