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arXiv:cond-mat/0510731 · Journal of Physics A Mathematical and General

Spin, bond and global fluctuation–dissipation relations in the non-equilibrium spherical ferromagnet

Alessia Annibale, Peter Sollich A

Published Mar 8, 2006• 30 citations
arXivHTMLPDF

Abstract

We study the out-of-equilibrium dynamics of the spherical ferromagnet after a quench to its critical temperature. We calculate correlation and response functions for spin observables which probe length scales much larger than the lattice spacing but smaller than the system size, and find that the asymptotic fluctuation–dissipation ratio (FDR) X ∞ is the same as for local observables. This is consistent with our earlier results for the Ising model in dimension d = 1 and d = 2. We also check that bond observables, both local and long range, give the same asymptotic FDR. In the second part of the paper the analysis is extended to global observables, which probe correlations among all N spins. Here, non-Gaussian fluctuations arising from the spherical constraint need to be accounted for, and we develop a systematic expansion in to do this. Applying this to the global bond observable, i.e. the energy, we find that non-Gaussian corrections change its FDR to a nontrivial value which we calculate exactly for all dimensions d > 2. Finally, we consider quenches from magnetized initial states. Here, even the FDR for the global spin observable, i.e. the magnetization, is nontrivial. It differs from the one for unmagnetized states even in d > 4, signalling the appearance of a distinct dynamical universality class of magnetized critical coarsening. For lower d , the FDR is irrational even to first order in 4 − d and d − 2, the latter in contrast to recent results for the transverse FDR in the n -vector model.

DOI
10.1088/0305-4470/39/12/002

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We study the out-of-equilibrium dynamics of the spherical ferromagnet after a quench to its critical temperature. We calculate correlation and response functions for spin observables which probe length scales much larger than the lattice spacing but smaller than the system size, and find that the asymptotic fluctuation–dissipation ratio (FDR) X ∞ is the same as for local observables. This is consistent with our earlier results for the Ising model in dimension d = 1 and d = 2. We also check that bond observables, both local and long range, give the same asymptotic FDR. In the second part of the paper the analysis is extended to global observables, which probe correlations among all N spins. Here, non-Gaussian fluctuations arising from the spherical constraint need to be accounted for, and we develop a systematic expansion in to do this. Applying this to the global bond observable, i.e. the energy, we find that non-Gaussian corrections change its FDR to a nontrivial value which we calculate exactly for all dimensions d > 2. Finally, we consider quenches from magnetized initial states. Here, even the FDR for the global spin observable, i.e. the magnetization, is nontrivial. It differs from the one for unmagnetized states even in d > 4, signalling the appearance of a distinct dynamical universality class of magnetized critical coarsening. For lower d , the FDR is irrational even to first order in 4 − d and d − 2, the latter in contrast to recent results for the transverse FDR in the n -vector model.
We study the non-equilibrium dynamics of the spherical ferromagnet quenched to its critical temperature, as a function of the magnetization of the initial state. The two limits of unmagnetized and fully magnetized initial conditions can be understood as corresponding to times that are respectively much shorter and much longer than a magnetization timescale, as in a recent field theoretical analysis of the n -vector model. We calculate exactly the crossover functions interpolating between these two limits, for the magnetization correlator and response and the resulting fluctuation–dissipation ratio (FDR). For d > 4 our results match those obtained recently from a Gaussian field theory. For d < 4, non-Gaussian fluctuations arising from the spherical constraint need to be accounted for. We extend our framework from the fully magnetized case to achieve this, providing an exact solution for the relevant integral kernel. The resulting crossover behaviour is very rich, with the asymptotic FDR X ∞ depending non-monotonically on the scaled age of the system. This is traced back to non-monotonicities of the two-time correlator, themselves the consequence of large magnetization fluctuations on the crossover timescale. We correct a trivial error in our earlier calculation for fully magnetized initial states; the corrected FDR is consistent with renormalization group expansions to first order in 4 − d for the longitudinal fluctuations of the O ( n ) model in the limit n → ∞ .
We calculate analytically the fluctuation-dissipation ratio (FDR) for Ising ferromagnets quenched to criticality, both for the long-range model and its short-range analog in the limit of large dimension. Our exact solution shows that, for both models, if the system is unmagnetized while if the initial magnetization is nonzero. This indicates that two different classes of critical coarsening dynamics need to be distinguished depending on the initial conditions, each with its own nontrivial FDR. We also analyze the dependence of the FDR on whether local and global observables are used. These results clarify how a proper local FDR (and the corresponding effective temperature) should be defined in long-range models in order to avoid spurious inconsistencies and maintain the expected correspondence between local and global results; global observables turn out to be far more robust tools for detecting nonequilibrium FDRs.
Within the universality class of ferromagnetic vector models with O(n) symmetry and purely dissipative dynamics, we study the non-equilibrium critical relaxation from a magnetized initial state. Transverse correlation and response functions are exactly computed for Gaussian fluctuations and in the limit of infinite number n of components of the order parameter. We find that the fluctuation–dissipation ratios (FDRs) for longitudinal and transverse modes differ already at the Gaussian level. In these two exactly solvable cases we completely describe the crossover from the short-time to the long-time behaviour, corresponding to a disordered and a magnetized initial condition, respectively. The effects of non-Gaussian fluctuations on longitudinal and transverse quantities are calculated in the first order in the -expansion ( = 4−d) and reliable three-dimensional estimates of the two FDRs are obtained.
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.
Within the universality class of ferromagnetic vector models with O( n ) symmetry and purely dissipative dynamics, we study the non-equilibrium critical relaxation from a magnetized initial state. Transverse correlation and response functions are exactly computed for Gaussian fluctuations and in the limit of infinite number n of components of the order parameter. We find that the fluctuation–dissipation ratios (FDRs) for longitudinal and transverse modes differ already at the Gaussian level. In these two exactly solvable cases we completely describe the crossover from the short-time to the long-time behaviour, corresponding to a disordered and a magnetized initial condition, respectively. The effects of non-Gaussian fluctuations on longitudinal and transverse quantities are calculated in the first order in the -expansion ( = 4− d ) and reliable three-dimensional estimates of the two FDRs are obtained.
The two-time nonequilibrium correlation and response functions in 1D kinetic classical spin systems with non-conserved dynamics and quenched to their zero-temperature critical point are studied. The exact solution of the kinetic Ising model with Glauber dynamics for a wide class of initial states allows for an explicit test of the universality of the non-equilibrium limit fluctuation-dissipation ratio X_{\infty}. It is shown that the value of X_{\infty} depends on whether the initial state has finitely many domain walls or not and thus two distinct dynamic universality classes can be identified in this model. Generic 1D kinetic spin systems with non-conserved dynamics fall into the same universality classes as the kinetic Glauber-Ising model provided the dynamics is invariant under the C-symmetry of simultaneous spin and magnetic-field reversal. While C-symmetry is satisfied for magnetic systems, it need not be for lattice gases which may therefore display hitherto unexplored types of non-universal kinetics.