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arXiv:cond-mat/0310545 · Physical Review E

Reply to “Comment on ‘Fluctuation-dissipation relations in the nonequilibrium critical dynamics of Ising models’ ”

Peter Mayer, Ludovic Berthier, Juan P. Garrahan, Peter Sollich A

Published Jul 2, 2004• 25 citations
arXivHTMLPDF

Abstract

We have recently shown that in nonequilibrium spin systems at criticality the limit ${X}^{\ensuremath{\infty}}$ of the fluctuation-dissipation ratio $X(t,{t}_{w})$ for $t⪢{t}_{w}⪢1$ can be measured using observables such as magnetization or energy [Phys. Rev. E 68, 016116 (2003)]. Pleimling argues in a Comment [preceding paper, Phys. Rev. 70, 018101 (2004)] on our paper that for such observables correlation and response functions are dominated by one-time quantities dependent only on $t$, and are therefore not suitable for a determination of ${X}^{\ensuremath{\infty}}$. Using standard scaling forms of correlation and response functions, as used by Pleimling, we show that our data do have a genuine two-time dependence and allow $X(t,{t}_{w})$ and ${X}^{\ensuremath{\infty}}$ to be measured, so that Pleimling's criticisms are easily refuted. We also compare with predictions from renormalization-group calculations, which are consistent with our numerical observation of a fluctuation-dissipation plot for the magnetization that is very close to a straight line. A key point remains that coherent observables make measurements of ${X}^{\ensuremath{\infty}}$ easier than the traditionally used incoherent ones, producing fluctuation-dissipation plots whose slope is close to ${X}^{\ensuremath{\infty}}$ over a much larger range.

DOI
10.1103/physreve.70.018102

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We have recently shown that in nonequilibrium spin systems at criticality the limit X\ensuremath∞{X}^{\ensuremath{\infty}}X\ensuremath∞ of the fluctuation-dissipation ratio X(t,tw)X(t,{t}_{w})X(t,tw​) for t⪢tw⪢1t⪢{t}_{w}⪢1t⪢tw​⪢1 can be measured using observables such as magnetization or energy [Phys. Rev. E 68, 016116 (2003)]. Pleimling argues in a Comment [preceding paper, Phys. Rev. 70, 018101 (2004)] on our paper that for such observables correlation and response functions are dominated by one-time quantities dependent only on ttt, and are therefore not suitable for a determination of X\ensuremath∞{X}^{\ensuremath{\infty}}X\ensuremath∞. Using standard scaling forms of correlation and response functions, as used by Pleimling, we show that our data do have a genuine two-time dependence and allow X(t,tw)X(t,{t}_{w})X(t,tw​) and X\ensuremath∞{X}^{\ensuremath{\infty}}X\ensuremath∞ to be measured, so that Pleimling's criticisms are easily refuted. We also compare with predictions from renormalization-group calculations, which are consistent with our numerical observation of a fluctuation-dissipation plot for the magnetization that is very close to a straight line. A key point remains that coherent observables make measurements of X\ensuremath∞{X}^{\ensuremath{\infty}}X\ensuremath∞ easier than the traditionally used incoherent ones, producing fluctuation-dissipation plots whose slope is close to X\ensuremath∞{X}^{\ensuremath{\infty}}X\ensuremath∞ over a much larger range.
Recently Phys. Rev. E 68, 016116 (2003)]] proposed an alternative way to compute numerically the fluctuation-dissipation ratios in nonequilibrium critical systems. Using well-known facts of nonequilibrium critical dynamics I show that the leading contributions of the quantities they consider are in fact one-time quantities which are independent of the waiting time. The ratio of these one-time quantities determines the slope of the straight lines observed in the fluctuation-dissipation plots of Mayer et al.
In this response we reply to the criticism of Nordblad et al. on our measurements of the dynamic repsonse of dilute metallic spin glasses. We argue that we have, in our experiments on dilute metallic spin glasses, separated the thermalization dynamics from the dynamic response to a change in applied field. This is contrary to experiments with observation times that exceed the characteristic thermalization time. On the basis of this observation we show that our interpretation of our experimental results is completely consistent with a recent model calculation of the dynamic response of the infinite-range Ising spin-glass model.
We reply to the comment on our published paper `` Universal Fluctuations in Correlated Systems'',Phys. Rev. Lett. Vol; 84, p3744 (2000), by B. Zheng and S. Trimper, cond-mat/0109003. We argue that their results confirm our conjecture, that the probability distribution for order parameter fluctuations in the 2D and 3D Ising models at a temperature T∗(L)T^{\ast}(L)T∗(L) slightly below TCT_CTC​ for the infinite, system approximates the universal functional form of the 2D-XY model in its low temperature phase. We discuss the relevance of the temperature interval TC−T∗T_C-T^{\ast}TC​−T∗, considered to be large by Zheng and Trimper and explain why the observed phenomena is a critical phenomena.
The value of the nonequilibrium exponent aaa is measured in the two-dimensional Ising model quenched to below criticality from the dynamical scaling of the zero-field-cooled and the intermediate susceptibility. Our results fully reconfirm the expected value a=1∕2a=1∕2a=1∕2 but are inconsistent with the value a=1∕4a=1∕4a=1∕4, advocated by Corberi, Lippiello, and Zannetti, Phys. Rev. E 72, 028103 (2005).
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 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.