Convergence to extremal processes in random environments and extremal ageing in SK models (original) (raw)
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Universality and extremal aging for dynamics of spin glasses on subexponential time scales
Communications on Pure and Applied Mathematics, 2012
We consider Random Hopping Time (RHT) dynamics of the Sherrington-Kirkpatrick (SK) model and p-spin models of spin glasses. For any of these models and for any inverse temperature β > 0 we prove that, on time scales that are sub-exponential in the dimension, the properly scaled clock process (time-change process) of the dynamics converges to an extremal process. Moreover, on these time scales, the system exhibits aging like behavior which we called extremal aging. In other words, the dynamics of these models ages as the random energy model (REM) does. Hence, by extension, this confirms Bouchaud's REM-like trap model as a universal aging mechanism for a wide range of systems which, for the first time, includes the SK model.
Strong ergodicity breaking in aging of mean-field spin glasses
Proceedings of the National Academy of Sciences
Out-of-equilibrium relaxation processes show aging if they become slower as time passes. Aging processes are ubiquitous and play a fundamental role in the physics of glasses and spin glasses and in other applications (e.g., in algorithms minimizing complex cost/loss functions). The theory of aging in the out-of-equilibrium dynamics of mean-field spin glass models has achieved a fundamental role, thanks to the asymptotic analytic solution found by Cugliandolo and Kurchan. However, this solution is based on assumptions (e.g., the weak ergodicity breaking hypothesis) which have never been put under a strong test until now. In the present work, we present the results of an extraordinary large set of numerical simulations of the prototypical mean-field spin glass models, namely the Sherrington–Kirkpatrick and the Viana–Bray models. Thanks to a very intensive use of graphics processing units (GPUs), we have been able to run the latter model for more than 264 spin updates and thus safely e...
On pre-asymptotic aging in finite dimensional spin glasses
Unifying Concepts in Granular Media and Glasses, 2004
We study the off-equilibrium dynamics in the pre-asymptotic aging regime of the two, three and four dimensional Edwards-Anderson spin glass model. We define an exotic correlation function and the corresponding response, and we compare the resulting fluctuationdissipation ratio (FDR) with the usual one obtained by the ordinary spin-spin correlation function and relative response. In all systems we find that that after a short transient the two corresponding fluctuation-dissipation ratios coincide at equal times. Our findings support the idea that the FDR defines a unique set of effective temperatures for degrees of freedom evolving on the same time scale. In addition we show that in 2D, as it happens for the usual FDR, the new dynamic FDR at finite time can be relates to the static overlap probability function (OFF) of systems of suitable finite size.
We review recent theoretical progress on glassy dynamics, with special emphasis on the importance and universality of the aging regime, which is relevant to many experimental situations. The three main subjects which we address are: (i) Phenomenological models of aging (coarsening, trap models), (ii) Analytical results for the low-temperature dynamics of mean-field models (corresponding to the mode-coupling equations); and (iii) Simple non-disordered models with glassy dynamics. We discuss the interrelation between these approaches, and also with previous work in the field. Several open problems are underlined -in particular the precise relation between mean-field like (or mode-coupling) descriptions and finite dimensional problems.
Heterogeneous Aging in Spin Glasses
Physical Review Letters, 2002
We introduce a set of theoretical ideas that form the basis for an analytical framework capable of describing nonequilibrium dynamics in glassy systems. We test the resulting scenario by comparing its predictions with numerical simulations of short-range spin glasses. Local fluctuations and responses are shown to be connected by a generalized local out-of-equilibrium fluctuation-dissipation relation. Scaling relationships are uncovered for the slow evolution of heterogeneities at all time scales.
Evidence of aging in spin-glass mean-field models
Physical Review B, 1994
We study numerically the out of equilibrium dynamics of the hypercubic cell spin glass in high dimensionalities. We obtain evidence of aging effects qualitatively similar both to experiments and to simulations of low dimensional models. This suggests that the Sherrington-Kirkpatrick model as well as other mean-field finite connectivity lattices can be used to study these effects analytically.
Ageing in spin-glasses in three, four and infinite dimensions
Journal of Physics A: Mathematical and General, 2003
The SUE machine is used to extend by a factor of 1000 the timescale of previous studies of the aging, out-of-equilibrium dynamics of the Edwards-Anderson model with binary couplings, on large lattices (L = 60). The correlation function, C(t+ t w , t w), t w being the time elapsed under a quench from high-temperature, follows nicely a slightly-modified power law for t > t w. Very tiny (logarithmic), yet clearly detectable deviations from the full-aging t/t w scaling can be observed. Furthermore, the t < t w data shows clear indications of the presence of more than one time-sector in the aging dynamics. Similar results are found in four-dimensions, but a rather different behaviour is obtained in the infinite-dimensional z = 6 Viana-Bray model. Most surprisingly, our results in infinite dimensions seem incompatible with dynamical ultrametricity. A detailed study of the link correlation function is presented, suggesting that its aging-properties are the same as for the spin correlation-function.
Numerical Study of Aging Phenomena in Short-Ranged Spin Glasses
Progress of Theoretical Physics Supplement, 2000
Aging phenomena of short-range Ising spin glass models have been investigated using Monte Carlo simulations. It is found that in the low-temperature spin-glass phase the mean domain size exhibits a crossover from a power-law growth associated with the critical fluctuation at the transition temperature to slower growth inherent in the low-temperature phase. The temperature dependence of the growth law of the domain size can be almost explained by this crossover. We also find that the spin-autocorrelation function in the quasi-equilibrium regime follows the expected scaling behavior from the droplet theory expressed in terms of the mean domain size and a characteristic length scale of droplet excitations. typeset using PTPT E X.sty <ver.1.0>
Deviations from Full Aging in Numerical Spin Glass Models
2003
The deviations from full or pure aging behavior, i.e. perfect t/twt/t_wt/tw scaling of the correlation and response functions of aging glassy systems, are not well understood theoretically. Recent experiments of Rodriguez et al. (Phys. Rev. Lett. 91, 037203 (2003)) have shown that full aging applies to the thermoremanent magnetization in the limit of infinite cooling rate during the initial quench.