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arXiv:cond-mat/9607162

Non-equilibrium Monte Carlo dynamics of the Sherrington-Kirkpatrick mean field spin glass model

Andrea Baldassarri

Published Jul 23, 1996• 1 citation
arXivHTMLPDF

Abstract

We present a numerical study of the non-equilibrium dynamics of the Sherrington-Kirkpatrick model. We analize the overlap distribution between the configurations visited at the time t and in particular its scaling behaviour with the size of the system. We find two different non-equilibrium dynamical regimes. The first is a proper Out of Equilibrium Regime, that is the relevant regime for the dynamics of an infinite system. The second is an Intermediate Regime that separates the Out of Equilibrium Regime from equilibrium. After studying the crossover beetween the two regimes, we focus on the Out of Equilibrium Regime and we reveal some of the geometrical features of phase space. According to recent analitical work, we find that the asymptotic out of equilibrium energy density is the equilibrium one. The same happens to the staggered-magnetisation, suggesting there is a deep geometrical similarity between equilibrium and out of equilibrium configurations. A procedure, that we call clonation, shows that the dynamics follows sort of canyons, as opposed to a system of independent traps.

DOI
10.48550/arxiv.cond-mat/9607162

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We present a numerical study of the non-equilibrium dynamics of the Sherrington-Kirkpatrick model. We analize the overlap distribution between the configurations visited at the time t and in particular its scaling behaviour with the size of the system. We find two different non-equilibrium dynamical regimes. The first is a proper Out of Equilibrium Regime, that is the relevant regime for the dynamics of an infinite system. The second is an Intermediate Regime that separates the Out of Equilibrium Regime from equilibrium. After studying the crossover beetween the two regimes, we focus on the Out of Equilibrium Regime and we reveal some of the geometrical features of phase space. According to recent analitical work, we find that the asymptotic out of equilibrium energy density is the equilibrium one. The same happens to the staggered-magnetisation, suggesting there is a deep geometrical similarity between equilibrium and out of equilibrium configurations. A procedure, that we call clonation, shows that the dynamics follows sort of canyons, as opposed to a system of independent traps.
We present a numerical study of the nonequilibrium dynamics of the Sherrington-Kirkpatrick model. We analyze the behavior of the system for different system sizes N and time scales t, in order to identify the correct out-of-equilibrium regime. We find evidence of aging behavior in the dynamics of the system, compatible with a weak-ergodicity breaking scenario. In addition, we obtain a pictorial characterization of the out-of-equilibrium phase space that presents some analogies with the hierarchy of states found in equilibrium.
Numerical data on the probability distribution of the equilibrium relaxation time of the Sherrington-Kirkpatrick model are obtained by means of dynamical Monte Carlo simulation, for several values of the system size NNN and temperature TTT. Proper care is taken that the thermal fluctuations on the relaxation time estimates are totally negligible compared to the disorder induced fluctuations. The probability distribution of ln⁡τ−ln⁡τ‾\lnτ-\overline{\lnτ}lnτ−lnτ scales with the scaling variable N1/3(Tc−T)N^{1/3} (T_c-T)N1/3(Tc​−T) strengthening the belief that ln⁡τ‾∝N1/3\overline{\lnτ}\propto N^{1/3}lnτ∝N1/3 in the whole spin glass phase.
We present results of a Monte Carlo study of the equilibrium dynamics of the one dimensional long-range Ising spin glass model. By tuning a parameter , this model interpolates between the mean field Sherrington–Kirkpatrick model and a proxy of the finite dimensional Edward-Anderson model. Activated scaling fits for the behavior of the relaxation time as a function of the number of spins N (Namely ) give values of that are not stable against inclusion of subleading corrections. Critical scaling ( ) gives more stable fits, at least in the non mean field region. We also present results on the scaling of the time decay of the critical remanent magnetization of the Sherrington–Kirkpatrick model, a case where the simulation can be done with quite large systems and that shows the difficulties in obtaining precise values for dynamical exponents in spin glass models.
The authors use Monte Carlo simulations to investigate the possibility of obtaining equilibrium values of quantities which relax slowly to equilibrium by studying the evolution of their non-equilibrium behaviour. As an example they study the order parameter distribution P(q) for the infinite-range Ising spin glass during the approach to equilibrium. This is evaluated both by computing the overlap between two initially uncorrelated copies of the system and from the correlation between the same set of spins at two different times. They find that the value of the most slowly relaxing part of the distribution, P(q=0), decreases monotonically with time from the first method, while a monotonically increasing result is obtained from the second technique. Consequently they expect that one could obtain the equilibrium value of P(q=0) for larger sizes by extrapolation of non-equilibrium data.
We show that the dynamical order parameters can be re-expressed in terms of the distribution of the staggered auto-correlation and response functions. We calculate these distributions for the out of equilibrium dynamics of the Sherrington-Kirkpatrick model at long times. The results suggest that the landscape this model visits at different long times in an out-of-equilibrium relaxation process is, in a sense, self-similar. Furthermore, there is a similarity between the landscape seen out of equilibrium at long times and the equilibrium landscape. The calculation is greatly simplified by making use of the superspace notation in the dynamical approach. This notation also highlights the rather mysterious formal connection between the dynamical and replica approaches. We also perform numerical simulations which show good agreement with the analytical results for the out of equilibrium dynamics.