Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Numerical study of the out-of-equilibrium phase space of a mean-field spin glass model

Andrea Baldassarri

Published Dec 1, 1998• 10 citations
Publisher

Abstract

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.

DOI
10.1103/physreve.58.7047

Related papers

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

Andrea Baldassarri

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.

arXiv:cond-mat/9607162•Jul 23, 1996•1 citation

Evidence of aging in spin-glass mean-field models

Leticia F. Cugliandolo, Jorge Kurchan B, Félix Ritort

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.

arXiv:cond-mat/9307001•Physical review. B, Condensed matter•Mar 1, 1994•81 citations

Relaxational Modes and Aging in the Glauber Dynamics of the Sherrington-Kirkpatrick Model

Hajime Yoshino, Koji Hukushima, Hajime Takayama

The relaxational modes and aging of the Glauber dynamics of the mean-field spin-glass model, SK model, are studied by a numerical diagonalization technique and Monte Carlo simulations. We found that the aging process of the model is understood as hierarchical growth of quasi-equilibrium domain in the phase-space. The fascinating features of aging effects of spin-glasses revealed by experiments 1) have arose much theoretical interest in recent years, which includes various phenomenologies 2) 3) , analytic predictions 4) and numerical simulations. 5) 6) In order to seek for concrete information to provide theoretical base, we study the relaxational

Progress of Theoretical Physics Supplement•Jan 1, 1997•10 citations

Numerical study of the dynamics of some long range spin glass models

Alain H. Billoire

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.

arXiv:1505.00676•Journal of Statistical Mechanics Theory and Experiment•Jul 30, 2015•2 citations

AGING IN A ONE-DIMENSIONAL EDWARDS–ANDERSON SPIN GLASS MODEL WITH LONG-RANGE INTERACTIONS

Marcelo A. Montemurro, Francisco A. Tamarit

In this work we study, by means of numerical simulations, the out-of-equilibrium dynamics of the one-dimensional Edwards–Anderson model with long-range interactions of the form ± Jr-α. In the limit α → 0 we recover the well known Sherrington–Kirkpatrick mean-field version of the model, which presents a very complex dynamical behavior. At the other extreme, for α → ∞ the model converges to the nearest-neighbor one-dimensional system. We focus our study on the dependence of the dynamics on the history of the sample (aging phenomena) for different values of α. The model is known to have mean-field exponents already for values of α = 2/3. Our results indicate that the crossover to the dynamic mean-field occurs at a value of α < 2/3.

International Journal of Modern Physics C•Mar 1, 2003•2 citations

Distribution of timescales in the Sherrington–Kirkpatrick model

Alain H. Billoire

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 $N$ and temperature $T$. 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 $\lnτ-\overline{\lnτ}$ scales with the scaling variable $N^{1/3} (T_c-T)$ strengthening the belief that $\overline{\lnτ}\propto N^{1/3}$ in the whole spin glass phase.

arXiv:1003.6086•Journal of Statistical Mechanics Theory and Experiment•Nov 23, 2010•8 citations
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.
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.
The relaxational modes and aging of the Glauber dynamics of the mean-field spin-glass model, SK model, are studied by a numerical diagonalization technique and Monte Carlo simulations. We found that the aging process of the model is understood as hierarchical growth of quasi-equilibrium domain in the phase-space. The fascinating features of aging effects of spin-glasses revealed by experiments 1) have arose much theoretical interest in recent years, which includes various phenomenologies 2) 3) , analytic predictions 4) and numerical simulations. 5) 6) In order to seek for concrete information to provide theoretical base, we study the relaxational
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.
In this work we study, by means of numerical simulations, the out-of-equilibrium dynamics of the one-dimensional Edwards–Anderson model with long-range interactions of the form ± Jr-α. In the limit α → 0 we recover the well known Sherrington–Kirkpatrick mean-field version of the model, which presents a very complex dynamical behavior. At the other extreme, for α → ∞ the model converges to the nearest-neighbor one-dimensional system. We focus our study on the dependence of the dynamics on the history of the sample (aging phenomena) for different values of α. The model is known to have mean-field exponents already for values of α = 2/3. Our results indicate that the crossover to the dynamic mean-field occurs at a value of α < 2/3.
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.