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

arXiv:1003.6086 · Journal of Statistical Mechanics Theory and Experiment

Distribution of timescales in the Sherrington–Kirkpatrick model

Alain H. Billoire

Published Nov 23, 2010• 8 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1742-5468/2010/11/p11034

Related papers

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

Dynamics in the Sherrington-Kirkpatrick Ising spin glass at and above Tg

Alain H. Billoire, Ian A. Campbell

A detailed numerical study is made of relaxation at equilibrium in the Sherrington-Kirkpatrick Ising spin glass (ISG) model, at and above the critical temperature ${T}_{g}$. The data show a long time stretched exponential relaxation $q(t)\ensuremath{\sim}\mathrm{exp}{\ensuremath{-}{[t/\ensuremath{\tau}(T)]}^{\ensuremath{\beta}(T)}}$ with an exponent $\ensuremath{\beta}(T)$ tending to $\ensuremath{\approx}$1/3 at ${T}_{g}$. The results are compared to those which were observed by A. T. Ogielski, [Phys. Rev. B 32, 7384 (1985)] in the three-dimensional ISG model and are discussed in terms of a phase space percolation transition scenario.

arXiv:1105.1902•Physical Review B•Aug 12, 2011•15 citations

Statics and dynamics of the infinite-range Ising spin glass model

N D Mackenzie, A. P. Young

The Sherrington-Kirkpatrick (SK) spin glass model (1975) is studied by Monte Carlo simulation. Below T c there is a spectrum of relaxation times which diverge like exp(cN 1 4/) as N, the number of spins, tends to infinity. In zero field, h, there is one more timescale, tau eg , which is the time to turn over all the spins. Averaging over samples (ln tau eg ) j (=ln tau eg ) varies as N 1 2/. However, ln tau eg is not a self-averaging quantity and there are large sample-to-sample variations even for N to infinity . For h>>h*=T/N 1 2/, where T is the temperature, fluctuations on timescale tau eg may not occur. Above the Almeida-Thouless line (1978) relaxation times are finite and the SK solution is correct. Below this line, the results agree well with Parisi's theory (1983) if the latter is correctly interpreted.

Journal of Physics C Solid State Physics•Sep 30, 1983•49 citations

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

Finite-size scaling analysis of the distributions of pseudo-critical temperatures in spin glasses

Alain H. Billoire, L. A. Fernández, A. Maiorano +3 more

Using the results of large scale numerical simulations we study the probability distribution of the pseudo-critical temperature for the three-dimensional Edwards–Anderson Ising spin glass and for the fully connected Sherrington–Kirkpatrick model. We find that the behaviour of our data is nicely described by straightforward finite-size scaling relations.

arXiv:1108.1336•Journal of Statistical Mechanics Theory and Experiment•Oct 20, 2011•22 citations

Free-energy barriers in the Sherrington-Kirkpatrick model

Elmar Bittner, Wolfhard Janke E

The Sherrington-Kirkpatrick spin-glass model is investigated by means of Monte Carlo simulations by employing a combination of the multi-overlap algorithm with parallel tempering methods. We investigate the finite-size scaling behaviour of the free-energy barriers which are visible in the probability density of the Parisi overlap parameter. Assuming that the mean barrier height diverges with the number of spins N as N α , our data show good agreement with the theoretical value α = 1/3.

Europhysics Letters (EPL)•Mar 23, 2006•20 citations
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.
A detailed numerical study is made of relaxation at equilibrium in the Sherrington-Kirkpatrick Ising spin glass (ISG) model, at and above the critical temperature Tg{T}_{g}Tg​. The data show a long time stretched exponential relaxation $q(t)\ensuremath{\sim}\mathrm{exp}{\ensuremath{-}{[t/\ensuremath{\tau}(T)]}^{\ensuremath{\beta}(T)}}$ with an exponent $\ensuremath{\beta}(T)$ tending to $\ensuremath{\approx}$1/3 at Tg{T}_{g}Tg​. The results are compared to those which were observed by A. T. Ogielski, [Phys. Rev. B 32, 7384 (1985)] in the three-dimensional ISG model and are discussed in terms of a phase space percolation transition scenario.
The Sherrington-Kirkpatrick (SK) spin glass model (1975) is studied by Monte Carlo simulation. Below T c there is a spectrum of relaxation times which diverge like exp(cN 1 4/) as N, the number of spins, tends to infinity. In zero field, h, there is one more timescale, tau eg , which is the time to turn over all the spins. Averaging over samples (ln tau eg ) j (=ln tau eg ) varies as N 1 2/. However, ln tau eg is not a self-averaging quantity and there are large sample-to-sample variations even for N to infinity . For h>>h*=T/N 1 2/, where T is the temperature, fluctuations on timescale tau eg may not occur. Above the Almeida-Thouless line (1978) relaxation times are finite and the SK solution is correct. Below this line, the results agree well with Parisi's theory (1983) if the latter is correctly interpreted.
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.