Recipes for Metastable States in Spin Glasses

We study analytically the dynamics of a generalized p-spin model, starting with a thermalized initial condition. The model presents birth and death of states, hence the dynamics (even starting at equilibrium) may go out of equilibrium when the temperature is varied. We give a full description of this constrained out of equilibrium behavior and we clarify the connection to the thermodynamics by computing (sub-dominant) tap states, constrained to the starting equilibrium configuration.PACS numbers: 75.10. Nr, 64.70.Pf, 75.50.Lk Many interesting physical systems live for very long times out of equilibrium, and, in this regime, they display highly non trivial behaviors which are still to be understood (e.g. rejuvenation and memory effects in spin glasses). In general, these systems fall out of equilibrium when some external parameter is changed. For example, fragile glassforming liquids undergo a dramatic slowing down of their relaxational dynamics when the temperature is dropped below the glass transition temperature 1 . This effect is sharpened in certain mean-field models where, at a critical temperature T d , a transition occurs from an equilibrium kind of dynamics to an off-equilibrium aging one 2 . The phenomenon is ubiquitous and can be found also in very different fields: e.g. in local search algorithms for solving hard optimization problems the time-complexity may become extremely large by varying a macroscopic parameter 3 . A better understanding of the mechanisms leading to the dramatic slowing down in out of equilibrium dynamics is a subject of broad interest and wide applicability.In describing the dynamical slowing down (and possible eventual arrest) the common view suggests that at a low temperature a huge number of metastable states appears (with energies higher than the equilibrium one, E 0 ), making relaxation to equilibrium very slow, and even impossible if interactions are long ranged and metastable states lifetimes diverge in the thermodynamic limit. This picture has been verified by solving the out of equilibrium Langevin dynamics of a particularly simple mean-field model, the so-called fully connected spherical p-spin, whose Hamiltonian is 4where the N spins σ i are continuous variables subject to the spherical constraint i σ 2 i = N and the couplings are i.i.d. random variables with zero mean and variance p!/(2N p−1 ). In this model (hereafter p ≥ 3) if we consider a quench, that is if we choose an initial configuration of high energy and let the system relax at a fixed value of the temperature T < T d (T d being the dynamic transition temperature), the asymptotic dynamics remains trapped at the energy level of the highest and most numerous metastable states, the so-called threshold states. Time-translation invariance and the dynamic fluctuation-dissipation relation are violated and aging is observed in correlation and response functions 5 .These features are intriguing and experimentally relevant, since aging behaviour has been observed in many disordered systems. Nevertheless, in order t...

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“…Ref. 9), the global free energy of the system thus bearing a contribution from this state-related entropy, i.e. −T ln Z = F = f eq − T Σ(f eq ).…”

Section: The Modelmentioning

confidence: 99%

Off-equilibrium confined dynamics in a glassy system with level-crossing states

et al. 2006

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“…31 It is worth at this point to present a brief qualitative review of the results of the mean-field theory, based on the Hamiltonian ͑1͒. 4,19,20 The study of the thermodynamics of this system leads to the following results. At high temperatures the system is paramagnetic and ergodic.…”

Section: The Modelmentioning

confidence: 99%

“…15 In fact, recent progress in the comprehension of the dynamics of mean-field disordered systems 7,8 allowed for an extension of the mode coupling theory to the broken ergodicity phase. 16,17 Many studies 4,18,19,7,20,21 have pointed out the existence of a temperature T D where, despite the fact that no singularity is observed in the free energy, there is a statical breaking of ergodicity into an exponentially large number of metastable states. A thermodynamic singularity is present at a temperature T C smaller than T D .…”

Section: Introductionmentioning

confidence: 99%

Fragile-glass behavior of a short-rangep-spin model

1996

Self Cite

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We propose a short-range generalization of the p-spin interaction spin-glass model. The model is well suited to test the idea that an entropy collapse is at the bottom line of the dynamical singularity encountered in structural glasses. The model is studied in three dimensions through Monte Carlo simulations, which put in evidence fragile glass behavior with stretched exponential relaxation and super-Arrhenius behavior of the relaxation time. Our data are in favor of a Vogel-Fulcher behavior of the relaxation time, related to an entropy collapse at the Kauzmann temperature. We, however, encounter difficulties analogous to those found in experimental systems when extrapolating thermodynamical data at low temperatures. We study the spin-glass susceptibility, investigating the behavior of the correlation length in the system. We find that the increase of the relaxation time is accompanied by a very slow growth of the correlation length. We discuss the scaling properties of off-equilibrium dynamics in the glassy regime, finding qualitative agreement with the mean-field theory. ͓S0163-1829͑96͒04137-9͔

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“…These calculations are based on density functional 19,21,22 and self-consistent phonon approaches 18,23 which can be justified by elegant replica methodology 24,25,26,27,28,29,30,31 . Related theories have also recently appeared.…”

Section: Introductionmentioning

confidence: 99%

Intermolecular Forces and the Glass Transition

Hall

Wolynes

2007

J. Phys. Chem. B

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Random first order transition theory is used to determine the role of attractive and repulsive interactions in the dynamics of supercooled liquids. Self-consistent phonon theory, an approximate mean field treatment consistent with random first order transition theory, is used to treat individual glassy configurations, while the liquid phase is treated using common liquid state approximations.Free energies are calculated using liquid state perturbation theory. The transition temperature T * A , the temperature where the onset of activated behavior is predicted by mean field theory, the lower crossover temperature T * c where barrierless motions actually occur through fractal or stringy motions (corresponding to the phenomenological mode coupling transition temperature), and T * K , the Kauzmann temperature (corresponding to an extrapolated entropy crisis), are calculated in addition to T * g , the glass transition temperature that corresponds to laboratory cooling rates. Relationships between these quantities agree well with existing experimental and simulation data on van der Waals liquids. Both the isobaric and isochoric behavior in the supercooled regime are studied, providing results for ∆C V and ∆C p that can be used to calculate the fragility as a function of density and pressure, respectively. The predicted variations in the α-relaxation time with temperature and density conform to the empirical density-temperature scaling relations found by Casalini and Roland. We thereby demonstrate the microscopic origin of their observations. Finally, the relationship first suggested by Sastry between the spinodal temperature and the Kauzmann temperatures, as a function of density, is examined. The present microscopic calculations support the existence of an intersection of these two temperatures at sufficiently low temperatures.2

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Recipes for Metastable States in Spin Glasses

Cited by 307 publications

References 15 publications

Off-equilibrium confined dynamics in a glassy system with level-crossing states

Off-equilibrium confined dynamics in a glassy system with level-crossing states

Fragile-glass behavior of a short-rangep-spin model

Intermolecular Forces and the Glass Transition

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