Mean-field theory and fluctuations in Potts spin glasses. I

Cwilich, Gabriel; Kirpatrick, T R

doi:10.1088/0305-4470/22/22/023

Cited by 56 publications

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“…The mean field model has recently been studied in great detail by means of Monte Carlo methods and the results are in good qualitative agreement with the main predictions of the one-step replica symmetry breaking theory [1,2,4,6], although very strong finite size effects occur, the nature of which are not fully understood [12].…”

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confidence: 64%

“…In the case of infinite range interactions, i.e. mean field, these models can be solved exactly and it has been shown that they have a dynamical as well as a static transition at a temperature T D and T 0 , respectively [1,2,3,4,5,6,7,8,9,10,11,12,13]. At T D the relaxation times diverge but no singularity of any kind occurs in the static properties, whereas at T 0 a nonzero static glass order parameter appears discontinuously.…”

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confidence: 99%

“…Pf, 75.10.Nr, 75.50.Lk In recent years generalized spin-glass-type models such as the p-spin model with p ≥ 3 or the Potts glass with p > 4, have found a large attention [1,2,3,4,5,6,7,8,9,10,11,12,13] as prototype models for the structural glass transition [14,15,16,17,18,19]. In the case of infinite range interactions, i.e.…”

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Evidence against a glass transition in the 10-state short-range Potts glass

2002

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We present the results of Monte Carlo simulations of two different 10-state Potts glasses with random nearest neighbor interactions on a simple cubic lattice. In the first model the interactions come from a ±J distribution and in the second model from a Gaussian one, and in both cases the first two moments of the distribution are chosen to be equal to J 0 = −1 and ∆J = 1. At low temperatures the spin autocorrelation function for the ±J model relaxes in several steps whereas the one for the Gaussian model shows only one. In both systems the relaxation time increases like an Arrhenius law. Unlike the infinite range model, there are only very weak finite size effects and there is no evidence that a dynamical or a static transition exists at a finite temperature.PACS numbers: 64.70. Pf, 75.10.Nr, 75.50.Lk In recent years generalized spin-glass-type models such as the p-spin model with p ≥ 3 or the Potts glass with p > 4, have found a large attention [1,2,3,4,5,6,7,8,9,10,11,12,13] as prototype models for the structural glass transition [14,15,16,17,18,19]. In the case of infinite range interactions, i.e. mean field, these models can be solved exactly and it has been shown that they have a dynamical as well as a static transition at a temperature T D and T 0 , respectively [1,2,3,4,5,6,7,8,9,10,11,12,13]. At T D the relaxation times diverge but no singularity of any kind occurs in the static properties, whereas at T 0 a nonzero static glass order parameter appears discontinuously. Close to T D the time and temperature dependence of the spin autocorrelation function is described by the same type of mode coupling equations [3,7] that have been proposed by the idealized version of mode coupling theory (MCT) [15,16] for the structural glass transition which suggests a fundamental connection between these rather abstract spin models and real structural glasses. 2Now it is well known that for real glasses the divergence of the relaxation times predicted by MCT is rounded off since thermally activated processes, which are not taken into account by this version of the theory, become important [15,16]. This is in contrast to the mean field case because there these processes are completely suppressed since the barriers in the (free) energy landscape become infinitely high if T < T D . To what extend the static transition that exists in mean field can be seen also in the real system, is a problem whose answer is still controversial [9,17,18,19].In the present paper, we investigate whether the transitions present in the case of a mean field 10-state Potts glass are also found if the interactions are short The Hamiltonian of the short range model isThe Potts spins σ i on the lattice sites i of the simple cubic lattice take p discrete values, σ i ∈ {1, 2, . . . , p}, and the index j runs over the six nearest neighbors of site i. The exchange constants J ij are taken either from a bimodal distributionor a Gaussian distributionIn both cases the first two moments are chosen to be [· · · ] av is realized by averaging over 100 indep...

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Evidence against a glass transition in the 10-state short-range Potts glass

2002

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“…³ÑÅÎÂÔÐÑ ÓÇÊÖÎßÕÂÕÂÏ ÓÂÃÑÕ [29,30] ÔÒËÐ-ÔÕÇÍÑÎßÐÂâ ×ÂÊÂ ÐÂÃÎáAEÂÇÕÔâ ÕÑÅAEÂ, ÍÑÅAEÂ ÑÃÞÚÐÑÇ ×ÇÓÓÑÏÂÅÐËÕÐÑÇ ÖÒÑÓâAEÑÚÇÐËÇ ÐÇ AEÂÇÕ ÔÖÜÇÔÕÄÇÐÐÑÅÑ ÄÍÎÂAEÂ Ä ÔÄÑÃÑAEÐÖá àÐÇÓÅËá ÓÂÔÔÏÂÕÓËÄÂÇÏÑÌ ÔËÔÕÇÏÞ …”

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“…³ÑÅÎÂÔÐÑ ÓÇÊÖÎßÕÂÕÂÏ ÓÂÃÑÕÞ [29] n-ÓÇÒÎËÚÐÂâ ÔÄÑ-ÃÑAEÐÂâ àÐÇÓÅËâ ÄÃÎËÊË ÕÑÚÍË ÒÇÓÇØÑAEÂ ËÏÇÇÕ ÔÎÇAEÖáÜËÌ ÄËAE: ÇÑÃØÑAEËÏÞÇ ÖÔÎÑÄËâ Ô×ÑÓÏÖÎËÓÑÄÂÐÞ Ä ÔÎÇAEÖáÜÇÌ ÕÇÑÓÇÏÇ ®ÂÓÍÑÄÂ ë ¢ËÓÏÂÐ [55].…”

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Problems of probabilistic topology: the statistics of knots and non-commutative random walks

Nechaev¹

1998

Usp. Fiz. Nauk

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Simulation of Models for the Glass Transition: Is There Progress?

Binder

Baschnagel

Kob

et al. 2002

Bridging Time Scales: Molecular Simulations for the Next Decade

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The glass transition of supercooled fluids is a particular challenge for computer simulation, because the (longest) relaxation times increase by about 15 decades upon approaching the transition temperature Tg. Brute-force molecular dynamics simulations, as presented here for molten SiO2 and coarse-grained bead-spring models of polymer chains, can yield very useful insight about the first few decades of this slowing down. Hence this allows to access the temperature range around Tc of the so-called mode coupling theory, whereas the dynamics around the experimental glass transition is completely out of reach. While methods such as "parallel tempering" improve the situation somewhat, a method that allows to span a significant part of the region Tg ≤ T ≤ Tc is still lacking. Only for abstract models such as the infinite range 10-state Potts glass with a few hundred spins this region can be explored. However this model suffers from very strong finite size effects thus making it difficult to extrapolate the results obtained for the finite system sizes to the thermodynamic limit.For the case of polymer melts, two different strategies to use lattice models instead of continuum models are discussed: In the first approach, a mapping of an atomistically realistic model of polyethylene to the bond fluctuation model with suitable effective potentials and a temperature-dependent time rescaling factor is attempted. In the second approach, devoted to a test of the entropy theory, moves that are artificial but which lead to a faster relaxation ("slithering snake" algorithm) are used, to get at least static properties at somewhat lower temperatures than possible with a "realistic" dynamics. The merits and shortcomings of all these approaches are discussed.

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Mean-field theory and fluctuations in Potts spin glasses. I

Cited by 56 publications

References 35 publications

Evidence against a glass transition in the 10-state short-range Potts glass

Evidence against a glass transition in the 10-state short-range Potts glass

Problems of probabilistic topology: the statistics of knots and non-commutative random walks

Simulation of Models for the Glass Transition: Is There Progress?

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