Structural Glass Transition and the Entropy of the Metastable States

Monasson, Rémi

doi:10.1103/physrevlett.75.2847

Cited by 444 publications

(717 citation statements)

References 25 publications

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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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Section: Introductionmentioning

confidence: 99%

Intermolecular Forces and the Glass Transition

Hall

Wolynes

2007

J. Phys. Chem. B

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“…One alternative that has been widely discussed in the literature, is to solve the averaged dynamics (for a review of this approach, see: [18]). Here we shall discuss a different but related approach, which is a generalization of several works in the replica literature of the 80's and 90's [59,60,62] and in optimization problems [38].…”

Section: The J-pointmentioning

confidence: 99%

“…From the point of view of replica theory, this is just a generalization of the 'effective potential' method [59], and is also closely related to the discussion in Refs. [60,62]. We have to write a partition function:…”

Section: The J-pointmentioning

confidence: 99%

Landscape analysis of constraint satisfaction problems

2007

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We discuss an analysis of Constraint Satisfaction problems, such as Sphere Packing, K-SAT and Graph Coloring, in terms of an effective energy landscape. Several intriguing geometrical properties of the solution space become in this light familiar in terms of the well-studied ones of rugged (glassy) energy landscapes. A 'benchmark' algorithm naturally suggested by this construction finds solutions in polynomial time up to a point beyond the 'clustering' and in some cases even the 'thermodynamic' transitions. This point has a simple geometric meaning and can be in principle determined with standard Statistical Mechanical methods, thus pushing the analytic bound up to which problems are guaranteed to be easy. We illustrate this for the graph three and four-coloring problem. For Packing problems the present discussion allows to better characterize the 'J-point', proposed as a systematic definition of Random Close Packing [1], and to place it in the context of other theories of glasses.

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“…When the parameter m is equal to one (the number of replicas is actually one in the approach of [59]), then −βΦ(β, 1) = −βf (β) + Σ(f ) reduces to the usual free energy function considered in the RS approximation…”

Section: The Local Stability: a Simple Self-consistency Checkmentioning

confidence: 99%

“…In the interpretation of [59] m is the number of replicas of the system, thus the name "replicated free energy" for Φ(β, m). Note that we are using the word "replica" only to refer to the established terminology as no replicas are needed within the cavity formalism.…”

Section: The Local Stability: a Simple Self-consistency Checkmentioning

confidence: 99%

Phase transitions in the coloring of random graphs

2007

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We consider the problem of coloring the vertices of a large sparse random graph with a given number of colors so that no adjacent vertices have the same color. Using the cavity method, we present a detailed and systematic analytical study of the space of proper colorings (solutions).We show that for a fixed number of colors and as the average vertex degree (number of constraints) increases, the set of solutions undergoes several phase transitions similar to those observed in the mean field theory of glasses. First, at the clustering transition, the entropically dominant part of the phase space decomposes into an exponential number of pure states so that beyond this transition a uniform sampling of solutions becomes hard. Afterward, the space of solutions condenses over a finite number of the largest states and consequently the total entropy of solutions becomes smaller than the annealed one. Another transition takes place when in all the entropically dominant states a finite fraction of nodes freezes so that each of these nodes is allowed a single color in all the solutions inside the state. Eventually, above the coloring threshold, no more solutions are available. We compute all the critical connectivities for Erdős-Rényi and regular random graphs and determine their asymptotic values for large number of colors.Finally, we discuss the algorithmic consequences of our findings. We argue that the onset of computational hardness is not associated with the clustering transition and we suggest instead that the freezing transition might be the relevant phenomenon. We also discuss the performance of a simple local Walk-COL algorithm and of the belief propagation algorithm in the light of our results.

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Structural Glass Transition and the Entropy of the Metastable States

Cited by 444 publications

References 25 publications

Intermolecular Forces and the Glass Transition

Intermolecular Forces and the Glass Transition

Landscape analysis of constraint satisfaction problems

Phase transitions in the coloring of random graphs

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