Glasses and Replicas

Mézard, Marc; Parisi, Giorgio

doi:10.1002/9781118202470.ch4

Cited by 34 publications

(70 citation statements)

References 99 publications

(76 reference statements)

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“…However, we do not recover exactly the result of the small cage expansion reported in [59]. To obtain it, we must expand log g liq (r; T /m, ϕ) ∼ −βmv(r) + · · · , where v(r) is the liquid potential, to get…”

Section: Connections With Previous Workmentioning

confidence: 97%

“…Following Monasson [67], we consider m identical replicas of the system; from the knowledge of their free energy one can derive most of the physical observables in the liquid and glass phases [6,59]. A complementary approach is the Franz-Parisi potential (or "state following") method [68]: this only requires a minor modification of the replica structure, that we do not explicitly discuss in the following, see [11] for a discussion.…”

Section: A the Variational Problemmentioning

confidence: 99%

“…We wish now to perform an expansion which is valid for small A, motivated by the observation that vibrations in the glass are always quite small [6,59,69]. In [69] the expansion was organized by using A 1/2 as the small parameter.…”

Section: B "Small Cage" Expansionmentioning

confidence: 99%

“…The parameter A, over which we will have to minimize the free energy, represents the mean square displacement of particles due to vibrations in the glass phase [6,59].…”

Section: A the Variational Problemmentioning

confidence: 99%

“…In the rest of this work, we adopt a more technical language and we assume that the reader is familiar with liquid theory [27], with the general formulation of RFOT [19,22,66], with the replica method [67,68] and its application to structural glasses [6,59,60] and with the perturbative methods discussed in [6,69] on which the present work is strongly based.…”

Section: Introduction a The Exact Solution Of Infinite-dimensionamentioning

confidence: 99%

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Quantitative approximation schemes for glasses

Mangeat

Zamponi

2016

Phys. Rev. E

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By means of a systematic expansion around the infinite-dimensional solution, we obtain an approximation scheme to compute properties of glasses in low dimensions. The resulting equations take as input the thermodynamic and structural properties of the equilibrium liquid, and from this they allow one to compute properties of the glass. They are therefore similar in spirit to the Mode-Coupling approximation scheme. Our scheme becomes exact, by construction, in dimension d → ∞ and it can be improved systematically by adding more terms in the expansion. ContentsI. Introduction A. The exact solution of infinite-dimensional glassy hard spheres B. Extension to finite dimension: state of the art C. Aim and structure of this paper II. Derivation of the equations A. The variational problem B. "Small cage" expansion C. Determination of G(r), and the final set of equations D. Discussion III. Connections with previous work A. Mézard-Parisi 1999 B. Parisi-Zamponi 2005 C. Berthier-Jacquin-Zamponi 2011 IV. Extracting physical quantities from the replicated free energy A. A convenient expression of q m,A (r) B. The equation for A and the complexity C. The equilibrium transition densities: dynamical transition and Kauzmann transition D. The jamming line E. Correlation functions and non-ergodicity factor V. Numerical methods A. Integral equations of liquid theory B. Discrete Fourier transformation C. Algorithm for g liq D. More accurate expressions for three-dimensional hard spheres E. Non-ergodicity factor F. Summary VI. Results: hard spheres A. HNC and PY results as a function of dimension B. Three dimensional results: thermodynamics C. Three dimensional results: non-ergodicity factor VII. Conclusions2. The dynamical transition density [8,12], at which the liquid phase becomes infinitely viscous and ergodicity is broken by the emergence of many metastable states. This transition is similar to the one of Mode-Coupling Theory (MCT) [23] and is thus characterized by MCT critical dynamical scaling, controlled by the so-called MCT parameter λ that can be also computed [8,12]. The Kauzmann transition [6], where the number of metastable states becomes sub-exponential, giving rise to an "entropy crisis" and a second order equilibrium phase transition.4. The Gardner transition line, that separates a region where glass basins are stable from a region where they are broken in a complex structure of metabasins [8,10,11].5. The density region where jammed packings exist (also known as "jamming line" or "J-line" [6,24,25]), which is delimited by the threshold density and the glass close packing density [6].6. The equation of state of glassy states, computed by compression and decompression of equilibrium glasses [11].7. The response of the glass state to a shear strain [11,26]. 8. The long time limit of the mean square displacement in the glass (the so-called Edwards-Anderson order parameter) [9]. 9. The behavior of correlation function, structural g(r) and non-ergodicity factor of the glass [6,9,12].10. The probability distribution of the for...

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Section: Connections With Previous Workmentioning

confidence: 97%

Section: A the Variational Problemmentioning

confidence: 99%

Section: B "Small Cage" Expansionmentioning

confidence: 99%

“…The parameter A, over which we will have to minimize the free energy, represents the mean square displacement of particles due to vibrations in the glass phase [6,59].…”

Section: A the Variational Problemmentioning

confidence: 99%

Section: Introduction a The Exact Solution Of Infinite-dimensionamentioning

confidence: 99%

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Quantitative approximation schemes for glasses

Mangeat

Zamponi

2016

Phys. Rev. E

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The Random First‐Order Transition Theory of Glasses: A Critical Assessment

Biroli

Bouchaud

2012

Structural Glasses and Supercooled Liquids

107

164

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The aim of this chapter is to summarise the basic arguments and the intuition bolstering the RFOT picture for glasses, based on a finite dimensional extension of mean-field models with an exponentially large number of metastable states. We review the pros and cons that support or undermine the theory, and the directions, both theoretical and experimental, where progress is needed to ascertain the status of RFOT. We elaborate in particular on the notions of mosaic state and point-to-set correlations, and insist on the importance of fluctuations in finite dimensions, that significantly blur the expected cross-over between a Mode-Coupling like regime and the mosaic, activated regime. We discuss in detail the fundamental predictions of RFOT, in particular the possibility to force a small enough system into an ideal glass state, and present several new ones, concerning aging properties or non-linear rheology. Finally, we compare RFOT to other recent theories, including elastic models, Frustration Limited Domains or Kinetically Constrained models.Details that could throw doubt on your interpretation must be given, if you know them. You must do the best you can -if you know anything at all wrong, or possibly wrong -to explain it. If you make a theory, for example, and advertise it, or put it out, then you must also put down all the facts that disagree with it, as well as those that agree with it.

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Critical Phenomena in Glasses

Rizzo

2018

Handbook of Materials Modeling

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Glasses and Replicas

Cited by 34 publications

References 99 publications

Quantitative approximation schemes for glasses

Quantitative approximation schemes for glasses

The Random First‐Order Transition Theory of Glasses: A Critical Assessment

Critical Phenomena in Glasses

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