Quantitative approximation schemes for glasses

Abstract

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...

A characteristic energy scale in glasses

“…It was previously proposed in [85], and tested in [70,85], that the energy scale that characterizes soft, extended vibrational modes that emerge near the unjamming transition -the loss of solidity observed in soft-sphere packings as the confining pressure is reduced [66][67][68] -can be defined by the response to local force dipoles. The statistical and structural properties of soft, extended vibrational modes that emerge near the unjamming point are predicted by mean-field frameworks [86][87][88][89][90][91][92] and variational arguments [85,88]; intriguingly, they are qualitatively and quantitatively different from the statistical and structural properties of the quasilocalized excitations studied here for a generic structural glass far from the unjamming point [50][51][52][53], despite that the same protocol appears to capture the characteristic energies of both types of soft excitations. The essential nature of the relation between these two classes of soft excitations is still an open question that deserves further investigation.…”

Simple Theory for the Dynamics of Mean-Field-Like Models of Glass-Forming Fluids

“…In particular, if we used the potential of the mean force V mf (r) = −T ln g(r) rather than the true potential in Eqs. (7)(8), the resulting ergodicity-breaking transition would coincide with the dynamic transition predicted by a recent version of the replica theory for a standard finitedimensional system [25]. Thus, as far as location of the dynamic transition is concerned, the latter approach approximates the standard hard-sphere system by a Mari-Krzakala-Kurchan system with particles interacting via a potential of mean force.…”

Edwards statistical mechanics for jammed granular matter

“…Upon compression of the metastable states (taking some care in the preparation protocol (Charbonneau et al, 2017)) the pressure diverges at jamming densities φ j ∈ [φ th , φ GCP ]. The lower limit is the threshold density φ th ≈ 0.64 calculated in , although it should be noted that the values calculated with replica theory come with a large error bar due to the approximation of the liquid equation of state (Mangeat and Zamponi, 2016). The maximal density is the glass close packing φ GCP ≈ 0.68 corresponding to the infinite pressure limit of the ideal glass φ K .…”