Exact Theory of Dense Amorphous Hard Spheres in High Dimension. II. The High Density Regime and the Gardner Transition

Kurchan, Jorge; Parisi, G.; Urbani, Pierfrancesco; Zamponi, Francesco

doi:10.1021/jp402235d

Cited by 147 publications

(307 citation statements)

References 71 publications

Supporting

Mentioning

281

Contrasting

Unclassified

Order By: Relevance

“…Comparison with Replica Theory in d = ∞ A very recent replica computation (17,18,39,40) was used to compute exponents in d = ∞ to arbitrary precision, and results in γ = 0:41269, κ = 1:41574, and θ f = 0:42311. These values are consistent with our prediction κ = 2=ð1 + γÞ, which appears to be exactly satisfied.…”

Section: Comparison With Numericsmentioning

confidence: 99%

Force distribution affects vibrational properties in hard-sphere glasses

DeGiuli

Lerner

Brito

et al. 2014

Proc. Natl. Acad. Sci. U.S.A.

120

166

View full text Add to dashboard Cite

We theoretically and numerically study the elastic properties of hard-sphere glasses and provide a real-space description of their mechanical stability. In contrast to repulsive particles at zero temperature, we argue that the presence of certain pairs of particles interacting with a small force f soften elastic properties. This softening affects the exponents characterizing elasticity at high pressure, leading to experimentally testable predictions. Denoting P(f ) ∼ f θe , the force distribution of such pairs and ϕ c the packing fraction at which pressure diverges, we predict that (i) the density of states has a low-frequency peak at a scale ω*, rising up to it as D(ω) ∼ ω 2+a , and decaying above ω* as D(ω) ∼ ω −a where a = (1 − θ e )=(3 + θ e ) and ω is the frequency, (ii) shear modulus and mean-squared displacement are inversely proportional with 〈δR 2 〉 ∼ 1=μ ∼ (ϕ c − ϕ) κ , where κ = 2 − 2=(3 + θ e ), and (iii) continuum elasticity breaks down on a scale ℓ c ∼ 1= ffiffiffiffiffi δz p ∼ (ϕ c − ϕ) −b , where b = (1 + θ e )=(6 + 2θ e ) and δz = z − 2d, where z is the coordination and d the spatial dimension. We numerically test (i) and provide data supporting that θ e ≈ 0:41 in our bidisperse system, independently of system preparation in two and three dimensions, leading to κ ≈ 1:41, a ≈ 0:17, and b ≈ 0:21. Our results for the mean-square displacement are consistent with a recent exact replica computation for d = ∞, whereas some observations differ, as rationalized by the present approach.colloids | glass transition | marginal stability | boson peak | jamming T he emergence of rigidity near the glass transition is a fundamental and highly debated topic in condensed matter and is perhaps most surprising in hard-sphere glasses where rigidity is purely entropic in nature. The rapid growth of relaxation time around a packing fraction ϕ g ≈ 0:58 suggests that metastable states have appeared in the free-energy landscape, and that activation above barriers is required for the system to flow (1). This scenario is presumably what mode-coupling theory captures (2, 3) and can be rationalized via density functional theory (4) and via the replica method (5). Recently a real-space description of mechanical stability and elasticity in hard-sphere glasses has been proposed (6, 7), which is most easily tested at large pressure, deep in the glass phase. It is based on two results. First, in elastic networks and athermal packings of soft spheres (8-10), mechanical stability is controlled by the mean number of contacts per particle, or coordination z (as already discussed by Maxwell in ref. 11), and the applied compressive strain e (10). As one may intuitively expect, increasing coordination is stabilizing, whereas increasing pressure at fixed coordination is destabilizing. Second, within a long-lived metastable state the vibrational free energy of a hard-sphere system can be approximated as a sum of local interaction terms between pairs of colliding particles, which are said to be in contact. On a time scale that contains many...

show abstract

Section: Comparison With Numericsmentioning

confidence: 99%

Force distribution affects vibrational properties in hard-sphere glasses

DeGiuli

Lerner

Brito

et al. 2014

Proc. Natl. Acad. Sci. U.S.A.

120

166

View full text Add to dashboard Cite

show abstract

“…In a seminal paper 1 , an universal jamming phase diagram was proposed to unify the transition of structural arrest of different systems, including the glass and the jamming transitions. However, within a mean field or the infinite dimensionality limit a distinction between the glass transition of liquids and the jamming transition of granular materials was posed in an analytical study of frictionless hard sphere particles [3][4][5][6][7] . As illustrated in Fig.…”

Section: Introductionmentioning

confidence: 99%

Universal behaviour of the glass and the jamming transitions in finite dimensions for hard spheres

2017

View full text Add to dashboard Cite

We investigate the glass and the jamming transitions of hard spheres in finite dimensions d, through a revised cell theory, that combines the free volume and the Random First Order Theory (RFOT). Recent results show that in infinite dimension the ideal glass transition and jamming transitions are distinct, while based on our theory we argue that they indeed coincide for finite d. As a consequence, jamming results into a percolation transition described by RFOT, with a static length diverging with exponent ν = 2/d, which we verify through finite size scaling, and standard critical exponents α = 0, β = 0 and γ = 2 independent on d.

show abstract

“…The use of replicas allows one to compute all the quantities listed in section I A, except the time-dependent correlators (point 1). Note in particular that the dynamical transition and the MCT parameter λ can be computed using replicas [8,65]. In particular, this approach can describe low-temperature glasses and the jamming transition.…”

Section: Introduction a The Exact Solution Of Infinite-dimensionamentioning

confidence: 99%

Quantitative approximation schemes for glasses

Mangeat

Zamponi

2016

Phys. Rev. E

Self Cite

View full text Add to dashboard Cite

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...

show abstract

Exact Theory of Dense Amorphous Hard Spheres in High Dimension. II. The High Density Regime and the Gardner Transition

Cited by 147 publications

References 71 publications

Force distribution affects vibrational properties in hard-sphere glasses

Force distribution affects vibrational properties in hard-sphere glasses

Universal behaviour of the glass and the jamming transitions in finite dimensions for hard spheres

Quantitative approximation schemes for glasses

Contact Info

Product

Resources

About