Kurchan, Parisi, and Zamponi (J. Stat. Mech. 2012) derived the exact free energy of dense amorphous hard spheres in the limit of large space dimension d→∞d\to\infty, showing that a Gaussian cage ansatz gives the exact thermodynamics once arbitrary replica symmetry breaking is allowed, and they explicitly listed the follow-up problems this free energy leaves open. Later work in the same series resolved two of them: the one-step replica-symmetry-breaking saddle is unstable at high pressure toward a Gardner transition and a full replica-symmetry-breaking solution, which is marginally stable and isostatic at jamming. The surviving open core comprises the following three fronts: (i) a controlled computation of the non-Gaussian cage distribution — the self part of the van Hove function and its large-deviation tails — within the exact d→∞d\to\infty framework, together with the quantitative comparison to high-dimensional simulations that the source promised; (ii) the finite-dimensional corrections caused by molecular dissociation (exponentially small in dd, now generalized to replica mismatches associated with hopping) and their thermodynamic consequences; and (iii) a replica-free, density-functional derivation of the same exact thermodynamics that solves for the amorphous reference positions without assuming their structure factor equals that of the liquid.

Background

Hard spheres — particles with no interaction except excluded volume — are the minimal model of the glass transition: on densification a liquid can avoid crystallizing and instead becomes a glass, in which each particle is confined by its neighbors to a cage, until at jamming the packing becomes mechanically blocked. In the limit of large space dimension d→∞d\to\infty (with packing fraction scaled as ϕ∼d 2−d\phi\sim d\,2^{-d}) the virial expansion of the liquid is controlled and the Van der Waals mean-field entropy becomes exact, so an essentially first-principles statistical-mechanics calculation of the amorphous phase is possible. Kurchan, Parisi, and Zamponi carried out this program for the free energy in 'Exact theory of dense amorphous hard spheres in high dimension I. The free energy' (J. Stat. Mech. 2012). Their construction uses the replica method: mm identical copies of the liquid are coupled so that in a glass state each particle stays near its copies, the group forming a 'molecule' whose size measures the cage, i.e. the region a particle explores around its reference position (conveniently quantified by the mean-square displacement order parameter Δ\Delta). Treating the replicated system as a liquid of molecules and taking d→∞d\to\infty before optimizing, they obtained the exact free energy of the glass at any pressure. A striking finding is that a Gaussian ansatz for the cage shape (each particle distributed as a Gaussian around its reference position) reproduces the exact thermodynamics — provided arbitrary replica symmetry breaking is allowed — even though the true cage is not Gaussian. Here 'one-step replica symmetry breaking' (1RSB) is the simplest replica scheme for glasses: the glass phase is an exponential number of metastable amorphous states, counted by the complexity (configurational entropy), each with its own cage; higher levels of replica symmetry breaking (RSB) describe a hierarchy of substates within states, as in spin glasses. Simulations show that the cage — measured by the self part of the van Hove function, i.e. the distribution of a single particle's displacement from its long-time (reference) position — keeps non-Gaussian tails that do not diminish as dd grows, so exact Gaussian thermodynamics coexists with a non-Gaussian cage, an 'apparent paradox' the paper discusses: it explains why the thermodynamics is insensitive to the cage shape, but leaves the actual computation of the non-Gaussian tails open. The paper also notes that the molecular description assumes molecules never dissociate (each particle stays tagged to the reference position of its molecule); this is self-consistent in d→∞d\to\infty, but at any finite dd there is an exponentially small probability of dissociation, whose thermodynamic consequences the paper leaves uncomputed. Finally, the authors ask whether the same exact thermodynamics could be obtained without replicas at all, by a direct density-functional theory (DFT) construction that solves for the amorphous reference positions {Ri}\{R_i\}; existing DFT approaches must assume that the structure factor of the {Ri}\{R_i\} equals that of the equilibrium liquid — a hypothesis the authors argue is likely false and that the replica method was designed to avoid. The high-pressure part of this program was subsequently completed by the same collaboration: the equilibrium 1RSB glass was shown to become unstable at a Gardner transition (a continuous glass-glass transition, found by Gardner in spin glasses, below which each amorphous state splits into a hierarchy of marginal substates), and the resulting full-RSB solution was constructed; it is marginally stable, predicts that jammed packings are isostatic (the number of interparticle contacts just equals the number of degrees of freedom, z=2dz=2d, the signature of marginal mechanical stability), and yields the jamming critical exponents, in agreement with numerics. What survives as open is therefore not the thermodynamics of the d→∞d\to\infty glass, which is settled, but the exact description of the amorphous state that the Gaussian-exact free energy is blind to: its structure (the cage distribution and its tails), its stability away from d=∞d=\infty (molecular dissociation and its generalization to replica mismatches, i.e. particle-level hopping), and its formulation independence (a replica-free derivation).

Problem Statement

Within the exact d→∞d\to\infty replica theory of dense amorphous hard spheres founded by Kurchan-Parisi-Zamponi (J. Stat. Mech. 2012) — whose Gaussian-ansatz free energy is now exact and whose high-pressure phase is the established full-RSB solution — the following three fronts, explicitly named by the source paper as unresolved, remain open. The common objective is to determine the structure, finite-dimensional stability, and formulation independence of the amorphous glass state itself, which the Gaussian-exact free energy is blind to. The three fronts are independent open problems.

  1. Cage structure (source Conclusions, second paragraph). Compute, within the exact replica framework of the source (its general free-energy functional, at the appropriate level of replica symmetry breaking, with controlled d→∞d\to\infty saddle points and no assumed cage shape), the cage distribution in the glass phase — the self part of the van Hove function, i.e. the distribution of a single particle's displacement from its reference position — including its non-Gaussian large-deviation tails. The cage distribution and its large-deviation tails are to be determined at the dynamical glass transition and at jamming, and compared quantitatively with the persistent non-Gaussian tails observed in high-dimensional simulations.

  2. Finite-dimensional corrections from molecular dissociation (source Section II and Section VIII). Compute the corrections to the exact d=∞d=\infty thermodynamics arising away from d=∞d=\infty from molecular dissociation: the exponentially small probability that a particle escapes the molecular reference position of the replica construction (in modern language, replica mismatches associated with hopping), and its consequences for the dynamical and thermodynamic glass transitions and for jamming.

  3. Replica-free derivation (source Conclusions, third paragraph). Derive the same exact thermodynamic results without using replicas: a direct density-functional construction that solves self-consistently for the amorphous reference positions {Ri}\{R_i\} of the glass, without invoking the hypothesis that the structure factor of the {Ri}\{R_i\} equals that of the equilibrium liquid.

References

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