Equilibrium determination of the thermodynamic glass transition in the 50:50 binary soft-sphere mixture: large-N equilibration through the Γ ≈ 1.65–1.7 specific-heat anomaly and finite-size scaling

Grigera and Parisi (Phys. Rev. E 63, 045102(R), 2001) introduced a swap Monte Carlo algorithm for the 50:50 binary soft-sphere mixture (diameter ratio σB/σA = 1.2) and found, in systems of up to N = 800 particles, thermodynamic behavior compatible with a glass transition near Γ ≈ 1.7, close to the Kauzmann coupling ΓK ≈ 1.65 predicted by the hyper-netted-chain (HNC) replica calculation of Coluzzi et al. However, at N = 800 the lowest temperatures had not equilibrated and the key low-temperature values came from extrapolation, so the anomaly could not be definitively identified as a thermodynamic transition. The problem is the equilibrium determination: equilibrate substantially larger systems through the anomaly region with any simulation method that provably samples the equilibrium Boltzmann distribution, control crystallization of the metastable liquid, and perform finite-size analysis that distinguishes a genuine singularity from a rounded crossover. Later literature resolved the algorithmic prerequisites (local swap with an exact detailed-balance correction; continuously polydisperse swap-friendly models; mean-field and effective-potential theories of swap acceleration), but no study has equilibrated any three-dimensional atomistic glass-former near its putative Kauzmann point — every existing claim rests on temperature extrapolation, as the 2023 field review states explicitly — and a direct equilibrium demonstration of the Kauzmann transition was achieved only in 2026, and only in two dimensions. The surviving open core is the three-dimensional equilibrium thermodynamic determination.

Background

When a liquid is cooled below its melting point without crystallizing (a supercooled liquid), its structural relaxation time grows by many orders of magnitude until, at the laboratory glass transition, it falls out of equilibrium on the observation timescale. Whether this is merely a dynamical crossover or reflects an underlying thermodynamic phase transition is a central open question of glass physics. In the random first-order transition (RFOT) scenario, the mean-field replica theory of the glass transition, the configurational entropy $S_c$ — the entropy difference between the supercooled liquid and its vibrational ground states, counting the exponentially many metastable amorphous packings — extrapolates to vanish at a finite Kauzmann temperature $T_K$ . If this entropy crisis is reachable in equilibrium, the liquid would transform into an ideal glass, and the transition would appear as a thermodynamic anomaly, for instance in the configurational specific heat. Demonstrating (or excluding) such an equilibrium singularity in a real three-dimensional glass-former is beyond straightforward simulation, because with local particle moves the relaxation times near the putative $T_K$ exceed any computational budget.

A canonical test bed is the binary soft-sphere mixture: $N/2$ particles of species A and $N/2$ of species B interact through the purely repulsive inverse-power potential $V = \sum_{i<j} (\sigma_i + \sigma_j)^{12} / |\mathbf{r}_i - \mathbf{r}_j|^{12}$ , with diameter ratio $\sigma_B/\sigma_A = 1.2$ at number density $\rho = 1$ ; the two species strongly inhibit crystallization, although the liquid branch remains only metastable, and because the potential is homogeneous, all excess thermodynamic properties (those beyond the ideal-gas contribution) depend on temperature $T$ and density only through the single coupling constant $\Gamma = \rho\, T^{-1/4}$ , giving a one-dimensional phase diagram. Coluzzi, Mézard, Parisi, and Verrocchio computed the replica thermodynamics of this mixture within the hyper-netted-chain (HNC) approximation — an integral-equation closure for the liquid correlation functions — and predicted a Kauzmann point at $\Gamma_K \approx 1.65$ (J. Chem. Phys. 111, 9039, 1999).

Grigera and Parisi attacked the equilibration bottleneck with swap Monte Carlo (SW): in addition to standard single-particle displacements, the algorithm proposes exchanging the positions of two particles of different species; since the diameters enter the potential, the move changes the energy and is accepted with the Metropolis probability, which preserves detailed balance (the property guaranteeing that the Markov chain samples the correct Boltzmann equilibrium distribution). Swap moves let particles escape the cages transiently formed by their neighbors by exchanging places with differently sized partners, and thermalization of systems of hundreds of particles deep in the glassy regime became achievable. Their findings were: at $N = 34$ the configurational specific heat (the part of the specific heat coming from potential-energy fluctuations) shows a peak at $T \approx 0.12$ , i.e. $\Gamma \approx 1.7$ , close to the predicted $\Gamma_K$ ; at $N = 800$ the lowest temperatures 'clearly have not reached equilibrium', and the reported low-temperature values resulted from extrapolation; and the swap acceptance ratio itself collapses in the deeply supercooled region (about $10^{-4}$ at $\Gamma = 1.8$ ). They also remarked that applying SW to other binary systems would likely require modifications to achieve a reasonable swap acceptance ratio. What the paper leaves open is the definitive identification: is the specific-heat anomaly near $\Gamma \approx 1.7$ a genuine thermodynamic glass transition, i.e. an equilibrium entropy-crisis singularity at $\Gamma_K$ , or a crossover signature rounded by finite size and nonequilibrium effects? Deciding this requires equilibrating larger systems through the anomaly region, finite-size analysis (the systematic study of how observables converge with $N$ , which distinguishes a growing singular anomaly from a finite rounded peak), and control of the mixture's metastability against crystallization; the HNC approximation itself is known to perform poorly at low temperatures.

Problem Statement

Decide, on equilibrium grounds, whether the specific-heat anomaly near $\Gamma \approx 1.65$ –1.7 of the 50:50 binary soft-sphere mixture ( $\sigma_B/\sigma_A = 1.2$ , $\rho = 1$ ) is a thermodynamic glass transition — an equilibrium entropy-crisis (Kauzmann) singularity at the replica-predicted $\Gamma_K \approx 1.65$ — or a crossover. Settling the question requires, as one linked determination: (1) equilibrium thermodynamic data — the configurational specific heat and at least one independent observable such as the configurational entropy (the equation of state is not independent for this homogeneous potential: the excess pressure is a fixed multiple of the excess internal energy, whose temperature derivative is the configurational specific heat) — for substantially larger systems than $N = 800$ , at state points bracketing and entering the anomaly region, obtained with any simulation method that provably samples the equilibrium Boltzmann distribution (swap variants, local swap, molecular dynamics, or new algorithms; algorithmic development is part of the task); (2) a finite-size analysis over several system sizes that distinguishes a growing, genuinely singular anomaly from a rounded crossover; and (3) control of crystallization and metastability of the liquid in the equilibrated regime. The question is open to either resolution — a located equilibrium transition, or a demonstrated crossover — and is settled by an equilibrium determination of the equilibrium liquid thermodynamics of this mixture sufficient to resolve it. The algorithmic sub-questions — whether swap generalizes to other binary mixtures with larger size disparity, and how to raise the productive swap-acceptance rate while preserving equilibrium sampling — are out of scope; they survive as prerequisites and tools of the determination above, not as open problems in themselves.

References

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