Quantitative equilibrium test of the mean-field (random first-order transition) scenario for structural glass formers: overlap statistics, configurational entropy, and the Kauzmann temperature

Coluzzi and Parisi (1998) simulated a three-dimensional binary soft-sphere glass former with parallel-tempering Monte Carlo and found qualitative support for the mean-field scenario of the glass transition: below a dynamical crossover the energy relaxes in two steps (a mean-field-like power law followed by activated decay), and the equilibrium distribution $P(d)$ of the distance $d$ between independent equilibrium configurations becomes highly nontrivial in the glassy phase. They stated the validity of the mean-field scenario for short-range models such as structural glasses as an open question, and identified as the natural next step obtaining the behaviour of the measured quantities, in particular the equilibrium $P(d)$, averaged over different system sizes toward the thermodynamic limit, with controlled finite-size and crystallization effects. Swap Monte Carlo now equilibrates suitably polydisperse glass formers far below the mode-coupling temperature; configurational entropy has been measured down to the experimental glass regime, but its vanishing at a putative finite Kauzmann temperature remains an extrapolation in three dimensions; in two dimensions the Kauzmann transition has been directly observed to occur at zero temperature; and overlap statistics have been measured with and without a conjugate field, but the one-step replica-symmetry-breaking-predicted nontrivial equilibrium overlap distribution in the thermodynamic limit is still unobserved in three dimensions. The quantitative verification or refutation of the mean-field scenario for three-dimensional structural glasses in equilibrium therefore remains the open problem.

Background

When a liquid is cooled fast enough to avoid crystallization it becomes a glass, and whether an underlying equilibrium thermodynamic transition exists behind the laboratory glass transition is a central unresolved question of condensed-matter physics. The mean-field scenario, also called the random first-order transition (RFOT) theory and derived from exactly solvable infinite-range spin-glass models, makes the following predictions. Below a dynamical transition temperature $T_D$ (the temperature where idealized mode-coupling theory, a kinetic theory of density fluctuations, predicts a dynamical singularity) the system gets trapped for long times in metastable amorphous states. The number of such states grows exponentially with the number of particles $N$ as $\exp(N\Sigma(T))$, where $\Sigma(T)$ is the configurational entropy, or complexity. Upon cooling, $\Sigma(T)$ is predicted to vanish linearly at a lower temperature $T_K$, the Kauzmann temperature, where an 'ideal' glass transition to a single amorphous state occurs and the relaxation time is predicted to diverge in the Vogel-Fulcher form, $\tau\propto\exp[A/(T-T_K)]$ with constant $A$. Below $T_D$ the energy density relaxes in two steps: a mean-field-like convergence toward metastable states, $e(t)-e_D\propto t^{-\alpha}$, followed by an activated decay of the metastable states through barrier crossing. A natural order parameter is the overlap, or equivalently a distance $d$, between two independent equilibrium configurations of the same system; if the system explores many competing amorphous states, the equilibrium distribution $P(d)$ of this distance becomes highly nontrivial (non-Gaussian, multi-peaked), which is the finite-dimensional analogue of one-step replica symmetry breaking, the mean-field solution structure in which the Gibbs measure splits into an exponentially large number of pure states.

Coluzzi and Parisi (1998) tested this scenario on a three-dimensional binary mixture of soft spheres (a 50/50 mixture of two particle species interacting through the steeply repulsive soft-sphere potential $V_{ik}(r)\propto[(\sigma_i+\sigma_k)/r]^{12}$, with diameter ratio $\sigma_B/\sigma_A=1.2$ chosen to inhibit crystallization; by a scaling property of soft spheres all thermodynamics depends on a single coupling $\Gamma\propto T^{-1/4}$, so larger $\Gamma$ means lower temperature). Because ordinary Monte Carlo cannot equilibrate the glassy regime, they used parallel tempering (replica exchange) Monte Carlo, in which copies of the system at different temperatures probabilistically exchange temperatures, but were restricted to very small systems, $N\le 36$, simulated in a container with interacting walls (no periodic boundary conditions). They observed two-step energy relaxation with exponent $\alpha\approx 0.8$ for the power-law first step and an activated second step compatible with a stretched exponential (a decay $\exp[-(t/\tau)^{\beta}]$ with $0<\beta<1$), and a highly nontrivial, strongly $N$-dependent $P(d)$ in the glassy phase. They could not resolve the temperature dependence of $\alpha$ or the functional form of the activated step, could not extract $\Sigma(T)$ or $T_K$ at their resolution, and observed a crystallization artifact at the special size $N=32$, where particles can fill a face-centered-cubic lattice. They stated the validity of the mean-field scenario for short-range models such as structural glasses as an open question and identified as the natural next step obtaining the behaviour of the measured quantities averaged over different $N$ values, noting that a slightly different definition of distance seems to be more suitable for mixtures.

Later work replaced the distance $d$ by the standard overlap $Q$ (a normalized similarity between configurations, $Q=0$ for unrelated and $Q=1$ for identical states) and, decisively, introduced swap Monte Carlo, in which particles of different sizes exchange positions and which equilibrates suitable size-polydisperse glass formers far below $T_D$ (Ninarello, Berthier, and Coslovich, 2017). The test is posed for the original formulation and asks for the quantitative verdict that the 1998 study could not deliver.

Problem Statement

For a three-dimensional structural glass-forming model (the binary soft-sphere mixture studied by Coluzzi and Parisi, or a related model designed for efficient equilibrium sampling), make the test of the mean-field scenario quantitative, replacing the 1998 qualitative comparison with measured numbers, exponents, and scales. The single scientific objective is a quantitative verdict, from equilibrium data, on whether the mean-field scenario - the random first-order transition (RFOT) framework, whose one-step-replica-symmetry-breaking solution predicts an exponentially large number of competing long-lived amorphous states - describes finite-dimensional glass formers. The quantitative tests to perform are: (1) Equilibrium order-parameter statistics: measure the equilibrium distribution $P(d)$ of the distance $d$ between two independent equilibrium configurations of the same system (equivalently, the distribution of the normalized overlap $Q$, with $Q=0$ for unrelated and $Q=1$ for identical configurations), at system sizes and temperatures allowing controlled extrapolation to the thermodynamic limit, including its finite-size scaling, and determine whether it develops the predicted nontrivial multi-valley structure. (2) Equilibrium thermodynamics: determine the configurational entropy (complexity) $\Sigma(T)$ - the entropy density associated with the number $\exp(N\Sigma)$ of competing amorphous metastable states - from equilibrium data under an operational definition that is stated explicitly, together with its assumptions and its bearing on the finite-$T_K$ verdict, and decide whether it vanishes at a finite Kauzmann temperature $T_K$ or behaves otherwise (for example, a zero-temperature entropy crisis). (3) Dynamics-thermodynamics connection: quantitatively characterize the two-step relaxation below the dynamical crossover (the temperature $T_D$ where idealized mode-coupling theory, a kinetic theory of density fluctuations, predicts a dynamical singularity) - the exponent $\alpha$ of the power-law first step and its temperature dependence, and the functional form (for example, stretched-exponential) of the activated second step - and relate it to the measured equilibrium quantities; the 1998 study measured this two-step relaxation after an abrupt quench from a random initial configuration, and either such a quench-relaxation protocol or the corresponding equilibrium time-correlation functions are admissible. The test is passed or failed by quantitative agreement or disagreement of these observables with the mean-field predictions, not by qualitative resemblance. The mean-field scenario holds for the model studied only if the equilibrium order-parameter statistics, the thermodynamic behaviour of the configurational entropy, and the dynamics-thermodynamics relation all agree quantitatively with its predictions; a quantitative disagreement of any single observable with its prediction by itself refutes that prediction.

References

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