Thermodynamic glass transition and asymptotic generalized fluctuation–dissipation relations in binary glass formers

Parisi's 1997 Monte Carlo simulations of a 50:50 binary soft-sphere glass former found numerical indications of a thermodynamic transition to a glassy phase, below which the off-equilibrium correlation and response functions appeared asymptotically compatible with the Cugliandolo–Kurchan generalized fluctuation–dissipation relations originally derived for mean-field spin glasses, in their one-step-replica-symmetry-breaking (two-temperature) form. Whether these indications reflect the true asymptotic, thermodynamic-limit behavior of finite-dimensional binary glasses has never been settled. The problem is to establish, by controlled computation or theory, (i) whether the three-dimensional binary soft-sphere glass former studied by Parisi undergoes a genuine finite-temperature thermodynamic transition to an ideal glass phase, and (ii) whether its aging dynamics asymptotically obeys the generalized fluctuation–dissipation relation with a well-defined effective temperature predicted by one-step replica symmetry breaking. Later work confirmed the picture at finite times and in mean field, and a 2026 equilibrium simulation study resolved the two-dimensional case (Kauzmann temperature vanishing in the thermodynamic limit), but the three-dimensional thermodynamic question and the asymptotic fluctuation–dissipation question both remain open.

Background

When a liquid is cooled fast enough to avoid crystallization it becomes a supercooled liquid whose relaxation time grows by many orders of magnitude, until at the laboratory glass transition it falls out of equilibrium on the experimental timescale and becomes an amorphous solid, a glass. Whether, given unlimited time, this slowdown terminates in a true thermodynamic phase transition to an 'ideal glass' phase — the entropy-crisis picture going back to Kauzmann (1948), modernized as random first-order transition (RFOT) theory — or is purely kinetic, is one of the central open questions of condensed-matter physics. Binary mixtures of two particle sizes (for example soft spheres with repulsive $r^{-12}$ interactions, or Lennard-Jones mixtures) are the standard computer models of glass formers, because the size disparity frustrates crystallization.

Mean-field theory of disordered systems gives a sharp prediction for what such a transition would look like dynamically. In certain infinite-range spin-glass models (generalized $p$-spin glasses), below a dynamic transition temperature the system never equilibrates: it ages, meaning its two-time correlation $C(t_w+t,t_w)$ depends on the waiting time $t_w$ elapsed since the preparation quench. Cugliandolo and Kurchan showed that in this aging regime the fluctuation–dissipation theorem (FDT — the equilibrium law relating the linear response of an observable to a perturbation to the time derivative of its spontaneous correlation) is replaced by a generalized relation: a parametric plot of the integrated response $\chi$ versus the correlation $C$ consists of a segment with the equilibrium slope $1/T$ followed by a second straight segment with slope $1/T_{\mathrm{eff}}$, defining an effective temperature $T_{\mathrm{eff}}>T$ for the slow degrees of freedom. The shape of the fluctuation–dissipation ratio $X(C)$ encodes the replica symmetry breaking (RSB) structure of the phase space — one-step RSB (1RSB) for structural-glass-type models, where $X(C)$ drops from 1 to a lower value at the plateau correlation $q_{\mathrm{EA}}$ (the Edwards–Anderson-type order parameter: the plateau value of the slow two-time correlation in the aging state; defined in the source paper through the stress fluctuations and denoted there $q_A$), versus full RSB for spin glasses. The same 1RSB thermodynamics predicts an entropy crisis: the configurational entropy (the entropy associated with the number of distinct amorphous metastable states) vanishes at a Kauzmann temperature $T_K$, below which an ideal glass phase exists; the effective temperature of the aging sector is predicted to coincide with the temperature conjugate to this configurational entropy.

Parisi (1997) simulated a 50:50 binary soft-sphere mixture (radius ratio $\sigma_B/\sigma_A=1.2$, Monte Carlo dynamics, $N=18$–$66$ particles, control parameter $\Gamma=\beta^4$ with the known glass crossover near $\Gamma\approx1.45$). After quenches he monitored the energy and the stress autocorrelation, and measured the response to a perturbation (deforming the simulation box) to construct the response–correlation plot and the function $X(C)$. He reported: a low-temperature region where large-time extrapolations show simple power-law corrections; a stress autocorrelation larger than the analytic continuation of the high-temperature behavior; approximate simple aging; and an $X(C)$ whose waiting-time dependence seemed to decrease, with small-$t_w$ data resembling the 1RSB prediction. He concluded that the indications pointed to a thermodynamic transition to a glassy phase with off-equilibrium dynamics asymptotically compatible with the Cugliandolo–Kurchan relations, while stressing that the extrapolation of $X(C)$ to infinite waiting time is delicate and would require roughly a hundredfold increase in computer time (larger waiting times, larger systems, better statistics), closing with the remark that it would be interesting to see whether the indications would be confirmed by more accurate and lengthy computations.

The two facets are inseparable in the underlying theory: the dynamical signature (asymptotic two-temperature FDT) and the thermodynamic signature (transition at $T_c$, which in the modern RFOT reading corresponds to an entropy crisis at $T_K$) are two faces of the same 1RSB/RFOT scenario, so confirming or refuting the source paper's claim requires addressing both. Subsequent developments — finite-time fluctuation–dissipation measurements in binary Lennard-Jones mixtures, inherent-structure thermodynamics, replica computations of $T_K$, swap Monte Carlo equilibration of extremely supercooled liquids, and the recent full equilibrium characterization of a two-dimensional glass former — settle neither the three-dimensional thermodynamic question nor the asymptotic waiting-time limit.

Problem Statement

Establish whether the source's three-dimensional 50:50 binary soft-sphere glass former undergoes a genuine thermodynamic transition at a finite temperature to a glassy (ideal-glass) phase, and whether, below that transition, its off-equilibrium dynamics is asymptotically governed by the Cugliandolo–Kurchan generalized fluctuation–dissipation relations in their one-step-replica-symmetry-breaking form — the two claims for which Parisi's 1997 simulations of this model found numerical indications. Concretely, in three dimensions, for the 50:50 binary soft-sphere mixture with $\sigma_B/\sigma_A=1.2$ ($r^{-12}$ repulsive interactions, $\rho=1$, control parameter $\Gamma=\beta^4$):

1. Thermodynamic signature: establish whether a genuine equilibrium thermodynamic transition to a non-ergodic glass state occurs at a finite temperature in the thermodynamic limit — the transition being characterized by the entropy crisis (the equilibrium configurational entropy $S_{\mathrm{conf}}(T,N)$ vanishing at a finite Kauzmann temperature $T_K$) and by the source's own characterization (a nonzero stress-fluctuation order parameter $q_A$ with non-analytic behavior of equilibrium quantities at a transition temperature $T_c$) — or whether the apparent transition is avoided (for instance $T_K$ vanishing in the thermodynamic limit, or crossed over smoothly), identifying the transition temperature where it exists.

2. Dynamical signature: determine the asymptotic relation between two-time correlations and conjugate responses in the coupled limit of large waiting time $t_w$ and large system size $N$ — the two limits do not commute, so the thermodynamic limit must be taken before, or jointly with, $t_w\to\infty$, with $N$ growing with time so that finite-volume effects remain controlled, as the source paper explicitly warns. For the correlation–response pair of a slow observable and its conjugate perturbation — in the source, the stress asymmetry $A$ conjugate to a volume-preserving box deformation, with the overlap $Q$ noted as a promising alternative — does the fluctuation–dissipation ratio $X(C)$ take the one-step form (equilibrium FDT slope $1/T$ above the plateau correlation $q_{\mathrm{EA}}$, followed by a second straight segment set by a temperature-dependent effective temperature), independently of the observable, as mean-field one-step-RSB theory predicts, or do activated finite-dimensional processes deform it (observable-dependent ratios, coarsening-like limits, or no single effective temperature)?

The two signatures are two faces of one overarching objective — the realization of the mean-field one-step-RSB/RFOT scenario in finite-dimensional binary glasses, which predicts that the effective temperature of the aging sector coincides with the temperature conjugate to the configurational entropy — whose resolution requires both.

References

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