Consolidating configurational entropy measurements for three-dimensional glass-formers and the fate of the Kauzmann transition

The configurational entropy $S_{\mathrm{conf}}$, the entropy associated with the multiplicity of amorphous metastable states of a supercooled liquid, is the central thermodynamic observable of entropy-crisis theories of the glass transition, and its extrapolated vanishing defines the Kauzmann temperature $T_K$. Ozawa, Parisi, and Berthier (2018) introduced a generalized Frenkel-Ladd scheme that makes $S_{\mathrm{conf}}$ computable for continuously polydisperse glass-formers, and found that it agrees only approximately with the Franz-Parisi overlap free-energy approach: a slight discrepancy of functional form for polydisperse hard spheres and a much larger one for the binary Kob-Andersen mixture. They also showed that the measured $S_{\mathrm{conf}}(T)$ is compatible both with a finite $T_K$ and with an avoided-transition scenario. The open problem is to consolidate the independent configurational entropy measurement schemes into a mutually consistent thermodynamic characterization of three-dimensional glass-formers - resolving the hard-sphere functional-form discrepancy and the Kob-Andersen discrepancy - and thereby to determine unambiguously whether $S_{\mathrm{conf}}$ vanishes at a finite Kauzmann temperature or the transition is avoided. Later literature confirms partial progress (overlap-definition studies, several new independent methods, and a full method consolidation plus a zero-temperature Kauzmann transition demonstrated in two dimensions in 2026), but the three-dimensional consolidation and the three-dimensional Kauzmann question remain open.

Background

When a liquid is cooled fast enough to avoid crystallization, it falls out of equilibrium at the laboratory glass transition and becomes an amorphous solid. The leading theoretical account of this process, the random first-order transition (RFOT) theory, holds that underlying the kinetic transition there is a thermodynamic one: the number of amorphous metastable states available to the liquid becomes exponentially rare as temperature decreases, so that the 'configurational entropy' $S_{\mathrm{conf}}$ - the part of the liquid entropy that counts these distinct amorphous packings, as opposed to vibrations within one packing - decreases and, if extrapolated, vanishes at a finite Kauzmann temperature $T_K$. This is the modern form of the Kauzmann paradox: below $T_K$ the supercooled liquid would have entropy lower than the crystal unless an 'ideal glass' transition intervenes. Whether such a finite-temperature entropy crisis truly occurs in physical (three-dimensional) systems is a long-standing open question; the alternative is that the transition is avoided, with $S_{\mathrm{conf}}$ undergoing a qualitative change at a crossover temperature rather than vanishing.

Computer simulation is currently the only route to equilibrium $S_{\mathrm{conf}}$ measurements deeply below the experimental glass transition, but it faces two obstacles that motivate the source paper. First, realistic glass-formers are often continuously polydisperse - their particle diameters are drawn from a continuous distribution, as in colloidal suspensions - and for such systems the mixing entropy associated with permuting distinguishable particles diverges with the resolution of the diameter distribution, a variant of the Gibbs paradox that makes naive $S_{\mathrm{conf}}$ definitions ill-defined (Ozawa and Berthier, 2017; Baranau and Tallarek, 2018). Second, the available measurement schemes rest on different physical constructions and have never been shown to agree. The main schemes are: (i) the potential energy landscape (PEL) or inherent-structure approach, which counts the minima of the potential energy (Berthier et al., 2017); (ii) the Frenkel-Ladd route, in which the absolute entropy of a solid phase is obtained by thermodynamic integration from a reference 'Einstein crystal' in which each particle is tethered by a harmonic spring - Ozawa, Parisi, and Berthier (2018) generalized it to glasses by allowing Monte Carlo permutations of the particle diameters during the integration, which automatically supplies a finite, physically meaningful glass mixing entropy and captures non-vibrational contributions; the integration range in the spring stiffness $\alpha$ must be cut off at a state-dependent $\alpha_{\min}$ chosen inside a plateau of the mean-squared displacement; and (iii) the Franz-Parisi approach (Franz and Parisi, 1997; Berthier and Coslovich, 2014), which measures a Landau free energy $V(Q)$ as a function of the overlap $Q$ - a number between 0 and 1 quantifying how similar two independent equilibrium configurations are, given a tolerance of order the interparticle spacing - and identifies $S_{\mathrm{conf}}$ with the free-energy cost of confining the system to a single metastable state; this path, however, crosses an equilibrium first-order transition in the (field, temperature) plane and is computationally expensive. Deep equilibration of polydisperse models is made possible by swap Monte Carlo, which exchanges particle positions (diameters) and bypasses much of the glassy slowdown.

Comparing schemes (ii) and (iii) for three standard three-dimensional glass-formers (continuously polydisperse soft spheres, continuously polydisperse hard spheres, and the Kob-Andersen binary Lennard-Jones mixture, a canonical 80:20 two-component computer glass-former with non-additive cross interactions), the source paper found the two constructions to give comparable but not identical results: a slight discrepancy in the functional form of $S_{\mathrm{conf}}(T)$ and in the resulting $T_K$ for polydisperse hard spheres, and a still 'quite large' distinction for the Kob-Andersen model. Fitting the empirical form $S_{\mathrm{conf}}/N = A(1 - T_K^*/T^*)$ (with $T^*$ the temperature normalized by the crossover temperature of mode-coupling theory, a mean-field theory predicting that the liquid relaxation time diverges at a finite temperature) yielded finite $T_K^*$ for all three models, yet the data equally admit an avoided-transition scenario below the lowest equilibrated temperature. The authors state that consolidating the mutual consistency among different configurational entropy measurements would be an important step toward the complete thermodynamic characterization of the glass transition. They also flag, explicitly outside the scope of the configurational entropy measurement itself, that the entropy of polydisperse fluids in general remains plagued by the potential-infinity problems of the Gibbs paradox, and that extensions of their scheme to canonical-ensemble phase equilibria and to granular entropy are undeveloped. The source paper names, among the refinements that could affect the comparison of the two schemes, a state-dependent choice of the integration cutoff $\alpha_{\min}$ in the Frenkel-Ladd scheme and refined definitions of the overlap function and its coarse-graining length in the Franz-Parisi scheme.

Problem Statement

Establish mutual consistency among the independent computational measurements of the configurational entropy of three-dimensional glass-formers, and use it to determine the fate of the configurational entropy at low temperature. Concretely: (1) reconcile the generalized Frenkel-Ladd scheme of Ozawa, Parisi, and Berthier (2018), which computes $S_{\mathrm{conf}} = S_{\mathrm{tot}} - S_{\mathrm{glass}}$ with diameter permutations included in the glass entropy, with the Franz-Parisi overlap free-energy approach (and with potential-energy-landscape estimates), for the model systems where discrepancies were reported - the slight functional-form discrepancy and resulting $T_K$ shift for continuously polydisperse hard spheres, and the much larger discrepancy for the Kob-Andersen binary mixture - determining whether the schemes provably measure (or provably do not measure) the same quantity, or quantitatively accounting for their difference. (2) With the relation between the measurement schemes established, determine whether $S_{\mathrm{conf}}(T)$ in the three source models (continuously polydisperse soft spheres, continuously polydisperse hard spheres, and the Kob-Andersen binary mixture) vanishes at a finite Kauzmann temperature $T_K$ (as the empirical fit $S_{\mathrm{conf}}/N = A(1 - T_K^*/T^*)$ suggests) or whether the sharp transition is avoided and $S_{\mathrm{conf}}$ instead changes qualitatively at a crossover temperature, requiring equilibrium access to temperatures at or below the putative $T_K$. Resolve, or quantitatively account for, the two reported scheme discrepancies - the functional-form and resulting $T_K$ shift for continuously polydisperse hard spheres and the much larger discrepancy for the Kob-Andersen binary mixture - and determine, with quantified error bars and finite-size analysis, which scenario the three source models realize, together with the physical origin of any residual difference between measurement schemes (for instance whether it stems from non-vibrational contributions to the glass entropy, from the overlap definition, or from non-ergodic sampling). The problem is posed for three-dimensional glass-formers as in the source work; the two-dimensional case, recently settled separately (Jung et al., 2026: consolidation achieved and a Kauzmann transition found at a temperature vanishing in the thermodynamic limit), is a solved limiting variant, not the target.

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