A microscopic dynamical theory of the glass transition that is exact in the large-dimensional limit and reduces to mode-coupling phenomenology in three dimensions

Simulations of hard spheres in dimensions $d=3$ to $12$ (Charbonneau, Ikeda, Parisi, Zamponi, Phys. Rev. Lett. 107, 185702 (2011)) showed that the numerically measured dynamical transition density follows the static replica-theory scaling while the prediction of standard mode-coupling theory (MCT) grows far too fast with $d$ and would even cross the jamming threshold near $d\approx 13$. The authors concluded that a new dynamical theory must be constructed that interpolates between the successful MCT phenomenology at $d=3$ and the exact mean-field regime at $d\to\infty$, and asked whether it would be a minor modification of MCT or a fundamentally new formalism, as part of completing the random first-order transition (RFOT) framework. Subsequent work solved the equilibrium dynamics exactly at $d\to\infty$ (Maimbourg–Kurchan–Zamponi, 2016, with the ergodicity-breaking transition identical to the replica prediction) and produced a simple replica-consistent mean-field dynamical theory (Szamel, 2017), but no single microscopic theory yet connects these limits quantitatively to $d=3$. The construction of such an interpolating theory and its embedding in a complete RFOT remain open; the paper also noted that the activated (hopping) relaxation regime is inaccessible to high-dimensional simulation (it would require compaction rates $\gamma\lesssim e^{-d}$) and can therefore be approached only theoretically.

Background

When a liquid is supercooled or compressed, its structural relaxation time grows by many orders of magnitude until it falls out of equilibrium and forms a glass; for hard spheres — identical particles that cannot overlap — the control parameter is the packing fraction $\varphi$, the fraction of volume covered by the spheres, and the athermal endpoint of the process is jamming at the random close-packing density. Because spatial dimension $d$ suppresses local fluctuations much as mean-field theory does for ordinary phase transitions, the limit $d\to\infty$ offers a solvable starting point for the glass problem, with the natural scaling of density as $\hat\varphi=2^d\varphi/d$.

Two first-principles descriptions of glass formation exist and are supposed to be two halves of one theory. Mode-coupling theory (MCT) is dynamical: it writes closed, self-consistent equations for the time-dependent density autocorrelation function, closing them by a factorization approximation that expresses higher-order correlators through the static structure factor of the liquid; it predicts an ergodicity-breaking transition at a density $\varphi_{\mathrm{MCT}}$ above which density fluctuations never relax, with a universal two-step relaxation pattern that is quantitatively successful in $d=3$ (Götze, 2009). Replica theory (RT) is static: following the random first-order transition (RFOT) theory of the glass transition (Kirkpatrick–Wolynes 1987; Kirkpatrick–Thirumalai–Wolynes 1989; Mézard–Parisi 1999), it replicates the liquid in a quenched amorphous reference state and computes the entropy (complexity) of metastable glass states, predicting a dynamical transition density $\varphi_d$ where metastable amorphous states first appear, a Kauzmann density $\varphi_K$ where their complexity vanishes (ideal glass), and a glass close-packing density $\varphi_{\mathrm{GCP}}$ where they jam (Parisi–Zamponi, Rev. Mod. Phys. 82, 789 (2010)). RFOT asserts that MCT is the dynamical counterpart of RT, so in a genuinely mean-field limit the two must coincide.

They do not. At large $d$, RT is asymptotically exact and predicts $\hat\varphi_d=2^d\varphi_d/d\to 4.80678$, whereas standard MCT (with the structure factor computed in the Percus–Yevick approximation, a standard closure of liquid-state theory) predicts $\varphi_{\mathrm{MCT}}\sim 0.22\,d^2\,2^{-d}$, i.e. $\hat\varphi\sim 0.22\,d$ growing without bound. This discrepancy was identified by Schmid–Schilling (2010) and Ikeda–Miyazaki (2010), and Szamel (2010) traced its origin to the factorization approximation in the MCT closure. Charbonneau, Ikeda, Parisi, Zamponi (Phys. Rev. Lett. 107, 185702 (2011)) tested the two theories against event-driven simulations of hard spheres in $d=3$ to $12$: the measured dynamical transition density $\varphi_d$ and threshold jamming density $\varphi_{\mathrm{th}}$ follow the RT trends ($\varphi_K>\varphi_d$, $\varphi_{\mathrm{GCP}}>\varphi_{\mathrm{th}}$), while the MCT prediction rises so fast with $d$ that it would cross the numerical $\varphi_{\mathrm{th}}$ near $d\approx 13$ — paradoxical, since a liquid should fall out of equilibrium well before it jams. The paper concluded that key ingredients of high-dimensional physics are missing from MCT and that 'a new theory should be constructed, in order to interpolate between MCT in $d=3$ and the large $d$ mean-field regime', explicitly asking whether this would be 'a minor modification of MCT or a completely new theory', and noting that a complete RFOT combining RT with the modified dynamics — opening the way to a controlled large-$d$ expansion or a renormalization-group treatment — would be the first complete theory of the glass and jamming transitions for particle-based systems. It further observed that the activated (thermally assisted) relaxation regime, in which hopping over barriers restores ergodicity beyond the mean-field transition, is inaccessible to high-dimensional simulation because it would require compaction rates $\gamma\lesssim e^{-d}$, so only theoretical control can reach it.

Problem Statement

Construct a microscopic dynamical theory of the glass transition for particle systems — with hard spheres as the reference case — that simultaneously satisfies two limiting requirements:

1. It is exact in the limit $d\to\infty$: solving its self-consistency equations must reproduce the exact mean-field dynamics of dense liquids, in particular a mean-field (non-activated) ergodicity-breaking (dynamical) transition at $\hat\varphi_d=2^d\varphi_d/d=4.80678$ for hard spheres, identical to the replica-theory prediction for the equilibrium dynamics without hopping corrections, and consistent with the large-$d$ trend of simulated dynamical transition densities.

2. It reduces to the successful mode-coupling-theory phenomenology in $d=3$ (two-step relaxation, square-root cusp of the non-ergodicity parameter — the long-time plateau height of the density autocorrelation function — standard $\beta$-relaxation scaling), or else establishes precisely which modification of the standard MCT closure is required and in what sense standard MCT is recovered or superseded. This second alternative is a component of the resolution, not an independent completion standard: merely diagnosing which modification of the MCT closure is needed does not by itself satisfy requirement 1 or resolve the task, and the identified modification must be carried through into a theory that does.

As part of the same objective, the answer must also: (a) determine whether such a theory can be obtained by a minor modification of the standard MCT closure or requires a fundamentally new dynamical formalism; and (b) yield quantitative, testable predictions for the dynamical transition density and structural relaxation at intermediate dimensions (e.g. $d=4$ to $12$, where simulation data exist), embedding the dynamics and the static replica theory in one consistent RFOT framework. All four demands — requirements 1 and 2, and (a) and (b) — are to be met together: satisfying requirement 1 alone, or answering only the classification question (a), does not resolve the task. The source paper notes that such a complete RFOT would open the way towards a more systematic construction through a large-$d$ expansion or a renormalization-group approach. The activated (thermally assisted, hopping) relaxation regime beyond the mean-field dynamical transition is relevant background — at high $d$ activation barriers grow linearly with dimension, the regime is inaccessible to direct simulation (it would require compaction rates $\gamma\lesssim e^{-d}$), and recent generalized replica treatments indicate hopping shifts the dynamical transition itself — but the source paper raises it as a limitation of high-dimensional simulation rather than as part of the interpolation question.

References

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