On the relation between kinetically constrained models of glass dynamics and the random first-order transition theory

TL;DR

This paper establishes a formal mapping between RFOT models and kinetically constrained models (KCM), revealing their underlying similarities in glass transition dynamics, especially on Bethe lattices, despite their different apparent thermodynamic properties.

Contribution

The authors prove a formal master equation-level mapping between RFOT and KCM models, demonstrating their deep connection in glass transition phenomena.

Findings

RFOT models can be exactly mapped onto KCMs.

The mapping involves complex non-local functions for the order parameter.

The results suggest a close relationship between the two descriptions of glass transition.

Abstract

In this paper we revisit and extend the mapping between two apparently different classes of models. The first class contains the prototypical models described --at the mean-field level-- by the Random First Order Transition (RFOT) theory of the glass transition, called either "random XORSAT problem" (in the information theory community) or "diluted $p$-spin model" (in the spin glass community), undergoing a single-spin flip Glauber dynamics. The models in the second class are Kinetically Constrained Models (KCM): their Hamiltonian is that of independent spins in a constant magnetic field, hence their thermodynamics is completely trivial, but the dynamics is such that only groups of spin can flip together, thus implementing a kinetic constraint that induces a non-trivial dynamical behavior. A mapping between some representatives of these two classes has been known for long. Here we formally prove this mapping at the level of the master equation, and we apply it to the particular case of Bethe lattice models. This allows us to show that a RFOT model can be mapped exactly into a KCM. However, the natural order parameter for the RFOT model, namely the spin overlap, is mapped into a very complicated non-local function in the KCM. Therefore, if one were to study the KCM without knowing of the mapping onto the RFOT model, one would guess that its physics is quite different from the RFOT one. Our results instead suggest that these two apparently different descriptions of the glass transition are, at least in some case, closely related.