Determining the nonequilibrium criticality of a Gardner transition via a hybrid study of molecular simulations and machine learning

Abstract

Apparent critical phenomena, typically indicated by growing correlation lengths and dynamical slowing down, are ubiquitous in nonequilibrium systems such as supercooled liquids, amorphous solids, active matter, and spin glasses. It is often challenging to determine if such observations are related to a true second-order phase transition as in the equilibrium case or simply a crossover and even more so to measure the associated critical exponents. Here we show that the simulation results of a hard-sphere glass in three dimensions are consistent with the recent theoretical prediction of a Gardner transition, a continuous nonequilibrium phase transition. Using a hybrid molecular simulation–machine learning approach, we obtain scaling laws for both finite-size and aging effects and determine the critical exponents that traditional methods fail to estimate. Our study provides an approach that is useful to understand the nature of glass transitions and can be generalized to analyze other nonequilibrium phase transitions. Significance Understanding the nature of glass states remains as one of the grand challenges presently. A much-debated issue is whether or not a glass-to-glass transition, the Gardner transition, occurs in deeply annealed glass states, for which a number of clearly defined physical properties must follow, according to theories of phase transitions. Here, utilizing the current machine learning techniques, we show that finite-time and finite-size analyses of the massive numerical data, produced from molecular dynamics simulations of a hard-sphere glass model, support that the Gardner transition is a second-order phase transition in three dimensions. Our study also provides estimates of the critical exponents of the transition, which traditional approaches are unable to obtain.