摘要: 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. All study data are included in the article and [ SI Appendix ][1]. [1]: https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.2017392118/-/DCSupplemental