Generalized mode-coupling theory of the glass transition. II. Analytical scaling laws

Abstract

Generalized mode-coupling theory (GMCT) constitutes a systematically correctable, first-principles theory to study the dynamics of supercooled liquids and the glass transition. It is a hierarchical framework that, through the incorporation of increasingly many particle density correlations, can remedy some of the inherent limitations of the ideal mode-coupling theory (MCT). However, despite MCT’s limitations, the ideal theory also enjoys several remarkable successes, notably including the analytical scaling laws for the α- and β-relaxation dynamics. Here, we mathematically derive similar scaling laws for arbitrary-order multi-point density correlation functions obtained from GMCT under arbitrary mean-field closure levels. More specifically, we analytically derive the asymptotic and preasymptotic solutions for the long-time limits of multi-point density correlators, the critical dynamics with two power-law decays, the factorization scaling laws in the β-relaxation regime, and the time-density superposition principle in the α-relaxation regime. The two characteristic power-law-divergent relaxation times for the two-step decay and the non-trivial relation between their exponents are also obtained. The validity ranges of the leading-order scaling laws are also provided by considering the leading preasymptotic corrections. Furthermore, we test these solutions for the Percus–Yevick hard-sphere system. We demonstrate that GMCT preserves all the celebrated scaling laws of MCT while quantitatively improving the exponents, rendering the theory a promising candidate for an ultimately quantitative first-principles theory of glassy dynamics.

Multi-component generalized mode-coupling theory: predicting dynamics from structure in glassy mixtures

“…However, if we increase the level of the hierarchy to n max = 3 (orange) and then n max = 4 (green), the accuracy increases and the critical point of GMCT manifestly converges toward the simulations. This uniform convergence of the multicomponent theory with increasing n max is also consistent with earlier findings from single-component GMCT [29,[32][33][34].…”

“…Multi-component GMCT allows us instead to shift the closure to a larger , meaning that the correlations of order are not factorized and are more correctly described. This approach has already been shown to be beneficial in single-component glassy systems [ 27 – 29 , 33 , 34 ], and in this paper, we demonstrate that a similar improvement can be gained for binary systems.…”

“…This gradual but systematic improvement is also consistent with earlier GMCT studies for single-component systems. In future work, we will aim to push the boundaries of the highest level n we can numerically solve [ 33 , 34 ], in order to check whether the current GMCT framework might approach the exact scenario in the limit. This effort will also allow us to check if higher-order GMCT can capture the non-trivial glassy phenomenology contained in other successful microscopic theories such as stochastic beta relaxation theory [ 21 , 22 ].…”

“…In this paper, we have derived generalized MCT for multi-component systems, thus extending the earlier version of the theory [27][28][29]33,34] to the case of mixtures with an arbitrary number of species. The theory seeks to predict the microscopic relaxation dynamics of glassy mixtures in a fit-parameter-free manner using the static structure factors as its main input.…”

“…Similar to MCT, this generalized framework also requires only static structure as input and has no free parameters. Furthermore, GMCT also provides predictive insights into regimes of previously inaccessible dynamics for single-component glassy systems [ 29 , 33 , 34 ], and notably preserves the celebrated scaling laws of standard MCT [ 33 , 34 ].…”