As the Universe cooled down after the Big Bang, two distinct forms of far-from-equilibrium behavior become manifest: some regions were driven away from equilibrium by constant energy input, while others got stuck as they lost the ability to explore phase space and thus fell out of equilibrium. The first of these generic behaviors is covered by various sciences dealing with dynamics. The second has its roots deeply embedded in the study of the glassy state of matter. A glass – the prototypic and ubiquitous amorphous solid – is found whenever a liquid is cooled rapidly enough to avoid crystallization. It is based on very different organizational principles from a crystal and must be understood from a very different starting point. The nature of the glassy state forces us to consider more deeply the seemingly simple but non-trivial and unresolved question “What is a solid?”
The development of a theory that unravels the intricate interplay between disorder, statics and relaxation dynamics to describe the transition of a liquid to a glass, the glass transition, is one of the greatest challenge in physics today:
- As perhaps the most general “rough-energy landscape” problem, techniques and notions developed for glasses have had a large impact on fields as disparate as protein folding, transport in crowded biological environments, and constraint-satisfaction problems in computer science.
- Related to the rough landscape is the extreme slowing of dynamics, which forces many systems to fall from equilibrium and which emerges in science at all scales ranging from astro- and geophysics to chemistry and materials science.
- From a practical viewpoint, a deeper understanding of glassy solids will enable the development of new classes of materials.
Just as understanding glassiness leads us to face issues that standard solid-state physics is unable to handle, the glass transition defies ordinary phase transition theory. For example, the minimization of a function of 1023 variables (e.g., a system’s free energy with respect to its degrees of freedom) can usually be reduced to finding a few low-energy minima, corresponding to simple symmetry breakings. It thus suffices to focus on fluctuations around only a few competing states to describe the transition. In glasses, however, there is a gigantic degeneracy of low-energy minima so these conventional assumptions fail. Moreover for the glass transition, statics and dynamics are inextricably linked because glassy systems fail to explore fully their energy landscapes. As a result, new tools and concepts are required.
Cooling a liquid to low temperatures (the glass transition) and jamming a disordered collection of static particles by applying external pressure are two ways of forming a rigid solid. The jamming transition is athermal and static whereas in the glass transition thermal and dynamical effects are crucial. Previous work has produced the discovery in d ≥ 2 and exact solution in d = ∞ of the jamming transition, and an exact solution to statics of the mean-field glass transition of hard spheres in d = ∞. It has provided an understanding of the length scales governing glassy relaxation and their implications for glassy properties. The confluence of these breakthroughs brings new insight and opportunity to tackle two outstanding challenges: (1) to develop a unified theory of structure and excitations in glassy matter and (2) to develop a theory for the statics and relaxation dynamics of the glass transition. We plan to address (1) by connecting our finite-dimensional understanding of jamming with our infinite-dimensional mean-field solution. We will pursue (2) with a three-pronged strategy by expanding to higher temperature T about the solution for jamming, which describes T = 0 behavior, by bridging the gap between d = ∞ and d = 2, 3, which are the observable dimensions of interest and by understanding the dynamics of a marginally stable material.