Glass – the prototypic and ubiquitous amorphous solid – inhabits a complex and ramified energy landscape. Dealing with the plethora of relevant energy minima has emerged as one of the central problems of statistical physics. The mere existence of a glass reveals organizational principles that are dramatically different from those of ordered solids and poses the fundamental question: is there a limit of absolute disorder that is the opposite pole from the perfect order of a crystal? For crystals, the minimization of a function of 10^23 variables (e.g., a system’s free energy with respect to its degrees of freedom) is reduced to finding a few competing minima; for glasses, the gigantic degeneracy of low-energy minima leads to a spectacular failure of conventional approaches.
Because a supercooled liquid that solidifies into a glass results from the system’s failure to fully explore its energy landscape, statics and dynamics are inextricably linked. The transition into a glass is therefore an entirely different phenomenon from the transition into a crystal [1]. Just as the physics of the glass state defies concepts from ordinary solid-state physics, the glass transition defies ordinary phase-transition theory.
Cooling a liquid to low temperatures (the glass transition) and compressing a zero-temperature disordered collection of particles (the jamming transition) are two ways of obtaining amorphous solids [2]. Jamming is athermal and static, and is in some ways simpler than glass formation, where thermal and dynamical effects also play crucial roles [3], yet also presents a remarkably rich phenomenology.
Recent advances in the glass field have discovered and characterized the jamming transition for dimension d≥2 and obtained a solution for the statics and the dynamics of liquids in the mean-field, d->∞ limit [4]. These theoretical advances provide a unifying framework that includes a third transition, the Gardner transition, that separates jamming from glass formation [5]. The recent development of a field-theoretic framework to describe glassy dynamics and associated length scales for systems with many energy minima away from d=∞ [6] as well as a harmonic theory for jammed solids near zero temperature [7] further provide two broad insights, which make this workshop particularly timely: (i) the crucial role played by dimension as a control parameter for analyzing the statics and the dynamics of glassy matter, and (ii) the realization that real-space and phase-space approaches converge to the same critical exponents.
A number of these advances would have been impossible without the constant support, confrontation and validation from computational sciences. This research program has indeed benefited from and stimulated a vast program of molecular simulations, in order to both test the theoretical ideas and to provide a bridge between the mean-field results and physical systems in d=2,3 [8]. New computational concepts and algorithms are pushing forward our numerical capabilities far beyond what was once considered conceivable, and are currently playing a central role in solving the glass problem [9].