Abstract
We develop a novel approach that combines the ideas of inherent structures, free-volume theory and geometrical packing properties to derive a general theory to understanding the complex dynamics of glass-forming liquids, granular packings and amorphous solids. We show that the thermodynamical properties of these systems can be retrieved from the study of geometrical and topological properties of local configurations only. When applied to hard-sphere systems, the present theory describes the critical approach toward the random close-packing density with the configurational entropy that approaches zero, and the α-relaxation time that diverges according to the Vogel-Fulcher behavior following also the Adam-Gibbs relation. The equation of state derived for the stable fluid, the metastable fluid, and for the crystalline and amorphous solids, results in good quantitative agreement with the data available in the literature.