Relationship between Fragility, Diffusive Directions and Energy Barriers in a Supercooled Liquid

An analysis of diffusion in a supercooled liquid based solely in the density of diffusive directions and the value of energy barriers shows how the potential energy landscape (PEL) approach is capable of explaining the $\alpha$ and $\beta$ relaxations and the fragility of a glassy system. We find that the $\beta$ relaxation is directly related to the search for diffusive directions. Our analysis shows how in strong liquids diffusion is mainly energy activated, and how in fragile liquids the diffusion is governed by the density of diffusive directions. We describe the fragile-to-strong crossover as a change in the topography of the PEL sampled by the system at a certain crossover temperature $T_{\times }$.