  In crystals, molecules thermally vibrate around the periodic lattice sites.
Vibrational motions are well understood in terms of phonons, which carry heat
and control heat transport. The situation is notably different in disordered
solids, where vibrational excitations are not phonons and can be even
localized. Recent numerical work has established the concept of elastic
heterogeneity: disordered solids show inhomogeneous local mechanical response.
Clearly, the heterogeneous nature of elastic properties strongly influences
vibrational and thermal properties, and it is expected to be the origin of
anomalous features, including boson peak, vibrational localization, and
temperature dependence of thermal conductivity. These are all crucial
long-standing problems in materials physics, which we address in the present
work. We have considered a toy model able to stabilize different states of
matter, by introducing an increasing amount of size disorder. The phase diagram
generated by molecular dynamics simulations encompasses the perfect crystalline
state with a spatially homogeneous elastic moduli distribution, multiple
defective phases with increasing moduli heterogeneities, and eventually a
series of amorphous states. We have established clear correlations among the
heterogeneous local mechanical response, vibrational states, and thermal
conductivity. We provide evidence that elastic heterogeneity controls both
vibrational and thermal properties, and is a key concept to understand the
anomalous puzzling features of disordered solids.
