Using scale-dependent elasticity to probe and influence the distribution of localization lengths in the amorphous solid state

The equilibrium amorphous solid state – formed, for example, by randomly crosslinking a macromolecular fluid – is characterized by a universal distribution of localization lengths. This distribution obeys a scaling form near to the continuous transition out of the amorphous solid state: it has a single peak at a length-scale that diverges at the transition along with the width of the distribution. Naturally, in the long wavelength limit the elastic rigidity of this state does not depend on the distribution of localization lengths. However, the elastic behavior at progressively shorter length-scales becomes increasingly insensitive to the more weakly localized particles, and thus the scale-dependence of the elastic modulus provides a probe of the universal localization-length distribution. We show, in particular, how the response to short length-scale elastic deformations sheds light on the asymptotics of the distribution at short localization length-scales. In addition, we explore the extent to which elastic deformations at various length-scales can induce a rearrangement of statistical weight in the localization-length distribution.