Mesoscale heterogeneity in equilibrium amorphous solids: The role of the induced measure

Consider an equilibrium amorphous solid consisting, e.g., of a sufficiently crosslinked macromolecular network. Owing to the random architecture, the characteristics of the thermal motion of the constituents vary randomly throughout the medium. A statistical field theory is known to describe amorphous solidification. Encoded in the mean value of the associated field is information about the thermal motions of the constituents: (i) the fraction that are spatially localized, and (ii) how their localization varies, statistically, across the medium. Encoded in the field's correlations is more refined information about the thermal motions, e.g.: (i) how the localization characteristics of the constituents are correlated, (ii) how the correlations between the thermal fluctuations of the pair are distributed, and (iii) how these descriptors vary with the separation of the pair. We examine increasingly accurate approximations to the field correlations, incorporating first ungapped (i.e., elastic displacement) and then gapped branches of field fluctuations. We pay particular attention to the role of the induced measure, which arises from the nonlinear transformation to displacement fields. Hence, we obtain an increasingly accurate set of statistical distributions that govern the spatial correlations among the random characteristics of the thermal motion of the constituents of equilibrium amorphous solids.