Universal mesoscale heterogeneity and its spatial correlations in equilibrium amorphous solids

Random network media include (but are not limited to) macroscopic systems consisting of long, flexible macromolecules that have been permanently chemically bonded (i.e., crosslinked) to one another at random to form the network. Provided the medium has been crosslinked enough to have undergone amorphous solidification, the characteristics of the equilibrium thermal motion of its elements are spatially heterogeneous, varying randomly from point to point, owing to the randomness of the architecture. How these characteristics are distributed and spatially correlated is encoded in the wave-vector dependence of the field expectation values associated with the replica field theory that describes the amorphous solid state. By incorporating elastic-displacement fluctuations into the computation of these expectation values, a statistical characterization of the mesoscale heterogeneity of equilibrium amorphous solids is obtained. Along the way, particular attention is paid to the role of the induced measure that results from the nonlinear transformation from the fluctuating order-parameter field to the fluctuating elastic-displacement fields.