Statistical Field Theory of Equilibrium Amorphous Solids and the Intrinsic Heterogeneity Distributions that Characterize Them

A rich variety of amorphous solids are found throughout nature, science, and technology, including those formed via the vulcanization of long, flexible polymer molecules. A special class—those featuring a wide separation between the very long timescales on which constraining bonds release and the much shorter timescales on which unconstrained degrees of freedom relax—exhibit equilibrium states and are therefore amenable to equilibrium statistical mechanics. A review is given of the least detailed (and thus most general) approach to equilibrium amorphous solids: statistical field theory. The field at the center of this theory is motivated by the aim of characterizing the amorphous solid state. This field, and the theory that governs it, turn out to be rather unusual in essential ways. What the statistical field theory approach predicts—and can predict—is discussed, including the following: the emergence of the solid and its intrinsic heterogeneity; fluctuations and connections with percolation; symmetry breaking and elasticity; and correlations and the information they furnish. Emphasis is placed on the idea, particular to amorphous solids, that such solids are naturally characterized in terms of distributions that describe the spatial heterogeneity of the thermal motions of their constituents. This information is subtly encoded in the wave vector dependencies of the average field and its correlations. The review concludes with some reflections on the applicability—or otherwise—of the ideas and results it explores to a variety of amorphous solids and related systems.