Statistical mechanics of dislocation pileups

Dislocations experiencing applied stress in two dimensions can order when trapped in a single glide plane with aligned Burgers vectors. These dislocation queues, called dislocation pileups, are critical in the initiation and propagation of deformations in materials. We study the static and dynamical properties of this class of defect ordering, where the dislocations themselves form inhomogeneous quasilattices in one dimension, with spatially varying lattice spacings whose spatial profile depends on the form of the applied stress. We study these dislocation pileup lattices using an intriguing connection with recent generalizations of random matrix ensembles, and examine the crystallization of these dislocations at low temperatures. We use random matrix theory to probe the equilibrium statistical mechanics, which allows us to extract the spatial correlation functions and structure factors of two distinct types of dislocation pileups, those in uniform stress fields and those in stress fields linear in space. Our formalism provides an analytical formulation for these correlations generalizable to other inhomogeneous crystals in one dimension. Finally, we analyze the low temperature excitation spectrum of these dislocation pileups and the spatial properties of their excitation modes.