Long-range correlations in pinned athermal networks

Abstract

We derive analytic results for displacement fields that develop as a response to external forces on athermal networks in two dimensions. For a triangular lattice arrangement of particles interacting through soft potentials, we develop a Green's function formalism which we use to derive exact results for displacement fields produced by localized external forces. We show that in the continuum limit the displacement fields decay as 1/r at large distances r away from a force dipole. Finally, we extend our formulation to study correlations in the displacement fields produced in actively jammed systems. We show that uncorrelated active forces give rise to long-range correlations in displacements in athermal systems, with a non-trivial system size dependence.

Disorder Perturbation Expansion for Athermal Crystals

“…The perturbation expansion developed in this paper can easily be extended to involve systems with other types of microscopic disorder, such as spring networks with bond disorder, and also soft particle systems with external pinning forces [31]. Our framework is quite general, and can also be applied to any precompressed crystalline system with particles interacting through central potentials.…”

“…However, unlike at first order, the second order solutions are expected to display non-linear effects arising from the disorder as the first order displacement fields are coupled in the source terms in Eq. (31). We discuss such effects in detail in Section IX.…”

“…We first derive the displacement fields generated by the defect at the origin using the perturbation theory at linear order. It has been shown recently that mechanical equilibrium at linear order leads to non-trivial effects on fluctuations and correlations in athermal nearcrystalline systems [19,31]. The field in Eq.…”

“…This involves the displacement fields at first order {δx (1) , δy (1) }, which can be used to create the source term at second order using Eq. (31). The coefficients C αβγ ij that appear in these source terms are given in the Appendix.…”

“…In this context, models of disordered athermal materials that can be exactly solved are important tools with which to understand athermal disorder as well as the sensitive deviations from linear elasticity that such materials display. A particularly appealing class of systems are disordered athermal crystals where the periodicity of a reference configuration can be exploited [19,31], and consequently they offer a paradigm where exact results for disordered athermal systems can be obtained.…”