We analyze the fluctuations in particle positions and inter-particle forces in disordered jammed crystals in the limit of weak disorder. We demonstrate that such athermal systems are fundamentally different from their thermal counterparts, characterized by constrained fluctuations of forces perpendicular to the lattice directions. We develop a disorder perturbation expansion in polydispersity about the crystalline state, which we use to derive exact results to linear order. We show that constrained fluctuations result as a consequence of local force balance conditions, and are characterized by non-Gaussian distributions which we derive exactly. We analytically predict several properties of such systems, including the scaling of the average coordination with polydispersity and packing fraction, which we verify with numerical simulations using soft disks with one-sided harmonic interactions. arXiv:1910.06352v1 [cond-mat.soft]

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We derive exact results for correlations in the displacement fields {δ r} ≡ {δr µ=x,y } in near-crystalline athermal systems in two dimensions. We analyze the displacement correlations produced by different types of microscopic disorder, and show that disorder at the microscopic scale gives rise to longrange correlations with a dependence on the system size L given by δr µ δr ν ∼ c µν (r/L, θ). In addition, we show that polydispersity in the constituent particle sizes and random bond disorder give rise to a logarithmic system size scaling, with c µν (ρ, θ) ∼ const µν − a µν (θ) log ρ + b µν (θ)ρ 2 for ρ (= r/L) → 0. This scaling is different for the case of displacement correlations produced by random external forces at each vertex of the network, given byAdditionally, we find that correlations produced by polydispersity and the correlations produced by disorder in bond stiffness differ in their symmetry properties. Finally, we also predict the displacement correlations for a model of polydispersed soft disks subject to external pinning forces, that involve two different types of microscopic disorder. We verify our theoretical predictions using numerical simulations of polydispersed soft disks with random spring contacts in two dimensions.

We derive exact results for the fluctuations in energy produced by microscopic disorder in nearcrystalline athermal systems. Our formalism captures the heterogeneity in the elastic energy of polydispersed soft disks in energy-minimized configurations. We use this to predict the distribution of interaction energy between two defects in a disordered background. We show this interaction energy displays an average power-law behaviour δE ∼ ∆ −4 at large distances ∆ between the defects. These interactions upon disorder average also display the sixfold symmetry of the underlying reference crystal. Additionally, we show that the fluctuations in the interaction energy encode the athermal correlations introduced by the disordered background. We verify our predictions with energy minimized configurations of polydispersed soft disks in two dimensions.