Long range order in atomistic models for solids

The emergence of long range order at low temperatures in atomistic systems with continuous symmetry is a fundamental, yet poorly understood phenomenon physics. To address this challenge we study discrete microscopic model for an elastic crystal dislocations three dimensions, previously introduced by Ariza and Ortiz. rich enough to support some realistic features three-dimensional dislocation theory, most notably grains the Read– Shockley law grain boundaries, which rigorously derive simple, explicit geometry. We analyze positive temperatures, terms Gibbs distribution energy function given Ariza–Ortiz Hamiltonian plus contribution from cores. Our main result that exhibits positional temperatures. proof based on tools exterior calculus, together cluster expansion techniques.