Effective interface models with gradient interactions and the Cauchy-Born rule at positive temperature

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We consider gradient Gibbs fields in the context of both effective interface models and lattice spring models of nonlinearly elastic crystals. In case of strictly convex interaction potential, Funaki and Spohn have characterised the ergodic extremal gradient Gibbs states. Also the microcanonical Gibbs distribution for fixed volume can be derived via a large deviation principle in this case. In the context of lattice spring models a realistic interaction has to be nonconvex in view of frame indifference. Friesecke and Theil have shown for a model problem that despite the lack of convexity the Cauchy-Born rule holds in a certain parameter regime, i.e., the ground state for affine boundary conditions is given by an affine deformation. Their argument uses central ideas from the (continuum) calculus of variations, in particular the existence on interesting null Lagrangians. Our objective is to relax the convexity assumption of the interaction potential and to characterise the ergodic Gibbs states for both the interface model (which corresponds to a scalar independent variable) and lattice spring model (which corresponds to a Rm-valued independent variable).