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

我们认为梯度吉布斯场的上下文中的有效界面模型和晶格弹簧模型的非线性弹性晶体。在严格凸相互作用势的情况下，Funaki和Spohn已经描述了遍历极值梯度Gibbs态。在这种情况下，也可以通过大偏差原理导出固定体积的微正则吉布斯分布。在格点弹簧模型的背景下，一个现实的相互作用必须是非凸的框架无差异。Friesecke和Theil已经证明了一个模型问题，尽管缺乏凸性，但柯西-玻恩规则在一定的参数范围内成立，即，仿射边界条件的基态由仿射变形给出。他们的论点使用了（连续）变分法的中心思想，特别是有趣的零拉格朗日量的存在。我们的目标是放松相互作用势的凸性假设，并对界面模型（对应于标量自变量）和格点弹簧模型（对应于Rm值自变量）的遍历Gibbs态进行重新定义。

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).