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Cauchy-Born Rule from Microscopic Models with Non-convex Potentials

Stefan Adams, Simon Buchholz, Roman Kotecký, Stefan Müller

Abstract

We study gradient field models on an integer lattice with non-convex interactions. These models emerge in distinct branches of physics and mathematics under various names. In particular, as zero-mass lattice (Euclidean) quantum field theory, models of random interfaces, and as mass-string models of nonlinear elasticity.Our attention is mostly devoted to the latter with random vector valued fields as displacements for atoms of crystal structures,where our aim is to prove the strict convexity of the free energy as a function of affine deformations for low enough temperatures and small enough deformations. This claim can be interpreted as a form of verification of the Cauchy-Born rule at small non-vanishing temperatures for a class of these models. We also show that the scaling limit of the Laplace transform of the corresponding Gibbs measure (under a proper rescaling) corresponds to the Gaussian gradient field with a particular covariance. The proofs are based on a multi-scale (renormalisation group analysis) techniques needed in view of strong correlations of studied gradient fields. To cover sufficiently wide class of models, we extend these techniques from the standard case with rotationally symmetric nearest neighbour interaction to a more general situation with finite range interactions without any symmetry. Our presentation is entirely self-contained covering the details of the needed renormalisation group methods.

Cauchy-Born Rule from Microscopic Models with Non-convex Potentials

Abstract

We study gradient field models on an integer lattice with non-convex interactions. These models emerge in distinct branches of physics and mathematics under various names. In particular, as zero-mass lattice (Euclidean) quantum field theory, models of random interfaces, and as mass-string models of nonlinear elasticity.Our attention is mostly devoted to the latter with random vector valued fields as displacements for atoms of crystal structures,where our aim is to prove the strict convexity of the free energy as a function of affine deformations for low enough temperatures and small enough deformations. This claim can be interpreted as a form of verification of the Cauchy-Born rule at small non-vanishing temperatures for a class of these models. We also show that the scaling limit of the Laplace transform of the corresponding Gibbs measure (under a proper rescaling) corresponds to the Gaussian gradient field with a particular covariance. The proofs are based on a multi-scale (renormalisation group analysis) techniques needed in view of strong correlations of studied gradient fields. To cover sufficiently wide class of models, we extend these techniques from the standard case with rotationally symmetric nearest neighbour interaction to a more general situation with finite range interactions without any symmetry. Our presentation is entirely self-contained covering the details of the needed renormalisation group methods.

Paper Structure

This paper contains 78 sections, 75 theorems, 955 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Setting and Main Results
  3. General setup
  4. Main results for generalized gradient models
  5. Main results for discrete elasticity
  6. Main results for abstract perturbations
  7. The proof of the abstract perturbation result: informal preview and indication of new ideas
  8. The abstract perturbation theorem implies the results for the generalized gradient models
  9. Embedding of the initial perturbation
  10. Discrete Nonlinear Elasticity
  11. Continuous nonlinear elasticity and continuous null Lagrangians
  12. Discrete null Lagrangians
  13. Reformulation of discrete elasticity as a non-degenerate generalized gradient model
  14. Examples
  15. Explanation of the Method
  16. ...and 63 more sections

Key Result

Lemma \oldthetheorem

There exists a linear isomorphism $\Pi:\mathcal{G}_{{R_0}}\to \mathcal{V}_{Q_{{R_0}}}^\perp$ inducing a one-to-one correspondence between functions on $\mathcal{V}_{Q_{{R_0}}}^\perp$ and those on $\mathcal{G}_{{R_0}}$. Namely, for any $U:\mathcal{V}_{Q_{{R_0}}}^\perp\to {\mathbb{R}}$, there is $\mat

Figures (1)

  • Figure 1: Torus $(\mathbb{Z}/(L\mathbb{Z}))^d$ with $d = 2$, $L = 3$, and $N = 3$ identified with the square $\{z\in \mathbb{Z}^2: \lvert z\rvert_\infty\le 13\}$. The sets $\mathcal{B}_0$, $\mathcal{B}_1$, and $\mathcal{B}_2$ contain $27^2$, $9^2$, and $3^2$ blocks, respectively. There are six connected polymers in the figure: 0-polymers $X_1$ (the two pieces touching the boundary are connected on the torus), $X_5$, and $X_6$ (in the center), 1-polymers $X_3$ and $X_4$, and 2-polymer $X_2$ (again, the two pieces touching on the torus). Only the 1-polymers $X_1$ and $X_6$ are strictly disjoint from the other polymers---taking into account the connectedness on the torus, the polymer $X_2$ is connected with each of the polymers $X_3$, $X_4$, and $X_5$.

Theorems & Definitions (168)

  • Lemma \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem: Passage to subsequences
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • ...and 158 more