Nonlinear elastic free energies and gradient Young-Gibbs measures

TL;DR

This paper studies the large-volume behavior of Gibbs measures in nonlinear elasticity, establishing the existence of a quasiconvex free energy and characterizing local behavior via gradient Young-Gibbs measures.

Contribution

It introduces a new interpolation lemma for partition functions and develops exponential tightness estimates to handle unbounded state spaces in elasticity models.

Findings

Existence of a quasiconvex free energy as a large deviations rate functional.

Parametrization of local Gibbs measure behavior by gradient Young measures.

Development of exponential tightness estimates for unbounded potentials.

Abstract

We investigate, in a fairly general setting, the limit of large volume equilibrium Gibbs measures for elasticity type Hamiltonians with clamped boundary conditions. The existence of a quasiconvex free energy, forming the large deviations rate functional, is shown using a new interpolation lemma for partition functions. The local behaviour of the Gibbs measures can be parametrized by Young measures on the space of gradient Gibbs measures. In view of unboundedness of the state space, the crucial tool here is an exponential tightness estimate that holds for a vast class of potentials and the construction of suitable compact sets of gradient Gibbs measures.