Mathematical Analysis of Lattice gradient models & Nonlinear Elasticity

Abstract

Statistical Mechanics is considered as one of the most sound and confirmed theories in modern physics. In this thesis, we explore the possibility to view a large class of models under the point of view of statistical mechanics. The models are defined for simplicity on the standard lattice Z^d. However, most of the results apply unchanged to very general lattices. The Hamiltonians considered are of gradient type. Namely, as a function of the field ϕ, they depend only on all the pair differences ϕ(x)− ϕ(y), where x,y are elements of the lattice. Under suitable very general assumptions, we show that these models satisfy certain large deviation principles. The models considered contain in particular the typical models for Nonlinear Elasticity and Fracture Mechanics. Afterwards, we will concentrate on more specific models in which we show local properties of the free energy per particle. These models are sometimes known in the literature as mass-spring models. In particular, we will consider the space dependent case. For these models, we show the validity of the Cauchy-Born rule in a neighbourhood of the origin. The methods used to prove the Cauchy-Born rule are based on the Renormalization Group. We also show a new Finite Range Decomposition based on discrete Lp-theory.