KAM Theory for Equilibrium States in 1-D Statistical Mechanics Models

Abstract

We extend the Lagrangian proof of KAM for twist mappings [34,51] to show persistence of quasi-periodic equilibrium solutions in 1-D statistical mechanics models. The interactions in the models considered here do not need to be of finite range but they have to decrease sufficiently fast with the distance (a high enough power suffices). In general, these models do not admit an interpretation as a dynamical system. Even when they do, the Hamiltonian description may be very singular or the number of degrees of freedom may be very large, so that the Hamiltonian KAM theory does not apply. We formulate the main result in an “a-posteriori” way. We show that if we are given a quasi-periodic function which solves the equilibrium equation with sufficient accuracy, which has a Diophantine frequency and which satisfies some non-degeneracy conditions, then, there is a true solution of the equilibrium equation which is close to the approximate solution. As an immediate consequence, we deduce that quasi-periodic solutions of the equilibrium equation with one Diophantine frequency persist under small modifications of the model. The main result can also be used to validate numerical calculations or perturbative expansions. Indeed, the method of proof lends itself to very efficient numerical implementations. We also show that some perturbative expansions (Lindstedt series) can be computed to all orders and that they converge.