Exponentially long stability times for a nonlinear lattice in the thermodynamic limit

TL;DR

This paper demonstrates that a nonlinear 1D Klein-Gordon lattice exhibits exponentially long stability times in the thermodynamic limit, using a combination of Hamiltonian perturbation theory and probabilistic methods.

Contribution

It constructs a uniform-in-size adiabatic invariant for the Klein-Gordon lattice, establishing long stability times in the thermodynamic limit, which was not previously shown.

Findings

Stability times grow exponentially as perturbation parameters tend to zero.

The adiabatic invariant remains approximately constant with high probability.

Provides lower bounds on relaxation times based on autocorrelation estimates.

Abstract

In this paper, we construct an adiabatic invariant for a large 1--$d$ lattice of particles, which is the so called Klein Gordon lattice. The time evolution of such a quantity is bounded by a stretched exponential as the perturbation parameters tend to zero. At variance with the results available in the literature, our result holds uniformly in the thermodynamic limit. The proof consists of two steps: first, one uses techniques of Hamiltonian perturbation theory to construct a formal adiabatic invariant; second, one uses probabilistic methods to show that, with large probability, the adiabatic invariant is approximately constant. As a corollary, we can give a bound from below to the relaxation time for the considered system, through estimates on the autocorrelation of the adiabatic invariant.