An averaging theorem for FPU in the thermodynamic limit

TL;DR

This paper proves an averaging theorem for the Fermi-Pasta-Ulam (FPU) chain in the thermodynamic limit, showing that certain energy packets remain approximately invariant over long times under Gibbs measure.

Contribution

It constructs adiabatic invariants for large N FPU chains with mild packet restrictions, valid uniformly as N approaches infinity.

Findings

Energy packets are approximately conserved over long timescales.

Time autocorrelation of packet energies remains significant for extended periods.

Results hold uniformly in the number of particles, N, in the thermodynamic limit.

Abstract

Consider an FPU chain composed of $N\gg 1$ particles, and endow the phase space with the Gibbs measure corresponding to a small temperature $\beta^{-1}$. Given a fixed $K<N$, we construct $K$ packets of normal modes whose energies are adiabatic invariants (i.e., are approximately constant for times of order $\beta^{1-a}$, $a>0$) for initial data in a set of large measure. Furthermore, the time autocorrelation function of the energy of each packet does not decay significantly for times of order $\beta$. The restrictions on the shape of the packets are very mild. All estimates are uniform in the number $N$ of particles and thus hold in the thermodynamic limit $N\to\infty$, $\beta>0$.