FPU phenomenon for generic initial data

TL;DR

This paper investigates the FPU phenomenon at low energies, providing numerical evidence that the FPU model lacks mixing properties for generic initial data, especially in high-frequency modes, and discusses implications for the thermodynamic limit.

Contribution

It offers numerical evidence that the FPU model does not exhibit mixing at low energies for generic initial conditions, challenging previous assumptions.

Findings

High-frequency mode autocorrelations tend to positive values.

Nonmixing behavior likely persists in the thermodynamic limit.

Mixing may occur only at much longer time scales.

Abstract

The well known FPU phenomenon (lack of attainment of equipartition of the mode--energies at low energies, for some exceptional initial data) suggests that the FPU model does not have the mixing property at low energies. We give numerical indications that this is actually the case. This we show by computing orbits for sets of initial data of full measure, sampled out from the microcanonical ensemble by standard Montecarlo techniques. Mixing is tested by looking at the decay of the autocorrelations of the mode--energies, and it is found that the high--frequency modes have autocorrelations that tend instead to positive values. Indications are given that such a nonmixing property survives in the thermodynamic limit. It is left as an open problem whether mixing obtains within time--scales much longer than the presently available ones.