Some analytic results on the FPU paradox

TL;DR

This paper analyzes the FPU paradox by presenting rigorous results that explain the lack of thermalization, focusing on approaches like perturbation theory, Toda lattice, and adiabatic invariants, especially as the number of particles grows large.

Contribution

It introduces new rigorous analytic results that shed light on the FPU paradox, emphasizing approaches valid in the large particle limit.

Findings

Persistence of non-thermalization phenomena as N increases

Application of canonical perturbation theory and KdV equation

Construction of adiabatic invariants with high probability

Abstract

We present some analytic results aiming at explaining the lack of thermalization observed by Fermi Pasta and Ulam in their celebrated numerical experiment. In particular we focus on results which persist as the number $N$ of particles tends to infinity. After recalling the FPU experiment and some classical heuristic ideas that have been used for its explanation, we concentrate on more recent rigorous results which are based on the use of (i) canonical perturbation theory and KdV equation, (ii) Toda lattice, (iii) a new approach based on the construction of functions which are adiabatic invariants with large probability in the Gibbs measure.