# Stages of dynamics in the Fermi-Pasta-Ulam system as probed by the first Toda integral

## TL;DR

This paper studies the long-term behavior of FPU system trajectories using the Toda integral, revealing two key timescales and how they scale with energy and system size, providing insights into FPU dynamics and thermalization.

## Contribution

It introduces the use of the first Toda integral as a fluctuation-free observable to analyze FPU dynamics and identifies scaling laws for stability and equilibration times.

## Key findings

- Two fundamental timescales: stability and equilibrium.
- Exponential scaling of times with inverse energy.
- Crossover energy scales as a power law with system size.

## Abstract

We investigate the long term evolution of trajectories in the Fermi-Pasta-Ulam (FPU) system, using as a probe the first non--trivial integral $J$ in the hierarchy of integrals of the corresponding Toda lattice model. To this end we perform simulations of FPU--trajectories for various classes of initial conditions produced by the excitation of isolated modes, packets, as well as `generic' (random) initial data. For initial conditions corresponding to localized energy excitations, $J$ exhibits variations yielding `sigmoid' curves similar to observables used in literature, e.g. the `spectral entropy' or various types of `correlation functions'. However, $J(t)$ is free of fluctuations inherent in such observables, hence it constitutes an ideal observable for probing the timescales involved in the stages of FPU dynamics. We observe two fundamental timescales: i) the `time of stability' (in which, roughly, FPU trajectories behave like Toda), and ii) the `time to equilibrium' (beyond which energy equipartition is reached). Below a specific energy crossover, both times are found to scale exponentially as an inverse power of the specific energy. However, this crossover goes to zero with increasing the degrees of freedom $N$ as $\varepsilon _c \sim N^{-b}$, with $b \in [1.5, 2.5]$. For `generic data' initial conditions, instead, $J(t)$ allows to quantify the continuous in time slow diffusion of the FPU trajectories in a direction transverse to the Toda tori.