Ergodic and non-ergodic many-body dynamics in strongly nonlinear lattices

TL;DR

This paper investigates the dynamics of nonlinear oscillator chains with confining boundaries, revealing ergodic behavior at large sizes and chaos in two-dimensional cases, expanding understanding of many-body nonlinear systems.

Contribution

Introduces a new class of nonlinear lattice systems with confining boundaries, analyzing their ergodic and chaotic properties across different dimensions and sizes.

Findings

Mixed phase space persists at finite sizes

Large systems tend toward ergodicity with negligible regular regions

Two-dimensional stadia induce strong chaos even at small sizes

Abstract

The study of non-linear oscillator chains in classical many-body dynamics has a storied history going back to the seminal work of Fermi, Pasta, Ulam and Tsingou (FPUT). We introduce a new family of such systems which consist of chains of $N$ harmonically coupled particles with the non-linearity introduced by confining the motion of each individual particle to a box/stadium with hard walls. The stadia are arranged on a one dimensional lattice but they individually do not have to be one dimensional thus permitting the introduction of chaos already at the lattice scale. For the most part we study the case where the motion is entirely one dimensional. We find that the system exhibits a mixed phase space for any finite value of $N$. Computations of Lyapunov spectra at randomly picked phase space locations and a direct comparison between Hamiltonian evolution and phase space averages indicate that the regular regions of phase space are not significant at large system sizes. While the continuum limit of our model is itself a singular limit of the integrable sinh-Gordon theory, we do not see any evidence for the kind of non-ergodicity famously seen in the FPUT work. Finally, we examine the chain with particles confined to two dimensional stadia where the individual stadium is already chaotic, and find a much more chaotic phase space at small system sizes.