Ergodicity, mixing and recurrence in the three rotor problem

TL;DR

This paper investigates the ergodic and mixing properties of the classical three rotor problem, demonstrating that in a specific energy band, the system exhibits ergodicity, mixing, and exponential recurrence time distributions, indicating global chaos.

Contribution

The study provides numerical evidence of ergodicity and mixing in the three rotor problem within a specific energy band, highlighting phase transitions outside this band.

Findings

Distributions approach the Liouville measure over time

Trajectories become uniformly distributed, indicating mixing

Recurrence times follow an exponential distribution in the chaotic band

Abstract

In the classical three rotor problem, three equal point masses move on a circle subject to attractive cosine potentials of strength g. In the center of mass frame, energy E is the only known conserved quantity. In earlier work [Krishnaswami and Senapati, arXiv:1810.01317, Oct. 2018, arXiv:1811.05807, Nov. 2018], an order-chaos-order transition was discovered in this system along with a band of global chaos for 5.33g < E < 5.6g. Here, we provide numerical evidence for ergodicity and mixing in this band. The distributions of relative angles and angular momenta along generic trajectories are shown to approach the corresponding distributions over constant energy hypersurfaces (weighted by the Liouville measure) as a power-law in time. Moreover, trajectories emanating from a small volume are shown to become uniformly distributed over constant energy hypersurfaces, indicating that the dynamics is mixing. Outside this band, ergodicity and mixing fail, though the distributions of angular momenta over constant energy hypersurfaces show interesting phase transitions from Wignerian to bimodal with increasing energy. Finally, in the band of global chaos, the distribution of recurrence times to finite size cells is found to follow an exponential law with the mean recurrence time satisfying a scaling law involving an exponent consistent with global chaos and ergodicity.