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In the classical three rotor problem, three equal point masses move on acircle subject to attractive cosine potentials of strength g. In the center ofmass 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 witha band of global chaos for 5.33g < E < 5.6g. Here, we provide numericalevidence for ergodicity and mixing in this band. The distributions of relativeangles and angular momenta along generic trajectories are shown to approach thecorresponding distributions over constant energy hypersurfaces (weighted by theLiouville measure) as a power-law in time. Moreover, trajectories emanatingfrom a small volume are shown to become uniformly distributed over constantenergy hypersurfaces, indicating that the dynamics is mixing. Outside thisband, ergodicity and mixing fail, though the distributions of angular momentaover constant energy hypersurfaces show interesting phase transitions fromWignerian to bimodal with increasing energy. Finally, in the band of globalchaos, the distribution of recurrence times to finite size cells is found tofollow an exponential law with the mean recurrence time satisfying a scalinglaw involving an exponent consistent with global chaos and ergodicity.

Ergodicity, mixing and recurrence in the three rotor problem