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Ergodicity, mixing, and recurrence in the three rotor problem.
Govind S Krishnaswami1, Himalaya Senapati1
1Physics Department, Chennai Mathematical Institute, SIPCOT IT Park, Siruseri, Chennai 603103, India.
Chaos (Woodbury, N.Y.)
|May 3, 2020
Summary
This study numerically confirms ergodicity and mixing in a chaotic band of the three rotor problem. Trajectories uniformly distribute on energy hypersurfaces, demonstrating ergodic and mixing dynamics within this specific energy range.
Area of Science:
- Classical Mechanics
- Dynamical Systems
- Statistical Mechanics
Background:
- The three rotor problem involves three equal point masses on a circle with cosine potentials.
- Previous research identified an order-chaos-order transition and a global chaos band (5.33g≤E≤5.6g).
Purpose of the Study:
- To provide numerical evidence for ergodicity and mixing within the global chaos band of the three rotor problem.
- To investigate the behavior of trajectories and distributions outside this chaotic band.
Main Methods:
- Numerical simulations of the three rotor problem.
- Analysis of trajectory distributions (relative angles, angular momenta) over time.
- Examination of recurrence time distributions within the chaotic band.
Main Results:
- Ergodicity and mixing are numerically confirmed within the 5.33g≤E≤5.6g chaos band.
- Trajectory distributions approach Liouville measure distributions on constant energy hypersurfaces.
- Outside the band, ergodicity and mixing fail, with phase transitions observed in angular momentum distributions.
Conclusions:
- The identified band of global chaos in the three rotor problem exhibits ergodic and mixing dynamics.
- Recurrence time distributions in the chaos band follow an exponential law, consistent with ergodicity.
- The study highlights distinct dynamical behaviors inside and outside the global chaos band.
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