Energy extraction of a chaotic system in a cyclic process: a Szil\'ard Engine perspective

Inspired by the available examples of Microcanonical Szil\'ard Engines and by the original Szil\'ard Engine, we devise a system with two degrees of freedom whose ensemble average energy, starting with a microcanical ensemble, decreases after a cyclic variation of its external parameters. We use the Ergodic Adiabatic Theorem to motivate our cycle and numerical simulations to check the decrement in the average energy. We then compare our system to the aforementioned Szil\'ard Engines, Microcanonical or not, and speculate about symmetry breaking being the cause of energy extraction in cyclic processes, even when non-integrability and chaos are present.