We revisit the chaotic double pendulum and simulate its Kolmogorov entropy K as a function of the conserved total energy E . This system intuitively exhibits an energy range with negative temperatures that is in important respects related to the pioneering work of Onsager 1, due to increasingly periodic orbital motion as E grows larger. Thus, K in this range descends as E increases. Therein, at the energy required to put the pendulum into upright position, we experience a second-order phase transition with a discontinuous dK / dE curve into a high-energy phase with remarkable properties. Although the dynamics there is still strongly chaotic, the average potential energy suddenly levels off, thus admits conversion of higher potential energies into kinetic ones by simple means. This has beneficial consequences for molecular machines, as we may externally supply and control their mechanical power output and also compensate for friction.
Hans R. Moser (Sun,) studied this question.