Irreversibility of the pendulum revisited from Husimi's adiabaticity parameter

We revisit the irreversibility of the pendulum with time-dependent angular frequency, considered in a classical paper by K. Husimi. He introduced a parameter that measures the adiabaticity of a process utilizing an adiabatic invariant for the equation of motion of the pendulum. With this adiabaticity parameter, Husimi showed the irreversibility of the pendulum for a cyclic process, which is reminiscent of the Planck principle in thermodynamics, based on the microscopic mechanics. In this study, we generalize the argument by Husimi to a damped pendulum with friction, and highlight the role of conservation of a phase-space area on the Husimi's adiabaticity parameter. Moreover, we also investigate the second law of thermodynamics and its generalization for a general non-cyclic process as well as a cyclic process, and elucidate how the Husimi's adiabaticity parameter impacts on this law. In particular, for a general non-cyclic process, we show the law of entropy non-decrease for the pendulum without friction by using the property of the Husimi's adiabaticity parameter.