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  • Open Access

Irreversibility of the pendulum revisited from Husimi's adiabaticity parameter

Yuki Izumida*

  • Department of Complexity Science and Engineering, Graduate School of Frontier Sciences, The University of Tokyo, Kashiwa 277-8561, Japan

  • *Contact author: izumida@k.u-tokyo.ac.jp
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Phys. Rev. E 112, 064131 – Published 29 December, 2025

DOI: https://doi.org/10.1103/r7cg-vxz1

Abstract

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 noncyclic process as well as a cyclic process, and elucidate how the Husimi's adiabaticity parameter impacts this law. In particular, for a general noncyclic process, we show the law of entropy nondecrease for the pendulum without friction by using the property of the Husimi's adiabaticity parameter.

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Phys. Rev. E 112, 064131– Published 29 December, 2025
  • Received 7 September 2025
  • Accepted 9 December 2025
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