Irreversibility of the pendulum revisited from Husimi's adiabaticity parameter

TL;DR

The paper investigates irreversibility in a time-dependent damped pendulum by extending Husimi's adiabaticity parameter $Q_t^*$ to non-Hamiltonian dynamics with friction. It derives the internal energy relation $E_t=E_{t_0} (\omega_t/\omega_{t_0}) Q_t^*$ and proves the key bound $Q_t^* \ge W_t$, where the Wronskian $W_t$ encodes phase-space area expansion and satisfies $\dot W_t=-\Gamma_t W_t$. For slow frequency changes, $Q_t^*\approx W_t$, and cyclic protocols obey a Planck-principle-like bound $E_{t_1} \ge E_{t_0} Q_{t_1}^*$, with a generalization $E_{t_1} \ge E_{t_0} W_{t_1}$ when $\Gamma_t\neq 0$. The entropy analysis introduces volume entropy, giving $\Delta S_t=\ln\frac{Q_t^*+W_t}{2}$ and the bounds $\Delta S_t \ge \ln W_t$ (for friction) and $\Delta S_t^{\rm G}=\ln W_t$ for Gibbs entropy, connecting irreversibility to adiabaticity and area growth, and linking to Noether invariants in the quasistatic limit. The work thus provides a cohesive framework connecting microscopic dynamics, adiabatic invariants, and generalized second-law-type constraints for non-Hamiltonian, time-dependent systems.

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 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.

Figures (1)

• Figure 1: The time evolutions of the Husimi's adiabaticity parameter $Q_t^*$ (Eq. (\ref{['eq.Q_def']})) and the Wronskian $W_t$ (Eq. (\ref{['eq.def_W']})) as the functions of $X_t$ and $Y_t$ obeying Eqs. (\ref{['eq.X']}) and (\ref{['eq.Y']}), respectively, for three different $(t_1, \Gamma)$: (a) $(t_1, \Gamma)=(10, 0.05)$, (b) $(t_1, \Gamma)=(1, 0.05)$, and (c) $(t_1, \Gamma)=(10, 0.5)$. We used $t_0=0$, $\omega_{t_0}=1.2$, and $\omega_{t_1}=2$ as the other parameters.