The Jarzynski equality in van der Pol and Rayleigh oscillators

TL;DR

This study investigates the applicability of the Jarzynski equality in non-Hamiltonian, deterministic oscillators like van der Pol and Rayleigh models, finding approximate validity across various conditions despite their non-reversible nature.

Contribution

It demonstrates that the Jarzynski equality approximately holds in van der Pol and Rayleigh oscillators, which are not microscopically reversible, expanding understanding of JE's applicability.

Findings

JE approximately holds for a wide range of relaxation times

Work distribution is non-Gaussian with a U-shaped structure for strong damping

Semi-quantitative explanation of $ au$ dependence using a simple limit-cycle model

Abstract

We have studied the Jarzynski equality (JE) in van der Pol and Rayleigh oscillators which are typical deterministic non-Hamiltonian models, but not expected to rigorously satisfy the JE because they are not microscopically reversible. Our simulations that calculate the contribution to the work $W$ of an applied ramp force with a duration $\tau$ show that the JE approximately holds for a fairly wide range of $\tau$ including $\tau \rightarrow 0 $ and $\tau \rightarrow \infty$, except for $\tau \sim T$ where $T$ denotes the period of relaxation oscillations in the limit cycle. The work distribution function (WDF) is shown to be non-Gaussian with the $U$-shaped structure for a strong damping parameter. The $\tau$ dependence of $R$ $(=- k_B T \ln<e^{-\beta W}>)$ obtained by our simulations is semi-quantitatively elucidated with the use of a simple expression for limit-cycle oscillations, where the bracket $<\cdot>$ expresses an average over the WDF. The result obtained in self-excited oscillators is in contrast with the fact that the JE holds in the Nos\'{e}-Hoover oscillator which also belongs to deterministic non-Hamiltonian models.