Jarzynski equality in van der Pol and Rayleigh oscillators

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 reversible. Our simulations that calculate the contribution to the work W of an applied ramp force with a duration τ show that the JE approximately holds for a fairly wide range of τ including τ → 0 and τ → ∞, except for τ ~ 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 τ dependence of R ( = -k(B)(Tln(e)(-βW)) obtained by our simulations is semiquantitatively elucidated with the use of a simple expression for limit-cycle oscillations, where the bracket (·) 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 Nosé-Hoover oscillator, which also belongs to deterministic non-Hamiltonian models.