Phase oscillator

A model retaining an oscillation's phase while neglecting amplitude dynamics. Interactions can synchronize relative phases through phase locking; noise can cause phase slips. The Adler phase equation is a simple example for a coupled relative phase.

An event changing the unwrapped relative phase by one cycle, usually $2\pi$, compared with a phase-locked state. Thermally activated phase slips correspond to crossing neighboring barriers of a tilted washboard potential. Successive slips can produce long-term drift even while small fluctuations appear confined on shorter times.

A noise-driven barrier crossing between adjacent phase-locked wells. The Kramers escape rate is proportional to $e^{-\Delta V/T_{\mathrm{eff}}}$ when a mobility-one Langevin equation has white-noise covariance $2T_{\mathrm{eff}}\delta (t-t^{\prime })$. Both the barrier and local relaxation time determine whether an intrawell phase autocorrelation approximation is appropriate.

In a mobility-one noisy Adler phase equation, matching Kramers escape rate prefactors and a barrier difference $2\pi \omega$ give $p_{+}/p_{-}=e^{2\pi \omega /T_{\mathrm{eff}}}$. Positive mismatch favors forward winding. The ratio is a low-noise barrier-crossing result, not a claim that the particle remains in one well at arbitrarily long times.

The relative-phase drift $\dot{\theta }=\omega -ϵ\sin \theta$ describes competing frequency mismatch and phase coupling. For $0<\omega <ϵ$, the equilibrium points are $\arcsin (\omega /ϵ)$ and $\pi -\arcsin (\omega /ϵ)$, stable and unstable respectively. They merge in a saddle-node bifurcation at $\omega =ϵ$; greater mismatch gives running phase dynamics. Adding Gaussian white noise gives Overdamped Langevin dynamics in a tilted washboard potential.

For $0<\omega <ϵ$, set $\alpha =\arcsin (\omega /ϵ)$ and $\kappa =\sqrt{{ϵ}^2-{\omega }^2}$. The neighboring saddles of the tilted washboard potential have barriers $\Delta V_{+}=2\kappa -\omega (\pi -2\alpha )$ and $\Delta V_{-}=2\kappa +\omega (\pi +2\alpha )$ above the minimum at $\alpha$. Their difference is $2\pi \omega$; both have curvature $-\kappa$, so forward and backward Kramers escape rates have equal leading prefactors.

A phase that continues winding rather than tending to a locked value. In the noise-free Adler phase equation, $\omega >ϵ>0$ makes the drift positive at every phase, so there are no equilibrium points. The associated tilted washboard potential decreases monotonically; wells vanish at the locking threshold.