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The Adler phase equationdescribes a relative phase pulled by a coupling of strength $ϵ$ while a frequency mismatch $\omega$ drives it around the circle. For $|\omega |<ϵ$, an equilibrium satisfies $\sin \theta =\omega /ϵ$. Its linear growth rate is $-ϵ\cos \theta$, so the branch with positive cosine is stable and the opposite branch is unstable.

For $0<\omega <ϵ$, the stable and unstable values are $\arcsin (\omega /ϵ)$ and $\pi -\arcsin (\omega /ϵ)$ modulo $2\pi$. They merge at $\omega =ϵ$, a saddle-node bifurcation on the phase circle. The stable branch's relaxation rate is $\sqrt{{ϵ}^2-{\omega }^2}$, which tends to zero at the threshold. Greater positive mismatch gives running phase dynamics.

The drift is minus the derivative of the tilted washboard potential $V(\theta )=-\omega \theta -ϵ\cos \theta$. Although $V$ is not periodic, its derivative is; each increase of $2\pi$ lowers the potential by $2\pi \omega$. Adding Gaussian white noise produces overdamped Langevin dynamics. The activation barriers of the Adler phase equation then control rare thermally activated phase slips.

A thermally activated phase slip is a noise-driven crossing between adjacent phase-locked wells. In the mobility-one Langevin equation$T_{\mathrm{eff}}$ has the units of the phase potential $V$. A barrier of height $\Delta V$ is rarely crossed when $\Delta V\gg T_{\mathrm{eff}}$.

For a smooth nondegenerate minimum $a$ and neighboring saddle $b$, the overdamped Kramers escape rate is asymptoticallyThe mean waiting time is approximately $1/r$, whereas local relaxation within the well occurs on the scale $1/V^{\prime \prime }(a)$. An intrawell phase autocorrelation approximation is useful only on timescales where escape remains rare; the barrier height alone does not specify this hierarchy.

For the Adler phase equation, the activation barriers of the Adler phase equation differ by $2\pi \omega$ in the two winding directions, while the leading curvature prefactors agree. Their rate ratio is therefore $r_{+}/r_{-}=e^{2\pi \omega /T_{\mathrm{eff}}}$ in the low-noise approximation. The barrier-crossing description fails at the deterministic locking threshold, where the minimum and saddle merge and the assumed nonzero curvatures vanish.