Phase slip ($\Delta \theta =\pm 2\pi$)

A phase slip changes the continuously tracked relative phase by one cycle, usually $\pm 2\pi$, compared with a locked state. Tracking the angle only modulo $2\pi$ hides this accumulated winding: the wrapped phase can return to the same value after a full circuit. A jump caused solely by changing the plotting branch cut is not evidence of a dynamical phase slip.

In a noisy locked oscillator, fluctuations usually remain close to a stable phase, but a thermally activated phase slip crosses a barrier of the tilted washboard potential and settles in a neighboring well on the real-valued phase lift. The corresponding wrapped state can look unchanged even though the relative cycle count has increased or decreased by one.

If forward and backward slips have rare-event rates $r_{+}$ and $r_{-}$, their mean contribution to unwrapped phase drift is $2\pi (r_{+}-r_{-})$. This follows by multiplying the signed change per event by the expected event count per unit time. Equal rates can therefore give zero mean drift while individual slips still occur. The forward-backward bias of phase slips determines when a frequency mismatch makes one direction more likely.

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.

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.