 Thermally activated phase slip

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.