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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.

Near the stable Adler phase equation point $\alpha =\arcsin (\omega /ϵ)$, linearized noise obeys an Ornstein-Uhlenbeck process with relaxation rate $\kappa =\sqrt{{ϵ}^2-{\omega }^2}$. Its harmonic-model raw correlation is ${\alpha }^2+(T_{\mathrm{eff}}/\kappa )e^{-\kappa |\tau |}$. Nonlinear drift also shifts the local mean at order $T_{meff}$. This describes times after relaxation but before appreciable phase slips; it is not a stationary infinite-time correlation of the unwrapped phase at fixed nonzero noise.

Write the deterministic drift of the Adler phase equation as $f(\theta )=\omega -ϵ\sin \theta$. Its minimum is $\omega -ϵ$ and its maximum is $\omega +ϵ$. For $0<\omega <ϵ$, the zero condition has exactly two solutions in the specified interval:The linearization of a dynamical system at an equilibrium point gives $\delta \dot{\theta }=f^{\prime }({\theta }_{\ast })\delta \theta$, with $f^{\prime }=-ϵ\cos \theta$. Defining $\kappa =\sqrt{{ϵ}^2-{\omega }^2}$ gives $f^{\prime }({\theta }_s)=-\kappa <0$, so ${\theta }_s$ is stable; $f^{\prime }({\theta }_u)=+\kappa >0$, so ${\theta }_u$ is unstable. For $\omega >ϵ$, $f$ is positive everywhere and there are no equilibrium points: the phase runs continuously. This is the distinction between phase locking and running phase dynamics.

With mobility scaled to one, the effective force is $-V^{\prime }$. Therefore the tilted washboard potential isFor $\omega <ϵ$, its alternating local minima and maxima trap noise-free trajectories in wells. Minima coincide with ${\theta }_s+2\pi j$, and maxima with ${\theta }_u+2\pi j$. For $\omega >ϵ$, $V^{\prime }=ϵ\sin \theta -\omega <0$ everywhere: there are no wells and the particle slides down the tilt. The potential is defined on the unwrapped phase and obeys $V(\theta +2\pi )=V(\theta )-2\pi \omega$; it is not a single-valued periodic equilibrium potential on the circle.

At the transition $\omega =ϵ$, the two equilibrium points merge at $\theta =\pi /2$ in a saddle-node bifurcation. There $f\simeq ϵ(\theta -\pi /2{)}^2/2$, so the point is attracting from the left and repelling from the right; a zero linear derivative alone does not establish stable trapping.

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.

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.

A periodic corrugation plus a uniform tilt. For the Adler phase equation, $V(\theta )=-\omega \theta -ϵ\cos \theta$ gives force $-V^{\prime }=\omega -ϵ\sin \theta$ and $V(\theta +2\pi )-V(\theta )=-2\pi \omega$. It has wells when $|\omega |<ϵ$ and no extrema when $|\omega |>ϵ$. It is a potential on the unwrapped phase, not a single-valued periodic potential on the circle.