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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.

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.

Running phase dynamics means that an unwrapped phase keeps winding rather than approaching a locked value. For the deterministic Adler phase equation, $\omega >ϵ>0$ makes $\dot{\theta }=\omega -ϵ\sin \theta$ positive everywhere. There are no equilibrium points, and one full winding takesFor the integral, shift the angle so that the denominator is $\omega -ϵ\cos \varphi$ and use $u=\tan (\varphi /2)$. Splitting at the tangent's pole combines the two pieces intowhich has the displayed value. Thus the average winding speed is $2\pi /T=\sqrt{{\omega }^2-{ϵ}^2}$.

If $\omega =ϵ+\delta$ with small positive $\delta$, the average speed is asymptotic to $\sqrt{2ϵ\delta }$. Motion spends a long time near the vanished locking equilibria before completing each winding. The tilted washboard potential is monotone in this regime; noise-driven barrier hopping and deterministic running describe different mechanisms.