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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.

For the one-dimensional overdamped Langevin dynamics modellet $m$ be a nondegenerate local minimum of $\varphi$ and $M$ the neighbouring local maximum separating it from another basin. In the weak-noise limit $D\downarrow 0$, escape across the barrier has leading rateThe normalization has unit mobility. With physical potential energy $U$, constant mobility $\mathcal{M}$ and thermal noise $D=\mathcal{M}{k}_{B}T$, this becomes $\mathcal{M}\sqrt{U^{\prime \prime }(m)|U^{\prime \prime }(M)|}{e}^{-\Delta U/(k_{B}T)}/(2\pi )$.

A Laplace method applied to the one-dimensional mean exit-time integral explains both factors: concentration near the minimum supplies one Gaussian factor, and passage across the full saddle region supplies the other. Absorbing exactly at the saddle rather than beyond it changes this prefactor, as do degenerate barriers and inertial regimes. H. A. Kramers used escape of a thermally driven particle from a potential well as a model of a reaction rate.^[ref]

Langevin dynamics describes a dynamical variable driven by deterministic forces and random noise. Underdamped Langevin dynamics retains inertia, whereas overdamped Langevin dynamics neglects it. At thermal equilibrium, the fluctuation-dissipation relation for a Langevin particle relates Gaussian white noise strength to drag and temperature.

When momentum relaxes much faster than the imposed motion, force balance replaces the inertial equation: $\zeta \dot{x}=F(x,t)$ in a deterministic one-dimensional model. The linear friction coefficient sets the time scale. Thermal forcing, if included, gives overdamped Langevin dynamics.

For $0<\omega <ϵ$, let $x=\theta -{\theta }_s$ and $\kappa =\sqrt{{ϵ}^2-{\omega }^2}$. In a single well, sufficiently small fluctuations obey the linear overdamped Langevin dynamicsThis is an Ornstein-Uhlenbeck process. Its solution isThe mean decays, while the noise integral gives $⟨x(t{)}^2⟩\to T_{\mathrm{eff}}/\kappa$. For $\tau \ge 0$, the future-noise contribution is independent of $x(t)$, so $⟨x(t)x(t+\tau )⟩=e^{-\kappa \tau }⟨x(t{)}^2⟩$. The intrawell phase autocorrelation is thereforeThe connected unnormalized time autocorrelation is just the exponentially decaying second term. The constant first term is necessary because the requested raw correlation is not centered at the equilibrium point. This is the harmonic-well approximation. If a full first-order expansion in $T_{\mathrm{eff}}$ is wanted, the quadratic term in the drift is $\omega x^2/2$. Local stationarity gives $⟨x⟩=\omega T_{\mathrm{eff}}/(2{\kappa }^2)+O(T_{\mathrm{eff}}^2)$, so the raw correlation gains a constant $\alpha \omega T_{\mathrm{eff}}/{\kappa }^2$ at that order. The connected correlation is unchanged to first order; the boxed formula is the conventional linearized result rather than a complete nonlinear first-order raw-correlation expansion.

The required time window is long compared with ${\kappa }^{-1}$ but short compared with the escape time, with the lag also short compared with escape. Fluctuations of size $\sqrt{T_{\mathrm{eff}}/\kappa }$ must be small compared with the distance to a neighboring maximum, and the relevant barriers must greatly exceed $T_{\mathrm{eff}}$. For a fixed nonzero noise strength, the unwrapped phase eventually makes phase slips; its raw correlation does not have this stationary infinite-time limit. Thus “large time” here refers to local relaxation within the occupied well, not to the limit after arbitrarily many barrier crossings.

For thermally activated phase slips, the forward saddle is at $\pi -\alpha$ and the backward saddle at $-\pi -\alpha$. Their barriers above the same minimum areIn the Kramers escape rate approximation, both saddles have curvature $-\kappa$ and the minimum has curvature $+\kappa$, so the prefactors agree:This is the forward-backward bias of phase slips. It favors forward motion because each forward step lowers the tilted potential by $2\pi \omega$. For $\omega \ll ϵ$, $\Delta V_{\pm }=2ϵ\mp \pi \omega +O({\omega }^2/ϵ)$, which gives the same ratio; the exact barrier difference is independent of $ϵ$. These probabilities may be read as rates in a common short observation interval, or as normalized competing exit probabilities. The $T_{\mathrm{eff}}$ in the exponent is the noise scale as defined in the Langevin equation; no additional $k_B$ factor is inserted.

Write $E(t)=\delta A/\delta x(t)$ for the functional derivative of the action. Under time reversal in classical mechanics, $\dot{x}$ changes sign whereas $E$ is even by assumption. Thus the forward and reversed Gaussian white noise histories associated with the same geometric path areThe Onsager--Machlup path probability therefore givesHere the path probabilities are densities conditional on their respective initial states; they are not separately normalized bridges conditioned on both endpoints. The noise-to-path Jacobian determinant must be treated with a consistent discretization: it cancels when it is invariant under time reversal in classical mechanics. This is the usual additive-noise Langevin dynamics convention, including a constant-mass Underdamped Langevin dynamics or the time-reversal invariance of a path Jacobian in midpoint overdamped Langevin dynamics. A completely arbitrary velocity-dependent Lagrangian would require specifying that measure rather than deducing its cancellation solely from the parity of $E$.

For $q\ne 0$ and $\alpha >0$, each Fourier mode is an Ornstein-Uhlenbeck process with decay rate $r_q=\alpha q^2$. Its explicit Ornstein-Uhlenbeck solution isso the Itô isometry givesTo identify the overdamped Langevin dynamics, choose a friction ${\zeta }_q>0$, a harmonic oscillator potential $V_q={\zeta }_q\alpha q^2{h}_q^2/2$, and force noise $f_q={\zeta }_q{\eta }_q$. The fluctuation-dissipation relation for a Langevin particle then defines $k_B{T}_{\mathrm{eff}}={\zeta }_q{\sigma }^2/2$. Its Boltzmann distribution isThis is the stationary spectrum of a linear fluctuating interface, also following from the equipartition theorem. The numerical prefactor uses exactly the mode-noise normalization given in the paper. The nonzero modes can therefore have a stationary Gaussian distribution even though the unpinned zero mode cannot.

Equation (1) is Underdamped Langevin dynamics for a unit-mass particle in potential $U$, coupled to a heat bath of temperature $T$. The coefficient $\gamma >0$ is viscous friction, and the noise amplitude is fixed by the fluctuation-dissipation theorem. Its Fokker-Planck equation isFor $\gamma \gg 1$, velocity relaxes rapidly, so formallyThus the overdamped Langevin dynamics isand its density obeys

Thermal molecular motion causes the coarse droplet radius to fluctuate as well as drift down $F(R)$, so its effective equation must be an overdamped Langevin dynamics. With $⟨\Lambda (t)\Lambda (t^{\prime })⟩=\delta (t-t^{\prime })$, detailed balance with equilibrium density proportional to $e^{-F/(k_{B}T)}$ imposes the fluctuation-dissipation theorem. The Model A fluctuation-dissipation relation givesThe factor of mobility ensures that the diffusion in $R$ and the dissipative drift have the same equilibrium Gibbs distribution.

Underdamped Langevin dynamics retains both position $q$ and momentum $p$. For mass $m>0$, temperature $T>0$, friction rate $\gamma >0$ and smooth potential energy $U$, one common normalization isHere $W$ is vector Brownian motion. The noise amplitude is tied to friction by the fluctuation-dissipation relation for a Langevin particle. If the density is normalizable and the dynamics is nonexplosive, its invariant distribution is the Boltzmann distributionThe Hamiltonian flow preserves the total energy and phase-space volume; the momentum Ornstein-Uhlenbeck process preserves a centered normal distribution with coordinate variance $m{k}_{B}T$. These two facts verify the invariant density. In dimensionless units with $k_{B}T=1$ and momentum variance $\eta$, it is $\exp [-U(q)-|p{|}^2/(2\eta )]$, with the corresponding choice $m=\eta$.

Retaining inertia resolves the short-time ballistic-to-diffusive Brownian displacement. The overdamped Langevin dynamics approximation removes momentum relaxation when only much longer times are relevant.