Overdamped Langevin dynamics

Neglecting inertia in Langevin dynamics gives the overdamped limit.

For potential $F$ and mobility $\mathcal{M}$, overdamped Langevin dynamics isThe matching noise amplitude is the fluctuation-dissipation relation that makes the Boltzmann density proportional to $e^{-F/(k_{B}T)}$ invariant.

With local thermal noise matched to spatially varying drag, the overdamped probability density obeys the displayed diffusion equation. Its equivalent Itô diffusion is $dX=D^{\prime }(X)dt+\sqrt{2D(X)}dW$, since the two terms in the Fokker-Planck equation combine to ${\partial }_x(D{P}_x)$. Thus $d⟨X⟩/dt=⟨D^{\prime }(X)⟩$, even with no applied force. The drift points toward greater diffusivity, but a uniform density has zero flux: the drift is balanced by the spatial variation of the noise. Holding the force-noise amplitude fixed instead would not produce this isothermal Fickian equation.

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.

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]