 Kramers escape rate

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]