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By H. A. Kramers. Physica 7(4), 284–304.

Original scan: www.mit.edu/~kardar/research/seminars/translocation/Kramers1940.pdf

Publisher metadata and original abstract verified: thermally driven escape from a potential well models the rate of chemical reactions. The scan timed out in the research browser; no unverified precise equation locator is asserted. The wiki's unit-mobility asymptotic is given with its own model, nondegenerate-barrier and full-crossing qualifications.

H. A. Kramers modeled thermally activated escape from a potential well using Brownian motion with friction. His 1940 treatment linked chemical reaction rates to barrier crossing under both random forcing and dissipation.^[ref]

The exponential suppression reflects the energy needed to reach the barrier, but the prefactor depends on the frictional regime. Thus knowing a barrier height alone is insufficient to predict a reaction rate. The Kramers escape rate connects equilibrium thermal probabilities with the dynamics that carry a system out of a metastable state.

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]