Laplace method approximation of the mean first-passage time for a degree of freedom following a Fokker-Planck equation to overcome a barrier. Most easily solved in 1D, where the result is:

which is known as Kramers escape time. The exponential has the same form as the phenomenological Arrhenius equation, and the pre-factor is known as the inverse attempt frequency.

It is used to estimate reaction rates in Chemical kinetics where one defines a reaction coordinate approximating the evolution of the relevant molecules and their potential energy.

Barrier crossover time from probability distribution

Can use the (conditional) probability distribution $P(x,t|x_0,t_0)$ when a barrier is present to calculate the crossover time. This is done by considering the flux $J$ (see Fokker-Planck equation). The probability distribution for crossing the barrier is then:

$P(t) = \frac{J(\Delta x, t| - \Delta x, t_0)}{\int_0^\infty dt J(\Delta x, t| - \Delta x, t_0)}$

where we assume the barrier is at $0$, so that $- \Delta x$ and $\Delta x$ are at opposite sides.

The probability distribution for crossing the barrier above, is the same as the probability distribution for the first passage time. To understand why we use the flux (current, $J$) to calculate this, imagine many instances of the Brownian particle in the potential. We can approximate the above $P(t)$ by just considering frequencies, in the limit of infinite instances. Now, $J$ is just calculated by counting the number of times a particle is found within $dx$ of the point $\Delta x$, at time $t$, multiplied by its velocity at that moment.

Now, consider a first-passage path...

Well idea is that for every second passage passage path, there is a symmetric one with opposite velocity at measurement point (x,t), that thus cancels it in the sum: