Escape from a one-dimensional well

The starting point of the Kramers theory of activated rate processes is the onedimensional Markovian Langevin equation, Eq. (8.13) 

Equations (14.39) and (14.40) describe a one-dimensional Brownian particle moving under the influence of a systematic force associated with the potential V (x) and a random force and the associated damping that mimic the influence of a thermal environment. In applying it as a model for chemical reactions it is assumed that this 

We have already seen that Eqs (14.39) and (14.40) are equivalent to the Fokker-Planck equation, Eq. (8.144), 

In the present context, Eq. (14.41) is sometimes referred to as the Kramers equation. We have also found that the Boltzmann distribution 

As before, it is assumed that the potential V is characterized by a reactant region, a potential well, separated from the product region by a high potential barrier, see Fig. 14.2. We want to calculate the reaction rate, defined within this model as the rate at which the particle escapes from this well. 

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Transition state rate of escape from a one-dimensional well... 

Consider an activated rate process represented by the escape of a particle from a one-dimensional potential well. The Hamiltonian of the particle is... 

We will consider only one-dimensional surface diffusion. The potential w(q) is assumed to be periodic with a spacing a between wells. In the spatial diffusion limit, the particle might escape from a well, get trapped in an adjacent well, and after a long time escape with equal probability in either direction. In this case, the diffusion coefficient is proportional to the product of the spatial diffusion rate of escape from the well and the distance squared between the wells (116) ... 

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