Kramers theory for the rate constant

The theory of Brownian motion is a particular example of an application of the general theory of random or stochastic processes [2]. Since Kramers approach is based on a more general stochastic equation than the Langevin equation, we have reviewed some of the fundamental ideas and methods of the theory of stochastic processes in Appendix H. 

Kramers theory is based on the Fokker-Planck equation for the position and velocity of a particle. The Fokker-Planck equation is based on the concept of a Markov process and in its generic form it contains no specific information about any particular process. In the case of Brownian motion, where it is sometimes simply called the Kramers equation, it takes the form 

We shall now determine a solution to Eq. (11.16) with proper boundary conditions and use the result to determine the rate constant in a fluid. It is assumed that the barrier in going from reactant states at well (a) in Fig. 11.0.2 to the transition state at (b) is large compared to kBT the probability of being at (b) is therefore small and it will be reasonable to seek a steady-state solution to Eq. (11.16). Similarly, around the a-well, we also assume stationary conditions with reactants in thermal equilibrium with the solvent. Thus, we seek a solution to the equation with (dP/dt) = 0. 

At equilibrium, stationary conditions exist where P(r,v t) = P(r,v) and P(r,v) is given by equilibrium statistical mechanics  

It is easy to show that the function in Eq. (11.17) is indeed a solution to the stationary equation. We have 

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