Brownian motion barrier crossing

In the same year Hendrik Kramers published his landmark paper [117] on the theory of chemical reaction rates based on thermally activated barrier crossing by Brownian motion [77], These two papers clearly mark the domains of two related areas of chemical research. Kramers provided the framework for computing the rate constants of chemical reactions based on the molecular structures, energy, and solvent environment. (See Section 10.4.1.) Delbriick s work set the stage for predicting the dynamic behavior of a chemical reaction system, as a function of the presumably known rate constants for each and every reaction in the system. 

As is shown in [77,78], the last equation is valid if the stepwise Brownian motion can be approximated by a random walk, where the representing particle jumps randomly between the potential minima. It works well in the case of high potential barriers, or in other words, when the time during which the chain length retains unchanged, is much larger than the time of barrier crossing, i.e., the reaction time. 

So far we concentrated on the single particle crossing the barrier, but for a set of coupled equations of motion, as discussed in Section 9.8, the formalism is equally valid. The results of this chapter so far can be summarized using Eq. (9.33) as follows if the friction f = 0, we get classical TST if the friction acts on the reaction coordinate directly, we get Kramers s theory and if we let the friction act on a coordinate coupled to the, itself frictionless barrier transition coordinate, we get the Brownian oscillator model described in Section 9.8. 

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