Waite-Leibfried theory

A new principal element of the Waite-Leibfried theory compared to the Smoluchowski approach is the relation between the effective reaction rate K(t) and the intermediate order parameter x = Xab (r,i). In its turn, the Smoluchowski approach is just an heuristic attempt to describe the simplest irreversible bimolecular reactions A + B- B,A + B- B and A + B -> 0 and cannot be extended for more complicated reactions. The Waite-Leibfried approach is not limited by these simple reactions only it could be applied to the reversible reactions and reaction chains. However, in the latter case the particular linearity in the joint correlation function x = Xab (r, ) does not always mean linearity of equations since additional non-linearity caused by particle densities can arise. 

The said allows us to understand the importance of the kinetic approach developed for the first time by Waite and Leibfried [21, 22]. In essence, as is seen from Fig. 1.15 and Fig. 1.26, their approach to the simplest A + B —0 reaction does not differ from the Smoluchowski one However, coincidence of the two mathematical formalisms in this particular case does not mean that theories are basically identical. Indeed, the Waite-Leibfried equations are derived as some approximation of the exact kinetic equations due to a simplified treatment of the fluctuational spectrum a complete set of the joint correlation functions x(rJ) for kinds of particles is replaced by the only function xab (a t) describing the correlation of chemically reacting dissimilar particles. Second, the equation defining the correlation function X = Xab(aO is linearized in the function x(rJ)- This is analogous to the... 

The consistent derivation and analysis of the Waite-Leibfried equations is presented in Chapter 4. We show that this theory is the linear approximation of the exact many-particle formalism. Its relation to the Smoluchowski theory is also established. 

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