The shortened superposition approximation

In turn, equations (4.1.9) and (4.1.10) yield a general initial condition necessary for an analysis of the transition from the monomolecular to bimolecular kinetics. When the initial particle distribution is unknown (which is usually the case, e.g., after prolonged irradiation), it is assumed, as a mle, to be the simplest, i.e., the Poisson one, equation (4.1.12). 

Turning back now to the kinetics of the diffusion-controlled A -b B -4 0 recombination, and neglecting reactant interaction, let us write down several equations from an infinite hierarchy of equations (2.3.37), (2.3.45), and (2.3.53)  

Keeping in mind the physical meaning of single- and two-particle densities, equations (2.3.58) to (2.3.61), we can rewrite now equations (4.1.13) in a form similar to the standard chemical kinetics (Section 2.1). To do it, let us first transform the integral expression 

Passing from the integration over r/ to that over the variable f = ri - f[, we arrive at the equation 

In fact, the latter is a functional of the correlation function of dissimilar particles, i.e., to calculate K t) we need to know either Y r,t) or pi. In its turn, equation (4.1.16) demonstrates that these latter are coupled with three-point densities etc. Therefore, to solve the problem, we have to cut off the infinite equation hierarchy, thus only approximately describing the fluctuation spectrum. Usually it is done by means of the complete Kirkwood superposition approximation, equations (2.3.62) and (2.3.63), or the shortened approximation, equations (2.3.64) and (2.3.65). 

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