The nearest-available-neighbour approximation

The accuracy of the Kirkwood superposition approximation was questioned recently [15] in terms of the new reaction model called NAN nearest available neighbour reaction) [16-20]. Unlike previous reaction models, in the NAN scheme AB pairs recombine in a strict order of separation the closest pair in an initially random distribution is removed first, then the next one and so on. Thus for NAN, the recombination distance R, e.g., the separation of the closest pair of dissimilar particles at any stage of the recombination, replaces real time as the ordering variable time does not enter at all the NAN scheme. R is conveniently measured in units of the initial pair separation. At large R in J-dimensions, NAN scaling arguments [16] lead rapidly to the result that the pair population decreases asymptotically as (c is 

The kinetic equations [1-3, 12] were rewritten in [15] for a special choice of the recombination law cr(r), adequate to the NAN model, and solved numerically for d = 1. The general conclusion was drawn that the Kirkwood approximation is quite correct but leads to the error of the order of 10% for the critical exponent a in the asymptotic decay law n R) oc R ° . This quantity (10%) was suggested to be used as a measure of the accuracy of the Kirkwood approximation in the kinetics of the bimolecular reaction A -h B 0. 

Before analyzing critically the formalism [15], let us rewrite the basic equations of the NAN model in the superposition approximation. Equations for concentration n R) come from (4.1.28) using (4.1.19) with a formal substitution of time t for R  

The same should be done in the joint correlation functions introducing Y r,R) and X r,R). The relevant equations come from a set (5.1.14) to (5.1.16)  

The reaction rate a r) satisfying the NAN model should be substituted into equations (6.1.45) to (6.1.48) and these equations then simplified for the analytical solution. It was assumed in [15] that the NAN model corresponds to the choice of the function a r) = 5 r - R), where 5 x) is the Dirac delta function, which allowed the authors to simplify kinetic equations. Indeed, formal use of the main property of -function leads to K(R) = 2Y R, R) and simplifies integral terms in equations (6.1.47) and (6.1.48). However, it leads to a number of errors not noticed in [15]. In fact, direct use of the property of the function a r) = 5 r — R) in (6.1.49) is justified only if the function Y r,t) is an analytical one near the point r — R. This is not true in our case, since Y(r, R) is a solution of the singular equation (6.1.48) with a r) = 6 r—R) and consequently is also a generalized function (distribution). As it is well known from the theory, properties of products of such distributions a r)Y r, R) are not well defined. This is why we will introduce here less singular function a r) = w (r — R) where w x) is an analytical function of the argument x and parameter e — +0. The choice of cr(r) = 5 r — R) in [15] suits in particular the well-known representation 

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