Reaction asymptotic behaviour at long times

As it was shown above, in Section 6.1.1 using the reaction of immobile particles as example, the kinetic equations (5.1.2) to (5.1.4), being derived 

Indeed, the law (2.1.78) has been confirmed in [28-30] for different space dimensions. However, since instead of general equations (5.1.2) to (5.1.4), their black sphere analog (5.1.33) to (5.1.35) was analyzed, a set of non-linear equations lost its transparency. This is why let us start now with a complete set of equations for the superposition approximation, n t) = riQ t) = n(t) assuming that the recombination cr(r) is short-range and could be characterized by a finite radius tq so that the integral 

New reaction asymptotic law (2.1.78) emerges due to formation during the reaction course of a new spatial scale - the correlation length = Id. Similar to the case of immobile particles, we can expect here that at long times the coordinate r enters into the correlation function in a scaling form r] = r/Id, so that Y r,t) Y(t],t), Xi, r,t) — Xy(7y, r), where the second variable is r = In t. Let us consider consequences of the new scale formation. The reaction rate K t) (4.1.19) at long times can be estimated as 

It was used in deriving (6.2.2), that since the function Y(ry, r) changes considerably on the distances of the order of Id ro only, it can be taken out of the integral in (6.2.2) with the maximal contribution on the scale ro, along with the definition (6.2.1). In doing so, the value of Y rj,T) is taken atr] — 0. 

If the exponent a turns out to be less than its classical value found in the linear approximation, ao = 1 (d 2), the relevant reaction rate K t) is reducing at long times so that in the long-time limit K (oo) = 0 (the reaction rate s zerofication). On the other hand, when searching the asymptotic solution of non-linear equations, the asymptotic relation (6.2.4) permits to replace tK t)n t) for a. 

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