The A B - C reaction. Stochastic particle generation

Let us apply general stochastic equations (2.2.15) to the simple A+B C reaction with particle creation - the model problem discussed more than once ([84] to [93]). A relevant set of kinetic equations reads 

The most general method to solve the set (2.2.20), (2.2.21) seems to be their transformation in terms of different-order correlations. Let us present reactant concentrations as a sum of the mean value and fluctuating term 

To avoid bulky calculations, we restrict ourselves by the following problem statement particles A and B have equal diffusion coefficients. Da = Dq = D, fluctuating particle sources in in (2.2.20), (2.2.21) are 

Equation (2.2.24) means homogeneous generation of particles A and B with the rate p (per unit time and volume), whereas (2.2.25) comes from the statistical independence of sources of a different-kind particles. Physical analog of this model is accumulation of the complementary Frenkel radiation defects in solids. Note that depending on the irradiation type and chemical nature of solids (metal or insulator), dissimilar Frenkel defects could be either spatially correlated in the so-called gemmate pairs (see Chapter 3) or distributed at random. We will focus our attention on the latter case. 

This equation is linear and could be solved exactly, whereas the complementary equation for U(r, t) = Ca t, t) + C ir, t) is non-linear. We restrict ourselves to those problems whose solutions could be expressed through Z r,t). 

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