The Chapmen-Kolmogorov master equation

The approach developed in Section 2.2.1 is based on the independent treatment of the deterministic motion of a system and density fluctuations therein. The reaction description in terms of random process seems to be more consistent and logical. Equations (2.1.2) which were used above for a 

Since the formal chemical kinetics operates with large numbers of particles participating in reaction, they could be considered as continuous variables. However, taking into account the atomistic nature of defects, consider hereafter these numbers Ni as random integer variables. The chemical reaction can be treated now as the birth-death process with individual reaction events accompanied by creation and disappearance of several particles, in a line with the actual reaction scheme [16, 21, 27, 64, 65]. Describing the state of a system by a vector N = iVi. Ng, we can use the Chapmen-Kolmogorov master equation [27] for the distribution function P N, t) 

Entering this equation transition probability W(N N ) of the Markov process depends on the states N, N only. For mono- and bimolecular reactions these transition probabilities are not zero if vectors N and N differ by several projections only. To specify W N N ) in equation (2.2.37), one has to start from the equations (2.2.36) for the formal kinetics accompanied with some probabilistic arguments. 

To illustrate this approach, let us consider the A - - A B reaction. In this case the equation analogous to (2.1.10) reads (provided N = N, = N) 

A number of collisions in the iV-particle system is proportional to N N - 1) rather than entering equation (2.2.38). The only transition has non-zero probability - which takes away particles from a system 

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