Ask for help: the Reaction-Diffusion related problem

In container there are two kinds of molecules A and B which are distributed uniformly. Initially the quantities of A and B are $N_A$ and $N_B$ respectively A and B are distributed uniform, i.e., in any small space (of cause the space should be large enought that there are thousands of molecules in the space), the ratio of number A and B is equal to the ratio of total number of A and B.

Assume the molecules are fixed all the time (i.e., we don't consider diffusion) and in every peroid, there is at most one molecule $B_j$ around molecule $A_i (i=1,2,...,N_A)$ be transformed to molecule A: if the molecule $B_j$ is on the boundary/neighbor of molecule $A_i$, the molecule $B_j$ would be transformed to A, otherwise (i.e., all molecules neighboring $A_i$ are molecule A) no molecule around $A_i$ be transformed. In addition, if there are several $A_k$ and $B_j$ on the neighbor of molecule $A_i$, the closest $B_j$ would be transformed to A.

Now the problem is: Is there any analytical expression to show how many molecules (approximately) would be transformed to A in every peroid until all molecule B are transformed to molecule A?

Thanks, Tang Laoya

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It's difficult to understand your system. Are the molecules on a grid? I guess not, because you talk about closest neighbours - so how are they distributed, and what determines whether two molecules are neighbours? – Nathaniel Mar 28 '13 at 2:52
Hi Nathaniel, thanks for your kindly reply. The molecules are random distributed in the space, but on the large-size view the distribution is uniform. I say B_j is the neighbor of A_i if B_j is in the average distance of A and in the radial direction of B_j no other molecule is closer than B_j. – Tang Laoya Mar 28 '13 at 4:53