# Ask for help: the collide probability 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 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 method to get to know how many molecules would be transformed to A in every peroid until all molecule B are transformed to molecule A?

Thanks, Tang Laoya

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Your probabilistic model isn't clear to me. In the context of the container, "distributed uniformly" appears to refer only to space and not to phase space. But without any assumptions on the states of motion of the molecules, how do you want to draw any time-dependent conclusions? –  joriki Mar 26 '13 at 15:18
Hi Joriki, thanks for your kindly comments. Suppose initially the phase spaces of A and B are uniform, i.e., the mixture of A and B is uniform. During the entire process, the motions of A and B are the same, which could lead to the phase spaces of A and B are not uniform. –  Tang Laoya Mar 26 '13 at 15:58
What does "the phase spaces of $A$ and $B$ are uniform" mean? First, $A$ and $B$ are kinds of molecules; do you mean that all molecules of type $A$ and all molecules of type $B$ are independently uniformly distributed in phase space? Even if so, there's no uniform distribution over the entire phase space; just as you need a spatial restriction to the container to make sense of a uniform distribution of position, you'd need some restriction on the momenta to make sense of a uniform distribution of momenta. (Not to mention that uniform distributions of momenta would be rather unphysical.) –  joriki Mar 26 '13 at 16:10
Hi Joriki, thanks for your kindly reply. What I mean is that we can suppose A and B have the same volume and the same quality, so that they have the same probability to collide each other. Initially 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. However, after several peroids of colliding, they are not uniform anymore. –  Tang Laoya Mar 26 '13 at 16:20
Sorry, it's not becoming any clearer to me. I'm not even sure whether your question is physical or mathematical -- are you asking how the state distribution will evolve over time, physically, or do you know the distribution and want to calculate collision probabilities from it? If the latter, you'd need to specify the distribution. Or perhaps you're not interested in the distribution in phase space, only in collision probabilities -- but then it seems even less clear to me what distribution you're considering. Could you perhaps try to formalize your setting more? –  joriki Mar 26 '13 at 16:32