How often are the six molecules split evenly?

Question

Six molecules are split between a Left and a Right chamber. Every second, one of the six is chosen at random and moved to the other chamber. The state is the number of molecules in the Left chamber, from 0 to 6. In the long run, what fraction of time is the split exactly three molecules in each chamber?

Accepted Answer

This is the Ehrenfest model with N = 6. From a state with k on the Left, the count goes up with probability \frac{N-k}{N} (a Right molecule is picked) and down with probability \frac{k}{N} (a Left molecule is picked). Detailed balance for this birth-death chain gives \frac{\pi{k+1}}{\pik} = \frac{N-k}{k+1}, which are the ratios of the binomial coefficients \binom{6}{k}. So the long-run shares are proportional to \binom{6}{0}, \dots, \binom{6}{6} = 1, 6, 15, 20, 15, 6, 1, which add to 2^6 = 64. The stationary distribution is Binomial(6, 	frac12): \pik = \binom{6}{k}/64. The even split is k = …

How often are the six molecules split evenly?

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