How often do all the molecules crowd left?

Question

Four molecules are split between a Left and a Right chamber. Every second, one of the four molecules is chosen at random and moved to the other chamber. The state is the number of molecules currently in the Left chamber, from 0 to 4. In the long run, what fraction of time are all four molecules in the Left chamber?

Accepted Answer

This is the Ehrenfest model. From a state with k molecules on the Left, the next move depends on which molecule is picked: - Picking one of the N-k Right molecules moves it Left (state goes up by 1): probability \frac{N-k}{N}. - Picking one of the k Left molecules moves it Right (state goes down by 1): probability \frac{k}{N}. Here N = 4. Since the count only steps up or down by one, this is a birth-death chain, and detailed balance applies: the flow up from k equals the flow down from k+1, \pik \cdot \frac{N-k}{N} = \pi{k+1} \cdot \frac{k+1}{N} \implies \frac{\pi{k+1}}{\pik} = \frac{N-k}{k+1…

How often do all the molecules crowd left?

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