Gas molecules shuffling between two chambers

Question

Two connected chambers together hold N gas molecules. Each second, one molecule is chosen uniformly at random from all N and moved to the other chamber. Let Xt be the number of molecules in the left chamber. (a) In the long run, what is the distribution of Xt? (b) If the system starts with all N molecules on the left, what is the expected number of seconds until it returns to that all-on-the-left state?

Accepted Answer

This is the classic diffusion chain. From state k (left chamber has k), the next move picks one of the N molecules: with probability k/N a left molecule leaves (go to k-1), with probability (N-k)/N a right molecule arrives (go to k+1). (a) Stationary distribution. Guess that in equilibrium each molecule independently sits in the left chamber with probability 	frac12, i.e. X \sim 	ext{Binomial}(N, 	frac12), so \pik = \binom{N}{k}\frac{1}{2^N}. Verify with detailed balance \pik P(k 	o k+1) = \pi{k+1} P(k+1 	o k): \binom{N}{k}\frac{1}{2^N}\cdot\frac{N-k}{N} \stackrel{?}{=} \binom{N}{k+1}\fr…

Gas molecules shuffling between two chambers

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