Ehrenfest model

Example

Let $N \ge 1$ balls be distributed between two urns. At each step one of the $N$ balls is chosen uniformly at random and moved to the other urn. Let $X_n$ be the number of balls in the first urn after $n$ steps. Then $(X_n)$ is a Markov chain on $\set{0, 1, \dots, N}$ with transition probabilities $$P(k, k - 1) = \frac{k}{N}, \qquad P(k, k + 1) = \frac{N - k}{N}.$$ This is the Ehrenfest model, introduced by Paul and Tatiana Ehrenfest in 1907.

Check that the binomial distribution $\pi(k) = \binom{N}{k} 2^{-N}$ is a stationary distribution. It satisfies detailed balance: $\pi(k) P(k, k + 1) = \binom{N}{k} \frac{N - k}{N} 2^{-N} = \binom{N}{k + 1} \frac{k + 1}{N} 2^{-N} = \pi(k + 1) P(k + 1, k)$. Summing detailed balance over $k$ gives $\sum_k \pi(k) P(k, j) = \pi(j)$ for every $j$.