Ehrenfest urn model
= Ehrenfest urn model
{c}
{title2=$P_{i,i+1}=(k-i)/k,\quad P_{i,i-1}=i/k$}
The Ehrenfest urn model moves one uniformly chosen ball between two boxes at each step. For $k\geq1$ balls, the count in one box is a <Markov chain> on $\{0,\ldots,k\}$. Its <stationary distribution> is $2^{-k}\binom ki$, and <detailed balance> makes it a <reversible Markov chain>. The chain has period two, but its <mean recurrence time> to the all-in-one-box state is still $2^k$.