 Ehrenfest model

A model of random exchange between two containers, associated with Paul Ehrenfest and Tatyana Afanasyeva-Ehrenfest. In the discrete-time Ehrenfest urn model, choose one of $N$ balls uniformly at each step and change its container. The count in the first container decreases with probability $r/N$ and increases with probability $(N-r)/N$. A continuous-time version lets each particle change container independently at rate $\lambda$, giving birth-death process transition rates $r\lambda$ and $(N-r)\lambda$. Its sequence of visited states is the discrete urn Markov chain. Stopping when one container becomes empty produces an absorbing variant, rather than changing the transition rule of the unstopped model.