Bernoulli-Laplace diffusion model ($P(r,r+1)=(n-r{)}^2/n^2,P(r,r-1)=r^2/n^2$)

In the Bernoulli-Laplace diffusion model, two urns each contain $n\ge 1$ balls, with $n$ red and $n$ blue balls in total. Independently choose one ball uniformly from each urn and exchange them. The red count $R_k$ in the first urn is a Markov chain on $\{0,\dots ,n\}$, withThe upward event exchanges a blue ball from the first urn with a red ball from the second; the downward event reverses the colors. HenceStarting from $R_0=n$, the mean is $n/2+(n/2)(1-2/n{)}^k$, with the initial value understood separately if the geometric factor is zero. For $n\ge 2$, this mean tends to $n/2$. For $n=1$, however, the only two configurations alternate and the expected imbalance does not decay.

The Stationary distribution is the hypergeometric distributionIt counts uniformly allocated red/blue configurations. The ratio $\pi (r+1)/\pi (r)=[(n-r)/(r+1){]}^2$ verifies Detailed balance with the displayed transitions. The chain is irreducible; for $n\ge 2$ an interior holding Probability also makes it aperiodic, while the $n=1$ chain has period two. This model separates convergence of an average imbalance from the full probability law and its periodicity.