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If there are $i$ balls in box A, a ball currently in B is selected with probability $(k-i)/k$, increasing the count, and a ball in A is selected with probability $i/k$, decreasing it. For $k\ge 1$,with every other transition zero. This is the Ehrenfest urn model.

The stationary distribution isIt is normalized by the binomial theorem, and it obeys the detailed balance:All other pairs have zero flux in both directions. These identities prove stationarity and show that this is a reversible Markov chain in equilibrium. The finite chain is irreducible and has period two; reversibility does not require aperiodicity.