Particles independently change container at a rate λ.
If X(t) = i is defined to be the number of particles in one container at time t, then it is a birth–death process with transition rates and equilibrium distribution
Mark Kac proved in 1947 that if the initial system state is not equilibrium, then the entropy, given by is monotonically increasing (H-theorem).
It is expected that over time the number of particles in this container will approach
and stabilize near that state (containers will have approximately the same number of particles).
However from mathematical point of view, going back to the initial state is possible (even almost sure).
From mean recurrence theorem follows that even the expected time to going back to the initial state is finite, and it is
Using Stirling's approximation one finds that if we start at equilibrium (equal number of particles in the containers), the expected time to return to equilibrium is asymptotically equal to
If we assume that particles change containers at rate one in a second, in the particular case of
This supposes that while theoretically sure, recurrence to the initial highly disproportionate state is unlikely to be observed.