Consider that at the beginning all the particles are in one of the containers. It is expected that over time the number of particles in this container will approach N/2 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 2^N. 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 \textstyle\sqrt{\pi N/2}. If we assume that particles change containers at rate one in a second, in the particular case of N=100 particles, starting at equilibrium the return to equilibrium is expected to occur in 13 seconds, while starting at configuration 100 in one of the containers, 0 at the other, the return to that state is expected to take 4\cdot 10^{22} years. This supposes that while theoretically sure, recurrence to the initial highly disproportionate state is unlikely to be observed. ==Bibliography==