We all know the classical Joule free expansion experiment for an ideal gas: we trap an ideal gas inside half of a cylinder. Then, we open the door and the gas expands to the whole cylinder. After equilibrium has been reached, its volume doubles, temperature remains the same, pressure halves etc.
Another effect of this free expansion is that the entropy will increase. In this case, $ΔS = Nk\ln2$ where $N$ is the number of particles in the gas and $k$ is Boltzmann's constant.
We know from the second law of thermodynamics that this expansion is irreversible. In other words, if no work is done in this system, the gas will never on its own return to be confined in half the cylinder again. This, however, is a probabilistic statement. We know that entropy will not decrease in an isolated system because the probability of this happening is unbelievably, extremely low.
But... how really unlikely is it? In a more quantitative way?
Is there any way we can estimate the average time needed to wait until the molecules all return to that half even if they stay there only for an infinitesimal amount of time? I suppose the answer, in seconds, would be a number several orders of magnitude larger than the estimated age of the universe. Perhaps it will be larger than 10 to the Graham's number seconds or something like this. But this is just my intuition. Is there a way to solve this and find an actual estimation?