Decoherence (or any other interpretation of QM) will definitely lead to a pretty uniform distribution over this sort of time scale. Just as in the classical case, the underlying dynamics is extremely unstable within the bounds of conservation laws, with the additional problem that the final state for any given perturbation is a distribution instead of a single measurement.
If there is any actual asymmetry in the setup (e.g. one side of the box was 0.001 K warmer than the other, or the volumes of each side were 10^-9 m^3 different), you will probably get a very lopsided distribution for an observation of which side has more molecules regardless of initial perturbation.
If the setup is actually perfectly symmetric though (which seems fitting with the other idealizations in the scenario), the resulting distribution of outcomes will be 50:50, essentially independent of the initial state within the parameters given.
Decoherence (or any other interpretation of QM) will definitely lead to a pretty uniform distribution over this sort of time scale. Just as in the classical case, the underlying dynamics is extremely unstable within the bounds of conservation laws, with the additional problem that the final state for any given perturbation is a distribution instead of a single measurement.
If there is any actual asymmetry in the setup (e.g. one side of the box was 0.001 K warmer than the other, or the volumes of each side were 10^-9 m^3 different), you will probably get a very lopsided distribution for an observation of which side has more molecules regardless of initial perturbation.
If the setup is actually perfectly symmetric though (which seems fitting with the other idealizations in the scenario), the resulting distribution of outcomes will be 50:50, essentially independent of the initial state within the parameters given.