To preface my question, I know it is related to the Gibbs paradox, but I've read the wikipedia page on it and am still confused about how to resolve the question in the particular form I state below. Suppose a completely isolated ideal gas consisting of identical particles is confined to a volume V. Call this state 1. Now insert a wall dividing the volume in half, so that you now have two completely isolated system, one with N1 particles one with N2 particles. Call this state 2. Regardless of how the inserted wall partitions the particles, the multiplicity of the system in state 2 will always be less than the multiplicity of the system in state 1 because state 1 includes every configuration compatible with state 2 (i.e. configurations with N1 particles on one side and N2 on the other) as well as additional configurations (configurations with N1 + 1 on one side and N2 -1 on the other, etc). Hence, the entropy of the system always decreases upon insertion of a wall. I understand that the decrease may be small compared to scale of the system's entropy, but small or not, does the system's entropy decrease in this example?