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However, this represents only a subset of the states that are consistent with our macrovariables (which was, one particle in each box). The position distributions should actually be

Because the particles are macroscopically identical (our macrovariables do not distinguish between them), there is no reason to believe that our system exists in only a subset of the available microstates as suggested by fig. 2. Of course, the system is definitely in one of the square regions in the position distribution, it cannot evolve between the two. If we wanted to insist that the system is in one of the squares, we must have a way to prescribe this configuration using our macrovariables, but this is equivalent to requiring the particles be macroscopically distinguishable.

Therefore, both $\Omega_\text{sys}$ and $\Omega_\text{sys}'$ lead to the same entropy, and there is no entropy gain by removing the divider.

Essentially what we've shown is that the multiplicity of an monatomic ideal gas, calculated with or without correct Boltzmann counting (hereafter CBC), leads to an entropy which when maximized, predicts the same equilibrium states. One might justify the multiplicity calculated without CBC by Liouville's theorem and the ergodic hypothesis (to argue that macrostate probability is proportional to phase space volume). CBC multiplicity can then be rationalized as being empirically equivalent to the non-CBC multiplicity. Similar arguments could probably be made for legitimizing the use of CBC in other circumstances where CBC might not be "easily justified" from the foundational physical assumptions.

Jaynes, E. T. (1996). The Gibbs paradox.

However, this represents only a subset of the states that are consistent with our macrovariables (which was, one particle in each box). The position distribution should actually be

Because the particles are macroscopically identical (our macrovariables do not distinguish between them), there is no reason to believe that our system exists in only a subset of the available microstates as suggested by fig. 2. Of course, the system is definitely in one of the square regions in the position distribution, it cannot evolve between the two. If we wanted to insist that the system is in a specific square, we must have a way to prescribe this configuration using our macrovariables, but this is equivalent to requiring the particles be macroscopically distinguishable.

Therefore, both $\Omega_\text{sys}$ and $\Omega_\text{sys}'$ lead to the same entropy, and there is no entropy gained by removing the divider.

Essentially what we've shown is that the multiplicity of a monatomic ideal gas, calculated with or without correct Boltzmann counting (hereafter CBC), leads to an entropy which when maximized, predicts the same equilibrium states. One might justify the multiplicity calculated without CBC by Liouville's theorem and the ergodic hypothesis (to argue that macrostate probability is proportional to phase space volume). CBC multiplicity can then be rationalized as being empirically equivalent to the non-CBC multiplicity. Similar arguments could probably be made for legitimizing the use of CBC in other circumstances where CBC might not be "easily justified" from the foundational physical assumptions.

Jaynes, E. T. (1996). "The Gibbs paradox".

As a side note, the goal of statistical mechanics (seems to be) to use statistical arguments to determine thermodynamic potentials of new systems, such that they interact "consistently" with the existing thermodynamic potentials of studied systems.

As a side note, the goal of statistical mechanics (seems to be) to use probabilistic arguments to determine thermodynamic potentials of new systems, such that they interact "consistently" with the existing thermodynamic potentials of studied systems.

As another side note, going back to the Gibbs paradox example, when we calculated the multiplicity, without correct Boltzmann counting, after the divider is removed, we could have done it in two ways, either $$ \Omega_\text{sys}' = \Omega(V_1+V_2, N_1 + N_2) \,. $$ as was done initially, or $$ \Omega_\text{sys}' = \Omega(V_1+V_2, N_1)\Omega(V_1+V_2, N_2) \,. $$ The two can be shown to be equivalent. The first interpretation says, the system after the divider is removed, is the same as a box of volume $V_1+V_2$ with $N_1+N_2$ particles. The second interpretation says, the system after the divider is removed, is the same as expanding the volume of the left and right boxes, to fill the size of both boxes, and the left and right boxes kind of "coexist". This is a nice symmetry that does not exist when correct Boltzmann counting is used.

As another side note, going back to the Gibbs paradox example, when we calculated the multiplicity, without correct Boltzmann counting, after the divider is removed, we could have done it in two ways, either $$ \Omega_\text{sys}' = \Omega(V_1+V_2, N_1 + N_2) \,. $$ as was done initially, or $$ \Omega_\text{sys}' = \Omega(V_1+V_2, N_1)\Omega(V_1+V_2, N_2) \,. $$ The two can be shown to be equivalent (again, not exactly equivalent since the second calculation does not consider the microstates where a small group of particles, a single particle say, contains all the energy of the system, and the others are very slow moving). The first interpretation says, the system after the divider is removed, is the same as a box of volume $V_1+V_2$ with $N_1+N_2$ particles. The second interpretation says, the system after the divider is removed, is the same as expanding the volume of the left and right boxes, to fill the size of both boxes, and the left and right boxes kind of "coexist".