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When we first started introducing the basic concepts of statistical mechanics we assumed the conservation of energy. Conservation of momentum and angular momentum had to be disregarded because the rectangular box broke both translational invariance and rotation invariance (i. e. the Lagrangian of the system is not translationally and rotationally invariant). If instead of a rectangular box I used a spherical one rotational invariance would be assured, am I right? The infinite potential wall we usually use to modellize the box would then be the same after the rotation and therefore the Lagrangian of the system should be invariant under rotations. This implies conservation of angular momentum. This means that when dealing with averages and all other calculations we should consider an even smaller phase space than the one we usually have when we only consider energy conservation and therefore we would get different results. Is my reasoning correct?

I find it absurd to believe that the shape of the box could influence, say, the pressure of an ideal gas. Is this dependence a big thing on the overall results if we carried out the calculations? If no is there a way to prove that we can disregard the angular momentum conservation law a priori?

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In a perfect spherical well, your gas has a conserved total angular momentum. That means that at equilibrium, your gas will keep that total angular momentum and different values of the total angular momentum would give you different equilibrium ensembles.

If the total angular momentum of your gas is zero, then your gas behaves similarly to a gas in a rectangular box or any other shape.

If your gas has some non-zero total angular momentum, then at equilibrium, it will continue to be rotating in your spherical well. The centripetal acceleration/force needed to keep the gas rotating does change the pressure of your gas; think about a swirling fluid in a cylinder, it will exert more force on its walls.

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