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To answer your first two questions, yes, it is a known construction (physics education journal, 1973).

If the mirrors are made of ordinary matter and the photons have any sort of wavelike properties, then the whole setup (including the mirrors) will approach thermal equilibrium. If you're asking about comparing an output integral to an input integral without such physical considerations, then you get an imbalance. I'm a little rusty on this, but I've been told that thermodynamics as a theory cannot be derived from physical principles without some quantum mechanical input. In particular, conventional notions of entropy seem to require some kind of finiteness for microstates.

Edit: This page has a discussion of a very similar problem (but with spherical instead of flat pieces). The answer claims that if your body has small nonzero size, but idealized non-physical mirrors and photons, you still get a violation of the second law (contra your suggestion). More Edit: The previous sentence arose from a misreading - as Yuri Bakhtin's comments explain, nonzero size yields a homogeneous photon gas. I imagine the homogeneity is what you mean with the reference to Liouville measure.

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To answer your first two questions, yes, it is a known construction (physics education journal, 1973).

If the mirrors are made of ordinary matter and the photons have any sort of wavelike properties, then the whole setup (including the mirrors) will approach thermal equilibrium. If you're asking about comparing an output integral to an input integral without such physical considerations, then you get an imbalance. I'm a little rusty on this, but I've been told that thermodynamics as a theory cannot be derived from physical principles without some quantum mechanical input. In particular, conventional notions of entropy seem to require some kind of finiteness on for microstates.

Edit: This page has a discussion of a very similar problem (but with spherical instead of flat pieces). The answer claims that if your body has small nonzero size, but idealized non-physical mirrors and photons, you still get a violation of the second law (contra your suggestion).