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I'm reading some lecture notes on statistical physics and thermodynamics and I'm stuck at an expression for a partition function which I really don't understand.

The chapter is on mean field theory and the discussion is about hard spheres on a lattice. The interaction of the hard sphere is [itex]\beta U = \infty[/itex] if [itex]r<\sigma[/itex] and [itex]\beta U = 0[/itex] if [itex]r>\sigma [/itex] as usual, where [itex]\sigma[/itex] is the diameter of the spheres.

Now it's said that a single hard sphere is treated exactly and the other spheres are located at their 'ideal' lattice positions. This supposedly leads to [itex]Z_N=\prod_{i=1}^N V_i[/itex] where [itex]V_i[/itex] is the free volume in which the center of mass of particle i can move.

I really don't see this. I'm expecting an [itex]h^3[/itex] or [itex]\Lambda ^3[/itex] to appear somewhere, obtained by integrating over the momenta of such a sphere, but it isn't there. Why is this the partition function?

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# Partition function for hard spheres on a lattice

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