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This is not a complete answer, since I can't seem to work out the $r_1=r_2$ case, but I think this should get you most of the way to an answer. To start, its useful to give another definition for the grand potential in terms of the grand partition function and write the derivatives implicitly using this definition: $$\Omega=\frac{1}{\beta}\ln(\Xi)\tag{1}\...

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