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    Occupation number if two particles can occupy the same state

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    The grand canonical partition function is the product over all the single particle states of:

    sum over n_r of exp[-beta (e_r - mu)n_r] (1)

    The summation variable n_r is the occupation number. For Fermion this ranges from zero to 1, for Bosons from zero to infinity. In this problem we consider hypothetical Goofions for which n_r ranges from zero to 2. The above term then becomes:

    sum over n_r of exp[-beta (e_r - mu)n_r] = 1 + exp[-beta(e_r - mu)] + exp[-2 beta(e_r - mu)]

    The probability that some given single particle state with energy e is occupied by n Goofions is the ratio:

    exp[-n beta(e - mu)]/{1 + ...

    Solution Summary

    If two particles can occupy the same state then neither Bose-Einsten, nor Fermi-Dirac statistic applies. We explain in detail how one can derive the occupation number in this case.