Theoretical analysis demonstrates ensemble equivalence in the spherical ferromagnet model under non-vanishing fields, indicating that apparent mathematical discrepancies hold no physical consequence.
A discrepancy between the methods of the grand and the canonical ensemble which arose in the study of the spherical model of a ferromagnet even for properties of individual spins is traced to an incorrect order of taking limits. The spin system must be considered for a small but not vanishing field first, and the limit of zero field must only be taken after the limit of an infinite number of particles has been taken. For this case no discrepancy between the two ensembles arises. A calculation proceeding entirely at zero field cannot be considered as logically admissible; discrepancies arising in this case are thus of no physical consequence.
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Yan et al. (1965) studied this question.
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