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We derive the thermodynamic entropy of the mean field $\phi^{6}$ spin modelin the framework of the micro-canonical ensemble as a function of the energyand magnetization. Using the theory of large deviations and Rugh'smicro-canonical formalism we obtain the entropy and its derivatives and studythe thermodynamic properties of $\phi^{6}$ spin model. The interesting point wefound is that like $\phi^{4}$ model the entropy is a concave function of theenergy for all values of the magnetization, but is non-concave as a function ofthe magnetization for some values of the energy. This means that the magneticsusceptibility of the model can be negative for some values of the energy andmagnetization in the micro-canonical formalism. This leads to the inequivalenceof the micro-canonical and canonical ensembles. It is also shown that thismean-field model displays a first-order phase transition due to the magneticfield. Finally we compare the results of the mean-field $\phi^{6}$ and$\phi^{4}$ spin models.

Nonequivalent Ensembles for the Mean-Field $\phi^{6}$ Spin Model