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

We derive the thermodynamic entropy of the mean field $ϕ^{6}$ spin model in the framework of the micro-canonical ensemble as a function of the energy and magnetization. Using the theory of large deviations and Rugh's micro-canonical formalism we obtain the entropy and its derivatives and study the thermodynamic properties of $ϕ^{6}$ spin model. The interesting point we found is that like $ϕ^{4}$ model the entropy is a concave function of the energy for all values of the magnetization, but is non-concave as a function of the magnetization for some values of the energy. This means that the magnetic susceptibility of the model can be negative for some values of the energy and magnetization in the micro-canonical formalism. This leads to the inequivalence of the micro-canonical and canonical ensembles. It is also shown that this mean-field model displays a first-order phase transition due to the magnetic field. Finally we compare the results of the mean-field $ϕ^{6}$ and $ϕ^{4}$ spin models.