Abstract
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.
Highlights
The micro-canonical and canonical ensembles are the two main probability distributions with respect to which the equilibrium properties of statistical mechanical models are calculated
For systems with short range interactions, the choice of the statistical ensemble is typically of minor importance and could be considered a finite size effect: differences between, say micro-canonical and canonical expectation values are known to vanish in the thermodynamic limit of large system size and the various statistical ensembles become equivalent
The entropy of the mean field φ6 model is concave as a function of the energy, but is non-concave as a function of the energy and magnetization
Summary
The micro-canonical and canonical ensembles are the two main probability distributions with respect to which the equilibrium properties of statistical mechanical models are calculated. Loop in 3D massless scalar electrodynamics [11], Ising model in the ferromagnetic phase [12], statistical mechanics of nonlinear coherent structures and kinks in the φφ model [13], growth kinetics in the φφ N-Component model [14], stability of Q-balls [15], the liquid states of pion condensate and hot pion matter [16], instantons and conformal holography [17], first order phase transitions in superconducting films [18], field theoretic description of ionic crystallization in the restricted primitive mode [19] This increasing interest in φφ model and its many application in physics, motivated us to study the statistical mechanics of φφ spin models. It is worth to mention that the φφ spin model has been studied in [20,21]
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