Clustering and ensembles inequivalence in the phi-4 and phi-6 mean-field Hamiltonian models

We investigate a model of globally coupled conservative oscillators. Two different algebraic potentials are considered that display in the canonical ensemble either a second ($\phi^{4}$) or both a second and a first order phase transition separated by tricritical points ($\phi^{6}$). The stability of highly clustered states appearing in the low temperature/energy region is studied both analytically and numerically for the $\phi^{4}$-model. Moreover, long-lived out-of-equilibrium states appear close to the second order phase transition when starting with "water-bag" initial conditions, in analogy with what has been found for the Hamiltonian Mean Field (HMF) model. The microcanonical simulations of the $\phi^{6}$-model show strong hysteretic effects and metastability near the first-order phase transition and a narrow region of negative specific heat.