Phase Diagram and Tricritical Point for Quasi-stationary States
in the Hamiltonian Mean-field Model

It is a common view that Hamiltonian systems with long-range interaction are trapped in non-equilibrium quasi-stationary states before reaching thermal equilibrium states. The life time of those quasi-stationary states is shown to increase according to the system size, which is observed by computer experiments as well. A way to analyze a Hamil- tonian system with long-range interaction is to use the Vlasov equation or collisionless Boltzmann equation. On the basis of the conservation laws for the Vlasov equation, non-equilibrium statistical mechanics has been set up, which provides Fermi-Dirac type distributions for quasi-stationary states. The distribution contains undetermined parameters, which are to be determined by solving simultaneous equations coming from the conservation laws. This statistical theory can be successfully applied to the Hamiltonian mean-field model, which is one of the simplest models with long-range interactions. In this model, the quasi-stationary states exhibit first- and second-order phase transitions and a tricritical point, while the equilibrium states have second-order ones only. The tricritical point has been observed in a previous article, but is not accurate enough, since distributions and values of the order parameter have been computed pointwise on a parameter space by means of solving simultaneous equations numerically. In order to look into the tricritical point, the simultaneous equations are expanded with respect to an order parameter and reduced to one equation for the order parameter. The reduced equation can be looked on as the derivative of a Landau’s pseudo free-energy, and the Landau’s phenomenological theory serves to explore the tricritical point, phase transition lines and the coexistence region. Since the present equation for the tricritical point is exact, the position of the tricritical point is identified exactly in the phase diagram.