```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 Landauâ€™s pseudo free-energy, and the Landauâ€™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.
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