  We consider a reduced dynamics for the first four fluid moments of the
onedimensional Vlasov-Poisson equation, namely, the fluid density, fluid
velocity, pressure and heat flux. This dynamics depends on an equation of state
to close the system. This equation of state (closure) connects the fifth order
moment-related to the kurtosis in velocity of the Vlasov distribution-with the
first four moments. By solving the Jacobi identity, we derive an equation of
state which ensures that the resulting reduced fluid model is Hamiltonian. We
show that this Hamiltonian closure allows symmetric homogeneous equilibria of
the reduced fluid model to be stable.
