The isovector equation-of-state in heavy ion collisions and neutron stars

We discuss the investigation of the symmetry energy as a function of density in heavy ion collisions, which is still rather uncertain from a theoretical point of view, but of importance in nuclear structure physics and also in astrophysics. We briefly review transport theory and non-equilibrium and fluctuation effects. We then report on investigations of both the low density EOS in the Fermi energy regime and the high density EOS in intermediate energy collisions. At low density we discuss, in particular, the role and treatment of fluctuations and the dynamical fragment formation. At the high density we make connections with neutron stars.