Asymmetric nuclear matter is investigated in the low density region below the nuclear saturation density. Microscopic calculations based on the Dirac Brueckner Hartree-Fock (DBHF) approach with realistic nucleon-nucleon potentials are used to adjust a low density functional. This functional is constructed on a density expansion of the relativistic mean field theory which allows a clear interpretation of the role of the mesons to the equation of state. It is shown that a correction term should be added to the functional in order to take into account the effects beyond the mean field. Two functionals with different corrections are obtained. Those functionals converge to predict a reduction of the spinodal zone in asymmetric nuclear matter by about 15-20\% and an isoscalar unstable mode closer to the constant Z/A direction than the functional without correction.