  We present a comparative study of the Hubbard and $t-J$ models far away from
half-filling. We show that, at such fillings the $t-J$ Hamiltonian can be seen
as an effective model of the repulsive Hubbard Hamiltonian over the whole range
of correlation strength. Indeed, the $|t/U| \in [0,+\infty [$ range of the
Hubbard model can be mapped onto the finite range $|J/t| \in [1, 0 ]$ of the
$t-J$ model, provided that the effective exchange parameter $J$ is defined
variationally as the local singlet-triplet excitation energy. In this picture
the uncorrelated limit U=0 is associated with the super-symmetric point
$J=-2|t|$ and the infinitely correlated $U=+\infty$ limit with the usual J=0
limit. A numerical comparison between the two models is presented using
different macroscopic and microscopic properties such as energies, charge gaps
and bond orders on a quarter-filled infinite chain. The usage of the $t-J$
Hamiltonian in low-filled systems can therefore be a good alternative to the
Hubbard model in large time-consuming calculations.
