  Using Molecular Dynamics simulations based on the effective hamiltonian
developed by Zhong, Vanderbilt and Rabe [Phys. Rev. Lett. {\bf 73}, 1861
(1994)] (and fitted on first-principles calculations only), the technique of
the thermodynamic integration is applied to barium titanate. It allows to
compute the difference of free energy between macroscopic states with different
polarizations, from the thermal averages of the forces acting on the local
modes. This is achieved by performing molecular dynamics under the constraint
of fixed polarization. The Landau free energy is thus interpreted as a
potential of mean force. The thermodynamic integration directly gives access
(numerically) to the Landau free energy of barium titanate as a function of
$\vec P$, without any assumption on its analytical form. This technique, mainly
used in computational chemistry, allows to make a direct connection between
phenomenological theories and atomic-scale simulations. It is applied and
validated to the case of BaTiO$_3$ under fixed volume. The results are compared
to available phenomenological potentials of the litterature.
