Correlations and local order parameter in the paraelectric phase of barium titanate

General features of the order parameter distribution in barium titanate in its paraelectric phase and in its ferroelectric phases (tetragonal and orthorhombic) are presented. The density of probability of the polarization Pi(Lx),(Ly),(Lz) ((u) over right arrow), defined by an average of the local order parameters over regions of various sizes and shapes (L-x x L-y x L-z), is examined by molecular dynamics simulations using a first-principles derived effective Hamiltonian. The free energies F-Lx,(Ly),(Lz) ((u) over bar) associated with these probabilities are computed by thermodynamic integration. The evolution of these quantities are explained through the computation of pair correlations, which are found, as stated in several previous works, very anisotropic, 'needle-like', with longitudinal correlations (->->) having much longer range than transverse ones (up arrow up arrow). The correlations explain why the density of probability of the order parameter evolves from a multiple-peaked distribution with maxima along [111] (in the single cell), along [100] for small needle-like regions, towards a single-peaked distribution for larger regions. A useful expression in which the shape-dependence of the free energy is manifest is provided.