  Barium titanate (BaTiO$_3$) is a prototypical ferroelectric perovskite that
undergoes the rhombohedral-orthorhombic-tetragonal-cubic phase transitions as
the temperature increases. In this work, we develop a classical interatomic
potential for BaTiO$_3$ within the framework of the bond-valence theory. The
force field is parameterized from first-principles results, enabling accurate
large-scale molecular dynamics (MD) simulations at finite temperatures. Our
model potential for BaTiO$_3$ reproduces the temperature-driven phase
transitions in isobaric-isothermal ensemble (NPT) MD simulations. This
potential allows the analysis of BaTiO$_3$ structures with atomic resolution.
By analyzing the local displacements of Ti atoms, we demonstrate that the phase
transitions of BaTiO$_3$ exhibit a mix of order-disorder and displacive
characters. Besides, from detailed observation of structural dynamics during
phase transition, we discover that the global phase transition is associated
with changes in the equilibrium value and fluctuations of each polarization
component, including the ones already averaging to zero, Contrary to the
conventional understanding that temperature increase generally causes
bond-softening transition, the x polarization component exhibits a
bond-hardening character during the orthorhombic to tetragonal transition.
These results provide further insights about the temperature-driven phase
transitions in BaTiO$_3$.
