  We report first-principles simulation of polarization mechanisms in
hydroxyapatite to explain the underlying mechanism behind the reported ion
conductivities and polarization under electrical poling at elevated
temperatures. It is found that ion conduction occurs mainly in the column of
OH$^-$ ions along the $c$-axis through a combination of the flipping of OH$^-$
ions, exchange of proton vacancies between OH$^-$ ions, and the hopping of the
OH$^-$ vacancy. The calculated activation energies are consistent with those
found in conductivity measurements and thermally stimulated depolarization
current measurements.
