  We examine the anisotropic properties of the exchange stiffness constant,
$\mathcal{A}$, for rare-earth permanent magnet, Nd$_2$Fe$_{14}$B, by connecting
analyses with two different scales of length, i.e., Monte Carlo (MC) method
with an atomistic spin model and Landau-Lifshitz-Gilbert (LLG) equation with a
continuous magnetic model. The atomistic MC simulations are performed on the
spin model of Nd$_2$Fe$_{14}$B constructed from ab-initio calculations, and the
LLG micromagnetics simulations are performed with the parameters obtained by
the MC simulations. We clarify that the amplitude and the thermal property of
$\mathcal{A}$ depend on the orientation in the crystal, which are attributed to
the layered structure of Nd atoms and weak exchange couplings between Nd and Fe
atoms. We also confirm that the anisotropy of $\mathcal{A}$ significantly
affects the threshold field for the magnetization reversal (coercivity) given
by the depinning process.
