  The large values and the constituent-quark-number (NCQ) scaling of the
elliptic flow of low-$p_T$ $D$ mesons imply that charm quarks, initially
produced through hard processes, might be partially thermalized through the
strong interactions with the quark-gluon plasma (QGP) in high-energy heavy-ion
collisions. To quantify the degree of thermalization of low-$p_T$ charm quarks,
we compare the $D^0$ meson spectra and elliptic flow from a hydrodynamic model
to the experimental data as well as transport model simulations. We use an
effective charm chemical potential at the freeze-out temperature to account for
the initial charm quark production from hard processes and assume that they are
thermalized in local comoving frame of the medium before freeze-out. $D^0$
mesons are sampled statistically from the freeze-out hyper-surface of the
expanding QGP as described by the event-by-event (3+1)D viscous hydrodynamic
model CLVisc. Both hydrodynamic and transport model can describe the elliptic
flow of $D^0$ mesons at $p_T<3$ GeV/$c$ as measured in Au+Au collisions at
$\sqrt{s_{NN}}=200$ GeV. Though the experimental data on $D^0$ spectra are
consistent with the hydrodynamic result at small $p_T\sim 1$ GeV/$c$, they
deviate from the hydrodynamic model at high transverse momentum $p_T>2$
GeV/$c$. The diffusion and parton energy loss mechanisms in the transport model
can describe the measured spectra reasonably well within the theoretical
uncertainty. Our comparative study indicates that charm quarks only approach to
local thermal equilibrium at small $p_T$ even though they acquire sizable
elliptic flow that is comparable to light-quark hadrons at both small and
intermediate $p_T$.
