  The propagation of heavy quarks in the quark-gluon plasma (QGP) has been
often treated within the framework of the Langevin equation (LV), i.e. assuming
the momentum transfer is small or the scatterings are sufficiently forward
peaked, small screening mass $m_D$. We address a direct comparison between the
Langevin dynamics and the Boltzmann collisional integral (BM) when a bulk
medium is in equilibrium at fixed temperature. We show that unless the cross
section is quite forward peaked ($m_D\cong T $) or the mass to temperature
ratio is quite large ($M_{HQ}/T \gtrsim 8-10$) there are significant
differences in the evolution of the $p-$spectra and consequently on nuclear
modification factor $R_{AA}(p_T)$. However for charm quark we find that very
similar $R_{AA}(p_T)$ between the LV and BM can be obtained, but with a
modified diffusion coefficient by about $\sim 15-50\%$ depending on the angular
dependence of the cross section which regulates the momentum transfer. Studying
also the momentum spread suffered by a single heavy quarks we see that at
temperatures $T\gtrsim \, 250\,\rm MeV$ the dynamics of the scatterings is far
from being of Brownian type for charm quarks. In the case of bottom quarks we
essentially find no differences in the time evolution of the momentum spectra
between the LV and the BM dynamics independently of the angular dependence of
the cross section, at least in the range of temperature relevant for
ultra-relativistic heavy-ion collisions. Finally, we have shown the possible
impact of this study on $R_{AA}(p_T)$ and $v_2(p_T)$ for a realistic simulation
of relativistic HIC. For larger $m_D$ the elliptic flow can be about $50\%$
larger for the Boltzmann dynamics with respect to the Langevin. This is helpful
for a simultaneous reproduction of $R_{AA}(p_T)$ and $v_2(p_T)$.
