  The out-of-plane fluctuations of carbon atoms in a graphene sheet have been
studied by means of classical molecular dynamic simulations with an empirical
force-field as a function of temperature. The Fourier analysis of the
out-of-plane fluctuations often applied to characterize the acoustic bending
mode of graphene is extended to the optical branch, whose polarization vector
is perpendicular to the graphene layer. This observable is inaccessible in a
continuous elastic model of graphene but it is readily obtained by the
atomistic treatment. Our results suggest that the long-wavelength limit of the
acoustic out-of-plane fluctuations of a free layer without stress is
qualitatively similar to that predicted by a harmonic model under a tensile
stress. This conclusion is a consequence of the anharmonicity of both in-plane
and out-of-plane vibrational modes of the lattice. The most striking anharmonic
effect is the presence of a linear term, $\omega_{A}=v_{A}k$, in the dispersion
relation of the acoustic bending band of graphene at long wavelengths
($k\rightarrow0$). This term implies a strong reduction of the amplitude of
out-of-plane oscillations in comparison to a flexural mode with a
$k^{2}$-dependence in the long-wavelength limit. Our simulations show an
increase of the sound velocity associated to the bending mode, as well as an
increase of its bending constant, $\kappa,$ as the temperature increases.
Moreover, the frequency of the optical bending mode, $\omega_{O}(\Gamma)$, also
increases with the temperature. Our results are in agreement with recent
analytical studies of the bending modes of graphene using either perturbation
theory or an adiabatic approximation in the framework of continuous layer
models.
