  We investigate the impact of thermal velocities in N-body simulations of
structure formation in warm dark matter models. Adopting the commonly used
approach of adding thermal velocities, randomly selected from a Fermi-Dirac
distribution, to the gravitationally-induced velocities of the simulation
particles, we compare the matter and velocity power spectra measured from CDM
and WDM simulations, in the latter case with and without thermal velocities.
This prescription for adding thermal velocities introduces numerical noise into
the initial conditions, which influences structure formation. At early times,
the noise affects dramatically the power spectra measured from simulations with
thermal velocities, with deviations of the order of $\sim \mathcal{O}(10)$ (in
the matter power spectra) and of the order of $\sim \mathcal{O}(10^2)$ (in the
velocity power spectra) compared to those extracted from simulations without
thermal velocities. At late times, these effects are less pronounced with
deviations of less than a few percent. Increasing the resolution of the N-body
simulation shifts these discrepancies to higher wavenumbers. We also find that
spurious haloes start to appear in simulations which include thermal velocities
at a mass that is $\sim$3 times larger than in simulations without thermal
velocities.
