  We investigate the non-linear evolution of the relic cosmic neutrino
background by running large box-size, high resolution N-body simulations. Our
set of simulations explore the properties of neutrinos in a reference
$\Lambda$CDM model with total neutrino masses between 0.05-0.60 eV in cold dark
matter haloes of mass $10^{11}-10^{15}$ $h^{-1}$M$_{\odot}$, over a redshift
range $z=0-2$. We compute the halo mass function and show that it is reasonably
well fitted by the Sheth-Tormen formula. More importantly, we focus on the CDM
and neutrino properties of the density and peculiar velocity fields in the
cosmological volume, inside and in the outskirts of virialized haloes. The
dynamical state of the neutrino particles depends strongly on their momentum:
whereas neutrinos in the low velocity tail behave similarly to CDM particles,
neutrinos in the high velocity tail are not affected by the clustering of the
underlying CDM component. We find that the neutrino (linear) unperturbed
momentum distribution is modified and mass and redshift dependent deviations
from the expected Fermi-Dirac distribution are in place both in the
cosmological volume and inside haloes. The neutrino density profiles around
virialized haloes have been carefully investigated and a simple fitting formula
is provided. The neutrino profile, unlike the cold dark matter one, is found to
be cored with core size and central density that depend on the neutrino mass,
redshift and mass of the halo, for halos of masses larger than $\sim
10^{13.5}h^{-1}$M$_{\odot}$. For lower masses the neutrino profile is best
fitted by a simple power-law relation in the range probed by the simulations.
Our findings are particularly important in view of upcoming large-scale
structure surveys, like Euclid, that are expected to probe the non-linear
regime at the percent level with lensing and clustering observations.
