  We calculate the one-point probability density distribution functions (PDF)
and the power spectra of the thermal and kinetic Sunyaev-Zeldovich (tSZ and
kSZ) effects and the mean Compton Y parameter using the Magneticum Pathfinder
simulations, state-of-the-art cosmological hydrodynamical simulations of a
large cosmological volume of (896 Mpc/h)^3. These simulations follow in detail
the thermal and chemical evolution of the intracluster medium as well as the
evolution of super-massive black holes and their associated feedback processes.
We construct full-sky maps of tSZ and kSZ from the light-cones out to z=0.17,
and one realization of 8.8x8.8 degree wide, deep light-cone out to z=5.2. The
local universe at z<0.027 is simulated by a constrained realisation. The tail
of the one-point PDF of tSZ from the deep light-cone follows a power-law shape
with an index of -3.2. Once convolved with the effective beam of Planck, it
agrees with the PDF measured by Planck. The predicted tSZ power spectrum agrees
with that of the Planck data at all multipoles up to l~1000, once the
calculations are scaled to the Planck 2015 cosmological parameters with
\Omega_m=0.308 and \sigma_8=0.8149. Consistent with the results in the
literature, however, we continue to find the tSZ power spectrum at l=3000 that
is significantly larger than that estimated from the high-resolution
ground-based data. The simulation predicts the mean fluctuating Compton Y value
of <Y>=1.18x10^{-6} for \Omega_m=0.272 and \sigma_8=0.809. Nearly half (~
5x10^{-7}) of the signal comes from halos below a virial mass of
10^{13}M_\odot/h. Scaling this to the Planck 2015 parameters, we find
<Y>=1.57x10^{-6}. The PDF and the power spectrum of kSZ from our simulation
agree broadly with the previous work.
