  Recently, it has been discovered that systems of Active Brownian particles
(APB) at high density organise their velocities into coherent domains showing
large spatial structures in the velocity field. Such a collective behavior
occurs spontaneously, i.e. is not caused by any specific interparticle force
favoring the alignment of the velocities. This phenomenon was investigated in
the absence of thermal noise and in the overdamped regime where inertial forces
could be neglected. In this work, we demonstrate through numerical simulations
and theoretical analysis that the velocity alignment is a robust property of
ABP and persists even in the presence of inertial forces and thermal
fluctuations. We also show that a single dimensionless parameter, such as the
P\'eclet number customarily employed in the description of self-propelled
particles, is not sufficient to fully characterize such a phenomenon neither in
the regimes of large viscosity nor small mass. Indeed, the size of the velocity
domains, measured through the correlation length of the spatial velocity
correlation, remains constant when the swim velocity increases while decreases
as the rotational diffusion becomes larger. We find that the spatial velocity
correlation depends on the inertia but, contrary to common belief, are
non-symmetrically affected by mass and inverse viscosity variations. We
conclude that in self-propelled systems, at variance with passive systems,
variations of the inertial time (mass over solvent viscosity) and mass act as
independent control parameters. Finally, we highlight the non-thermal nature of
the spatial velocity correlations that are fairly insensitive both to solvent
and active temperatures.
