  In this note we establish hypocoercivity and exponential relaxation to the
Maxwellian for a class of kinetic Fokker-Planck-Alignment equations arising in
the studies of collective behavior. Unlike previously known results in this
direction that focus on convergence near Maxwellian, our result is global for
hydrodynamically dense flocks, which has several consequences. In particular,
if communication is long-range, the convergence is unconditional. If
communication is local then all nearly aligned flocks quantified by smallness
of the Fisher information relax to the Maxwellian. In the latter case the class
of initial data is stable under the vanishing noise limit, i.e.\ it reduces to
a non-trivial and natural class of traveling wave solutions to the noiseless
Vlasov-Alignment equation.
  The main novelty in our approach is the adaptation of a mollified Favre
filtration of the macroscopic momentum into the communication protocol. Such
filtration has been used previously in large eddy simulations of compressible
turbulence and its new variant appeared in the proof of the Onsager conjecture
for inhomogeneous Navier-Stokes system. A rigorous treatment of well-posedness
for smooth solutions is provided. Lastly, we prove that in the limit of strong
noise and local alignment solutions to the Fokker-Planck-Alignment equation
Maxwellialize to solutions of the macroscopic hydrodynamic system with the
isothermal pressure.
