  In this paper we study the macroscopic behavior of the inertial spin (IS)
model. This model has been recently proposed to describe the collective
dynamics of flocks of birds, and its main feature is the presence of an
auxiliary dynamical variable, a sort of internal spin, which conveys the
interaction among the birds with the effect of better describing the turning of
flocks. After discussing the geometrical and mechanical properties of the IS
model, we show that, in the case of constant interaction among the birds, its
mean-field limit is described by a non-linear Fokker-Planck equation, whose
equilibria are fully characterized. Finally, in the case of non-constant
interactions, we derive the kinetic equation for the mean-field limit of the
model in the absence of thermal noise, and explore its macroscopic behavior by
analyzing the mono-kinetic solutions.
