  We introduce and analyse a continuum model for an interacting particle system
of Vicsek type. The model is given by a non-linear kinetic partial differential
equation (PDE) describing the time-evolution of the density $f_t$, in the
single particle phase-space, of a collection of interacting particles confined
to move on the one-dimensional torus. The corresponding stochastic differential
equation for the position and velocity of the particles is a conditional
McKean-Vlasov type of evolution (conditional in the sense that the process
depends on its own law through its own conditional expectation). In this paper,
we study existence and uniqueness of the solution of the PDE in consideration.
Challenges arise from the fact that the PDE is neither elliptic (the linear
part is only {\em hypoelliptic}) nor in gradient form. Moreover, for some
specific choices of the interaction function and for the simplified case in
which the density profile does not depend on the spatial variable, we show that
the model exhibits multiple stationary states (corresponding to the particles
forming a coordinated clockwise/anticlockwise rotational motion) and we study
convergence to such states as well. Finally, we prove mean-field convergence of
an appropriate $N$-particles system to the solution of our PDE: more precisely,
we show that the empirical measures of such a particle system converge weakly,
as $N \rightarrow \infty$, to the solution of the PDE.
