  A recent kinetic approach for Vicsek-like models of active particles is
reviewed. The theory is based on an exact Chapman-Kolmogorov equation in phase
space. It can handle discrete time dynamics and "exotic" multi-particle
interactions. A nonlocal mean-field theory for the one-particle distribution
function is obtained by assuming molecular chaos. The Boltzmann approach of
Bertin et al., Phys. Rev. E 74, 022101 (2006) and J. Phys. A: Math. Theor. 42,
445001 (2009), is critically assessed and compared to the current approach. In
Boltzmann theory, a collision starts when two particles enter each others
action spheres and is finished when their distance exceeds the interaction
radius. The average duration of such a collision, $\tau_0$, is measured for the
Vicsek model with continuous time-evolution. If the noise is chosen to be close
to the flocking threshold, the average time between collisions is found to be
roughly equal to $\tau_0$ at low densities. Thus, the continuous-time
Vicsek-model near the flocking threshold cannot be accurately described by a
Boltzmann equation, even at very small density because collisions take so long
that typically other particles join in, rendering Boltzmann's binary collision
assumption invalid. Hydrodynamic equations for the phase space approach are
derived by means of a Chapman-Enskog expansion. The equations are compared to
the Toner-Tu theory of polar active matter. New terms, absent in the Toner-Tu
theory, are highlighted. Convergence problems of Chapman-Enskog and similar
gradient expansions are discussed.
