  Random walkers characterized by random positions and random velocities lead
to normal diffusion. A random walk was originally proposed by Einstein to model
Brownian motion and to demonstrate the existence of atoms and molecules. Such a
walker represents an inanimate particle driven by environmental fluctuations.
On the other hand, there are many examples of so-called "persistent random
walkers", including self-propelled particles that are able to move with almost
constant speed while randomly changing their direction of motion. Examples
include living entities (ranging from flagellated unicellular organisms to
complex animals such as birds and fish), as well as synthetic materials. Here
we discuss such persistent non-interacting random walkers as a model for active
particles. We also present a model that includes interactions among particles,
leading to a transition to flocking, that is, to a net flux where the majority
of the particles move in the same direction. Moreover, the model exhibits
secondary transitions that lead to clustering and more complex spatially
structured states of flocking. We analyze all these transitions in terms of
bifurcations using a number of mean field strategies (all to all interaction
and advection-reaction equations for the spatially structured states), and
compare these results with direct numerical simulations of ensembles of these
interacting active particles.
