As this model aims at being minimal, it assumes that flocking is due to the combination of any kind of self propulsion and of effective alignment. Since the velocity of each particle is a constant, the net momentum of the system is not conserved during collisions. An individual i is described by its position \mathbf{r}_i(t) and the angle defining the direction of its velocity \Theta_i(t) at time t. The discrete time evolution of one particle is set by two equations: • At each time step \Delta t, each agent aligns with its neighbours within a given distance r with an uncertainty due to a noise \eta_i(t): • \Theta_i(t+\Delta t) = \langle \Theta_j \rangle_{|r_i-r_j| • The particle then moves at constant speed v in the new direction: • \mathbf{r}_i(t+\Delta t) = \mathbf{r}_i(t) + v \Delta t \begin{pmatrix} \cos\Theta_i(t) \\ \sin\Theta_i(t) \end{pmatrix} In these equations, \langle \Theta_j \rangle_{|r_i-r_j| denotes the average direction of the velocities of particles (including particle i) within a circle of radius r surrounding particle i. The average normalized velocity acts as the order parameter for this system, and is given by v_a = \frac{1}{Nv} \Biggl|\sum_{i=1}^{N} v_i\,\Biggr| . The whole model is controlled by three parameters: the density of particles, the amplitude of the noise on the alignment and the ratio of the travel distance v \Delta t to the interaction range r . From these two simple iteration rules, various continuous theories have been elaborated such as the Toner-Tu theory which describes the system at the hydrodynamic level. An
Enskog-like kinetic theory, which is valid at arbitrary particle density, has been developed. This theory quantitatively describes the formation of steep density waves, also called invasion waves, near the transition to collective motion. == Phenomenology ==