Kinetic theory of flocking: Derivation of hydrodynamic equations

It is shown how to explicitly coarse-grain the microscopic dynamics of the rule-based Vicsek model for self-propelled agents. The hydrodynamic equations are derived by means of an Enskog-type kinetic theory. Expressions for all transport coefficients are given. The transition from a disordered to a flocking state, which at large particle speeds appears to be a fluctuation-induced first-order phase transition, is studied numerically and analytically.

Figure 1

(a) The critical noise ${\eta }_C$ as a function of the average number of collision partners, $M={\rho }_0\pi R^2$, and the prediction of Eq. (35) for large $v_0$ from Ref. [7], (dashed line) compared with results from Refs. [4, 8, 11]. (b) Real part of the growth rate ${\omega }_R$ of a small longitudinal perturbation of the ordered state versus dimensionless wave number $k_{\Vert }$ at $M=5$, very close to the threshold, $({\eta }_C-\eta )/{\eta }_C=0.000\hspace{0.5em}57$. The inset shows a lower and an upper bound for the crossover length $L^{*}$ (in units of the mfp), beyond which the phase transition is expected to become discontinuous: