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We introduce a system of self-propelled agents (active Brownian particles)with velocity alignment in two spatial dimensions and derive a mean-fieldtheory from the microscopic dynamics via a nonlinear Fokker-Planck equation anda moment expansion of the probability distribution function. We analyze thestationary solutions corresponding to macroscopic collective motion with finitecenter of mass velocity (ordered state) and the disordered solution with nocollective motion in the spatially homogeneous system. In particular, wediscuss the impact of two different propulsion functions governing theindividual dynamics. Our results predict a strong impact of the individualdynamics on the mean field onset of collective motion (continuous vsdiscontinuous). In addition to the macroscopic density and velocity field weconsider explicitly the dynamics of an effective temperature of the agentsystem, representing a measure of velocity fluctuations around the meanvelocity. We show that the temperature decreases strongly with increasing levelof collective motion despite constant fluctuations on individual level, whichsuggests that extreme caution should be taken in deducing individual behavior,such as, state-dependent individual fluctuations from mean-field measurements[Yates {\em et al.}, PNAS, 106 (14), 2009].

Mean-field theory of collective motion due to velocity alignment