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Active Brownian particles with velocity-alignment and active fluctuations

We consider a model of active Brownian particles with velocity-alignment in two spatial dimensions with passive and active fluctuations. Hereby, active fluctuations refers to purely non-equilibrium stochastic forces correlated with the heading of an individual active particle. In the simplest case studied here, they are assumed as independent stochastic forces parallel (speed noise) and perpendicular (angular noise) to the velocity of the particle. On the other hand, passive fluctuations are defined by a noise vector independent of the direction of motion of a particle, and may account for example for thermal fluctuations. We derive a macroscopic description of the active Brownian particle gas with velocity-alignment interaction. Hereby, we start from the individual based description in terms of stochastic differential equations (Langevin equations) and derive equations of motion for the coarse grained kinetic variables (density, velocity and temperature) via a moment expansion of the corresponding probability density function. We focus here in particular on the different impact of active and passive fluctuations on the onset of collective motion and show how active fluctuations in the active Brownian dynamics can change the phase-transition behaviour of the system. In particular, we show that active angular fluctuation lead to an earlier breakdown of collective motion and to emergence of a new bistable regime in the mean-field case.