Optimal noise for vortex of self-propelled particles around an obstacle

We investigate the influence of thermal noise on the formation and dynamics of vortex of self-propelled particles (SPPs) around a circular obstacle using numerical simulations. In the absence of symmetry-breaking factors, spherical SPPs can spontaneously form a sustained vortex near the obstacle when particle activity is sufficiently high and the obstacle size lies within a suitable range. We show that noise not only induces transition between the vortex state and the random state but also plays a non-monotonic role in the behaviors of the vortex. In particular, there exists an optimal noise level that maximizes both the vortex speed and polarity alignment of the first particle layer, as well as the rate of their linear decay with distance from the obstacle. The underlying mechanism arises from the noise-dependent particle exchange process and the biased selection of incoming particles according to their polarities. In addition, the angular fluctuation of cluster thickness and the interlayer friction of particles also exhibit non-monotonic variations with noise. Pressure analysis demonstrates that the swimming pressure dominates and decreases near the obstacle due to tangential particle orientation, analogous to the reduced pressure in flowing fluids compared to stationary ones. Remarkably, the radial pressure shows a non-monotonic dependence on noise as well, reflecting the complex role of thermal noise in mediating the coupling between particle velocity and orientation. Our results highlight the constructive role of thermal noise in promoting collective vortex motion, thereby enriching the understanding of noise-induced effects in active matter systems.