How localized gaussian curvature modulates the rachet effect

We study how a localized Gaussian curvature influences the ratchet effect in active Brownian particles. In particular, we analyze the distribution of the particles around an asymmetric passive object placed on the surfaces of conventional and hyperbolic cones. Our analytical results show that the Gaussian curvature at the cone apex (reflecting its sharpness) generates a scale-free density modulation along the cone flank, with an exponent determined by the cone geometry. We further demonstrate how Gaussian curvature, in combination with boundary conditions, governs the preferred orientation of the obstacle and the resulting forces and torques acting on it. The analytical predictions are supported by numerical simulations.