Critical escape dynamics of an active particle moving on curved surfaces

Abstract

On the macroscopic scale, active (self-propelled) objects moving through complex environments are often subject to effects arising from both the curvature of the surrounding surface and external influences such as gravity. We study the Kramers escape problem for active Brownian particles in two settings: motion in an external potential driven solely by gradient forces, and motion on a curved manifold subject to an additional constant gravitational force. In the absence of translational noise, a sharp dynamical transition occurs when the self-propulsion velocity $v_0$ reaches a critical velocity $v_c$ required for an escape. Above this threshold, the escape rate $k$ follows a universal scaling form, $k \propto \exp\left[-\mathrm{const}(v_0-v_c)^{-γ}\right]$, with an exponent $γ=3/2$ for the escape in a potential and $γ=1/2$ for the escape on a curved manifold in the presence of gravity. These distinct exponents reveal how the interplay of activity, surface curvature, and gravity determines the critical escape dynamics. Our theoretical predictions are verified by numerical simulations.