Active particles in power-law potentials: steady state distributions and shape transitions

We study the stationary states of an active Brownian particle (ABP) and run-and-tumble particle (RTP) in two dimensional power-law potentials, in the limit where translational diffusion is negligible. The potential energy of the particle has the form $U(r)\propto r^{n}$, where $n\geq 2$ and even. In two dimensions, we derive the exact equations for the positional probability distribution $\phi({\bf r})$ of ABP ($n\geq 2)$ and RTP ($n=2$), whose solutions are obtained under the assumption that the particle's orientation angle is Gaussian. Both analytical and numerical results show that, in all cases, $\phi({\bf r})$ has compact support and undergoes a phase transition-like shape change as a function of the trap strength. For ABP, our theory predicts a continuous transition in shape for $n=2$ and a discontinuous transition for $n>2$, both of which agree with the simulation results. Simulations suggest the existence of both types of shape transition in the case of RTP as well. For ABP, in the strongly active regime, the orientational probability distribution is unimodal near the outer boundary but becomes bimodal towards the interior, signifying a transition from predominantly radial orientation to orbiting motion. In RTP, the analogous shape transition in the orientational distribution is almost absent.