  We study the stationary states of an over-damped active Brownian particle
(ABP) in a harmonic trap in two dimensions, via mathematical calculations and
numerical simulations. In addition to translational diffusion, the ABP
self-propels with a certain velocity, whose magnitude is constant, but its
direction is subject to Brownian rotation. In the limit where translational
diffusion is negligible, the stationary distribution of the particle's position
shows a transition between two different shapes, one with maximum and the other
with minimum density at the centre, as the trap stiffness is increased. We show
that this non-intuitive behaviour is captured by the relevant Fokker-Planck
equation, which, under minimal assumptions, predicts a continuous ``phase
transition" between the two different shapes. As the translational diffusion
coefficient is increased, both these distributions converge into the
equilibrium, Boltzmann form. Our simulations support the analytical
predictions, and also show that the probability distribution of the orientation
angle of the self-propulsion velocity undergoes a transition from unimodal to
bimodal forms in this limit. We also extend our simulations to a three
dimensional trap, and find similar behaviour.

Statistical Mechanics;Biological Physics