Orientational Distribution of an Active Brownian Particle: an analytical study

We use the Fokker Planck equation as a starting point for studying the orientational probability distribution of an Active Brownian Particle (ABP) in $(d+1)$ dimensions. This Fokker Planck equation admits an exact solution in series form which is, however, unwieldly to use because of poor convergence for short and intermediate times. A truncated version of this series is a reasonable approximation for long times. In this paper, we present an analytical closed form expression, which gives a good approximate orientational probability distribution, which is derived using saddle point methods for short times. However, it works well even for intermediate times. Thus, we have simple analytical forms for the ${\it entire}$ range of time scales for the orientational probability distribution of an ABP. Our predictions can be tested against future experiments and simulations probing orientational probability distribution of an ABP.