Orientational probability 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 (i.e. d angular dimensions and 1 radial dimension). This Fokker–Planck equation admits an exact solution in series form which is, however, unwieldy to use because of poor convergence for short and intermediate times. We present an analytical closed form expression, which gives a good short time approximate orientational probability distribution. The analytical formula is derived using the saddle point method for short times. However, it works well even for intermediate times. We also present a simple analytical form for the long time limit of the orientational probability distribution. Thus, we have obtained simple analytical forms for the orientational probability distribution of an ABP for the entire range of time scales. Our predictions can be tested against future experiments and simulations probing orientational probability distribution of an ABP. (paper)