Chiral active Brownian displacement ($M(t)=4Dt+2v^2\mathrm{Re}[t/z-(1-e^{-zt})/z^2],z=D_r-i\omega$)

A planar self-propelled particle with translational diffusion coefficient $D$, propulsion speed $v$, rotational diffusion $D_r$ and constant angular drift $\omega$ has orientation correlation $e^{-D_r|t|}\cos (\omega t)$. Integrating this correlation twice gives the displayed mean-square displacement. For $D_r>0$, its long-time diffusion coefficient is $D+v^2{D}_r/[2(D_r^2+{\omega }^2)]$. The full position-angle process has the Markov property, whereas position alone generally retains orientational memory. The angular second moment refers to an unwrapped angle.