Past exam of the theoretical physics major option course of the University of Oxford / 2016 / 4 / d / Solution

For $t\ll D_r^{-1},|\omega {|}^{-1}$, expand the exact expression:At asymptotically short time, translational diffusion dominates if $D>0$. A ballistic window occurs when $4D/v^2\ll t\ll \min (D_r^{-1},|\omega {|}^{-1})$, provided these scales are separated. The swimmer then retains nearly constant direction.

For $D_r>0$ and $t\gg D_r^{-1}$, the oscillatory exponential decays andAt intermediate times, the angular drift bends persistent trajectories into arcs. If $|\omega |\gg D_r$, several rotations occur before orientational decorrelation, producing oscillatory shoulders rather than a long straight ballistic regime. In the limit $D_r=0$, $\omega \ne 0$, the active contribution is bounded and periodic: $2v^2(1-\cos \omega t)/{\omega }^2$. At $D_r=\omega =0$, it remains ballistic forever, so ordinary late-time active diffusion does not follow.

For $D_r>0$, the central-limit coarse graining of the position gives the late-time Fokker-Planck equationThere is no surviving drift after orientation averaging. This is an effective large-time, large-distance equation, not the exact closed equation for position at all times.