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

Codex  0 2026-10-25

For

t ≪ D_{r}^{- 1}, ∣ ω ∣^{- 1}

, expand the exact expression:

M (t) = 4 D t + v^{2} t^{2} - \frac{v ^{2} D _{r}}{3} t^{3} + O (t^{4}) .

At asymptotically short time, translational diffusion dominates if

D > 0

. A ballistic window occurs when

4 D / v^{2} ≪ t ≪ min (D_{r}^{- 1}, ∣ ω ∣^{- 1})

, provided these scales are separated. The swimmer then retains nearly constant direction.

For

D_{r} > 0

and

t ≫ D_{r}^{- 1}

, the oscillatory exponential decays and

M (t) = 4 D_{eff} t + \frac{2 v ^{2} ( ω ^{2} - D _{r}^{2} )}{( D _{r}^{2} + ω ^{2} ) ^{2}} + o (1), D_{eff} = D + \frac{v ^{2} D _{r}}{2 ( D _{r}^{2} + ω ^{2} )} .

At intermediate times, the angular drift bends persistent trajectories into arcs. If

∣ ω ∣ ≫ D_{r}

, several rotations occur before orientational decorrelation, producing oscillatory shoulders rather than a long straight ballistic regime. In the limit

D_{r} = 0

,

ω \neq = 0

, the active contribution is bounded and periodic:

2 v^{2} (1 - cos ω t) / ω^{2}

. At

D_{r} = ω = 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 equation

\partial_{t} P (r, t) = D_{eff} \nabla^{2} P (r, t) .

There 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.

Figure 1.
Exact displacement of a persistent chiral swimmer: ballistic and diffusive windows, rotation-induced shoulders, and suppression of active diffusion by angular drift
. Details.

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