= Solution
The impact gives fragments a spread of orbital energy and <specific angular momentum>. Their resulting spread of <mean motion> lets <Keplerian shear> stretch a compact <dust clump> into an arc and then a ring, while the spread of orbital frequencies causes <phase mixing>. Size-dependent <radiation-pressure coefficients> immediately place small grains on different eccentric or even <radiation-pressure blowout orbits>; <Poynting–Robertson drag> and <stellar-wind drag> then alter their orbits on longer timescales. Further collisions grind or disperse the clump, and planetary perturbations can accelerate mixing.
These processes depend strongly on $D_f$. Small grains have larger radiation-force-to-gravity ratios and generally shorter collisional or drag lifetimes, while larger fragments remain closer to the parent orbit but can preserve a velocity-dispersion-driven clump for longer. The lifetime also depends on collision location, ejection velocities, optical depth, and orbital radius. A universal fixed $\Delta t$ is therefore a useful population-model approximation, not a literal property of every collision; a size- and event-dependent lifetime distribution is more realistic.
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