Atmospheric heavy elements have been observed in more than a quarter of white dwarfs (WDs) at different cooling ages, indicating ongoing accretion of asteroidal material, whilst only a few per cent of the WDs possess a dust disk, and all these WDs are accreting metals. Here, assuming that a rubble-pile asteroid is scattered inside a WDs Roche lobe by a planet, we study its tidal disruption and the long-term evolution of the resulting fragments. We find that after a few pericentric passages, the asteroid is shredded into its constituent particles, forming a flat, thin ring. On a timescale of Myr, tens of per cent of the particles are scattered onto the WD. Fragment mutual collisions are most effective for coplanar fragments, and are thus only important in $10^3-10^4$ yr before the orbital coplanarity is broken by the planet. We show that for a rubble pile asteroid with a size frequency distribution of the component particles following that of the near earth objects, it has to be roughly at least 10 km in radius such that enough fragments are generated and $ge10%$ of its mass is lost to mutual collisions. At relative velocities of tens of km/s, such collisions grind down the tidal fragments into smaller and smaller dust grains. The WD radiation forces may shrink those grains orbits, forming a dust disk. Tidal disruption of a monolithic asteroid creates large km-size fragments, and only parent bodies $ge100$ km are able to generate enough fragments for mutual collisions to be significant.
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