  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 WD's 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, and are therefore directly accreted
without first passing through a circularised close-in disk. 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. Hence, those large asteroids experience a disk phase before being
accreted.
