  Planetary systems can survive the stellar evolution, as evidenced by the
atmospheric metal pollution and dusty disks of single white dwarfs. Recent
observations show that 1 to 4 percent of single white dwarfs are accompanied by
dusty disks, while the occurrence rate of metal pollution is about 25 to 50
percent . The dusty disks and metal pollution have been associated with
accretion of remnant planetary systems around white dwarfs, yet the relation
between these two phenomena is still unclear. Here we suggest an evolutionary
scenario to link the two observational phenomena. By analyzing a sample of
metal polluted white dwarfs, we find that the mass accretion rate onto the
white dwarf generally follows a broken power law decay, which matches well with
the theoretical prediction, if assuming dust accretion is primarily driven by
Poynting-Robertson drag and the dust source is primarily delivered via
dynamically falling asteroids perturbed by a Jovian planet. The presence of
disks is mainly at the early stage (about 0.1 to 0.7 billion years) of the
whole process of metal pollution, which is detectable until about 8 billion
years, naturally explaining the fraction (about 2-16 percent) of metal-polluted
white dwarfs having dusty disks. The success of this scenario also implies that
the configuration of an asteroid belt with an outer gas giant might be common
around stars of several solar masses.
