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

A power law decay evolution scenario for polluted single white dwarfs