Of the many complex processes of granular materials, vibrational settling and compaction are common phenomena that have attracted much attention. In this work, we investigate vibrational, or tapping, compaction, and propose that the underlying kinetics involves clusters fragmenting and aggregating, and individual grains attaching and dissociating at cluster surfaces. The periodic vibrations cause cluster breakage and interchange between individual free grains and the clusters. The population balance equations for the concurrent kinetics are solved by a moment method, yielding easily solved differential equations. The compaction ratio defined in terms of the mass moments agrees well with experimental data [Knight, Phys. Rev. E. 51, 3957 (1995) Nowak, Phys. Rev. E. 57, 1972 (1998)] and other models. A change in tapping acceleration can produce reversible or irreversible transitions between densities, depending on the number of clusters that have evolved. © 2004 The American Physical Society.