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We investigate the superfluid-insulator quantum phase transition in adisordered 1D Bose gas in the mean field limit, by studying the probabilitydistribution of the density. The superfluid phase is characterized by avanishing probability to have zero density, whereas a nonzero probability marksthe insulator phase. This relation is derived analytically, and confirmed by anumerical study. This fragmentation criterion is particularly suited fordetecting the phase transition in experiments. When a harmonic trap isincluded, the transition to the insulating phase can be extracted from thestatistics of the local density distribution.

Fragmentation and the Bose-glass phase transition of the disordered 1D Bose gas