  We study the thermalization of a quenched disordered Bose-Hubbard model. By
considering the eigenstate distribution fluctuation, we show that the thermal
to many-body localized transition is always connected to a minimum of this
distribution fluctuation. We also observe a Mott-localized regime, where the
system fails to thermalize due to the strong on-site repulsion. At last, we
show how to detect this eigenstate distribution fluctuation in cold atom
systems, which is equivalent to measure the Loschmidt echo of the system. Our
work suggests a way to measure the thermal to localized transitions in
experiments, especially for a large system.
