  We perform a study of phase transitions of RN-AdS black hole at its Davies
points according to a generalized Ehrenfest classification of phase transition
established on the basis of fractional derivatives. Davies points label the
positions where heat capacity diverges. According to the usual Ehrenfest
classification, second-order phase transitions occur there. For RN-AdS black
hole, the Davies points can be classified into two types. The first type
corresponds to the extreme values of temperature and the second type
corresponds to the infection point(namely the critical point) of temperature.
Employing the generalized Ehrenfest classification, we find that the orders of
phase transition at the two types of Davies points are different. It is
$3/2$-order for the first type and $4/3$-order for the second type. Thus this
finer-grained classification can discriminate phase transitions that are
supposed to be in the same category, which may provide some new insights toward
a better understanding of black hole thermodynamics.
