  We define a mean-field crossover generated by the Maxwell construction as the
dividing interface for the vapor-liquid interface area and a highly accurate
density-profile equation is thus derived. By using a mean-field equation of
sate for the Lennard-Jones fluid incorporated with the density gradient theory,
we show that the intrinsic free energy peaks and the isobaric heat capacity
exhibits local maxima at the interface. We demonstrate that the mean-field
interface is the natural extension of the Widom line into the coexistence
region, hence the entire space is coherently divided into liquid-like and
gas-like regions in all three (temperature-pressure-volume) planes. Finally,
the mean-field theory is found holding all the information for composing the
phase diagrams over the entire phase space.
