  In mean-field theory, the non-local state of fluid molecules can be taken
into account using a statistical method. The molecular model combined with a
density expansion in Taylor series of the fourth order yields an internal
energy value relevant to the fourth-gradient model, and the equation of
isother-mal motions takes then density's spatial derivatives into account for
waves travelling in both liquid and vapour phases. At equilibrium, the equation
of the density profile across interfaces is more precise than the Cahn and
Hilliard equation, and near the fluid's critical-point, the density profile
verifies an Extended Fisher-Kolmogorov equation, allowing kinks, which
converges towards the Cahn-Hillard equation when approaching the critical
point. Nonetheless, we also get pulse waves oscillating and generating critical
opalescence.
