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