Bifurcation theory of the critical liquid-gas interface

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Abstract
A new theory is presented to determine the structure of the intrinsic density profile of the liquidgas interface in the neighborhood of the critical point. In contrast to earlier theories, no use is made of scaling or homogeneity conjectures or of the Andrews-Thomson-van der Waals "loop" of the subcritical isotherm. A nonlinear integral equation is derived for the critical intrinsic density profile, and it is found that when the overall average density of the fluid is kept fixed and equal to its critical value and the temperature is allowed to vary, a nontrivial solution of the nonlinear integral equation, which described the intrinsic density profile, bifurcates at the critical point from the trivial solution which describes a spatially uniform fluid. It is shown that in the neighborhood of the critical point, this nontrivial solution has a scaling form which can be determined exactly. The asymptotic behavior of this nontrivial solution can also be calculated exactly: The wings of the intrinsic density profile decay exponentially with an exponent proportional to the inverse correlation length of the spontaneous density fluctuations in either bulk phase. These results confirm the conjectures of Widom.