Perturbation calculation for the density profile across the flat liquid-vapor interface in the steady heat-flow state

We consider a one-dimensional steady state where only the heat flows across a flat liquid-vapor interface of a one-component fluid occupying a container in zero gravity. By means of the Ginzburg-Landau type free-energy, we study the density profile up to the first order in the perturbation expansion with respect to the imposed temperature gradient. The first-order term satisfies a nonhomogeneous differential equation, of which the Green's function is derived. We show that, when both phases share the same volume, difference in a quantity between the liquid bulk and the vapor bulk can be related to direction of the interface shift caused by the imposed temperature gradient. We also calculate the density profile of a van der Waals liquid by assuming its thermal conductivity to be proportional to the density. The relation is shown to be valid for this model liquid.