  Proto-neutron stars (PNS) are born hot, with temperatures exceeding a few
times $10^{10}$ K. In these conditions, the PNS crust is expected to be made of
a Coulomb liquid composed of an ensemble of different nuclear species. We
perform a study of the beta-equilibrated PNS crust in the liquid phase in a
self-consistent multi-component plasma (MCP) approach, thus allowing us to
consistently calculate the impurity parameter, often taken as a free parameter
in cooling simulations. We developed a self-consistent MCP approach at finite
temperature using a compressible liquid-drop description of the ions, with
surface parameters adjusted to reproduce experimental masses. The treatment of
the ion centre-of-mass motion was included through a translational free-energy
term accounting for in-medium effects. The results of self-consistent MCP
calculations are systematically compared with those performed in a perturbative
and in the one-component plasma treatment. We show that the inclusion of
non-linear mixing terms arising from the ion centre-of-mass motion leads to a
breakdown of the ensemble equivalence between the one-component and MCP
approach. Our findings illustrate that the abundance of light nuclei becomes
important, eventually dominating the distribution at higher density and
temperature. This is reflected in the impurity parameter, which, in turn, may
have a potential impact on NS cooling. For practical applications, we also
provide a fitting formula for the impurity parameter in the PNS inner crust.
Our results obtained within a self-consistent MCP approach show important
differences in the prediction of the PNS composition with respect to those
obtained with a one-component or a perturbative MCP approximation, particularly
in the deeper region of the crust. This highlights the importance of a full,
self-consistent MCP calculation for reliable predictions of the PNS crust
composition.
