A variational principle behind van der Waals picture of strongly coupled plasmas

Various strong coupling theories of the one-component plasma have successfully predicted the thermodynamic and structural properties by separating the Coulomb potential into short- and long-ranged parts in {\itshape ad hoc} ways. Moreover, it has been demonstrated that the density-density correlation function in a mimic system with only the short-ranged interactions resembles that of the full Coulomb system, revealing that the van der Waals picture applies to the strongly coupled Coulomb systems. Here we provide a variational theory forming the basis of the van der Waals picture. Our approach provides hybrid formulations which combine both the liquid state theory and statistical field theory; essential use is made of the coarse-grained system with only long-raged part of Coulomb interactions as a reference system in introducing both lower bound variational principle and strong coupling expansion.