Density field theory for a fluid interacting with the Yukawa potential. Role of the ideal entropy

We present a field theory to describe liquids where the field represents the density. In terms of this field, the Hamiltonian contains the ideal entropy and the interaction between the density fields. The approach is illustrated with the Yukawa interaction and presented in the grand canonical ensemble formalism. In this framework, first, we derive a relation specific to the field theory. This relation is equivalent to the 'equation of motion' in field theory for interacting quantum particles. Then, focusing on the effect of the fluctuations, we calculate thermodynamic quantities beyond the mean field. The pressure, the density and the compressibility at a given chemical potential in the quadratic approximation and beyond are given. The aim of this paper is to illustrate the importance and the role of the ideal entropy in this type of approach. The density and the compressibility at a given chemical potential are calculated perturbatively in various ways. Whether from their field theoretical definition, or deriving them from one another using the thermodynamical relations or also using the 'equation of motion', the results are in all ways of calculation consistent. However, the different calculations require different levels of expansion of the ideal entropy term involving in our case three and four body coupling constants. The consistency is then closely related to the form of the functional of the ideal entropy.