Field theoretical description of the liquid state. Exact relations. The role of the ideal entropy revisited

In a series of papers, we have used a field theoretical description of the liquid state for a study of ionic systems. The formalism constructed is based on a simple Hamiltonian including the interaction potential and the ideal entropy. We discuss and analyse the Hamiltonian by detailing the role of its different contributions and its physical content. We show that the simple Hamiltonian based on particle densities as the fields also reproduces exactly the usual liquid state theory formulated in a discrete particle description rather than continuous fields. In this perspective, the formalism is discussed in view of a well-known exact and fundamental relation of the liquid state theory: the contact theorem. We demonstrate the validity of this theorem within the field theoretical framework. We find that the specific form of the Hamiltonian, in particular of the ideal entropy functional, is essential. The analysis of this term shows that it introduces the basic principle of uncertainty and the principle of the indiscernibility of quantum physics which exists for particles in a way suited for a description in terms of fields. This discussion is illustrated in the case of an ionic solution at an interface.