The collective variables representation of simple fluids from the point of view of statistical field theory

TL;DR

This paper reexamines the collective variable representation of simple fluids using statistical field theory, deriving a two-loop expansion and comparing it with the Hubbard-Stratonovich transform, revealing new approximations at the second-loop level.

Contribution

It provides a detailed comparison of CV and Hubbard-Stratonovich representations, deriving a two-loop expansion and introducing new approximations for simple fluids.

Findings

Two-loop expansion for grand potential and free energy derived

Results from both representations coincide at each order of expansion

New expressions for pressure and free energy at second-loop level

Abstract

The collective variable representation (CV) of classical statistical systems such as simple liquids has been intensively developed by the Ukrainian school after seminal works by Prof. Ihor Yukhnovskii. The basis and the structure of the CV representation are reexamined here from the point of view of statistical field theory and compared with another exact statistical field representation of liquids based upon a Hubbard-Stratonovich transform. We derive a two-loop expansion for the grand potential and free energy of a simple fluid in both version of the theory. The results obtained by the two approaches are shown to coincide at each order of the loop expansion. The one-loop results are identical to those obtained in the framework of the random phase approximation of the theory of liquids. However, at the second-loop level, new expressions for the pressure and the free energy are obtained, yielding a new type of approximation.