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

Abstract

The collective variable representation (CV) of classical statistical systems such as, for instance, 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 versions 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 within the framework of the random phase approximation of the theory of liquids. However, at the second-loop level, new expressions for pressure and the free energy are obtained, yielding a new type of approximation.

Statistical field theory for liquid vapor interface

“…of (24) is nothing but the limit at q → 0 of the bulk correlation function and is not to be included in the capillary limit of the model. In any case, for sufficiently small values ofh 1 , the bulk contribution to equation (24) becomes negligible at q = 0. This is made more precise hereinafter concerning the behavior of S ∆z ∆z .…”

“…The important steps leading to the appropriate statistical field theory for a liquid are outlined in Appendix A; the interested reader is referred to [18,[23][24][25] for deeper details and to [29] for an alternative formulation. We consider a simple fluid whose pair interaction potential includes a hard sphere (HS) repulsive part and a soft attractive part, denoted by v(r).…”

“…In this appendix we review the so-called Kac-Siegert-Stratonovich-Hubbard-Edwards (KSSHE) [17][18][19][20][21][22] transformation devised to rewrite the GCPF of a classical fluid as the partition function of a bosonic statistical field theory; more details are given in refs [23][24][25].…”

“…The latter is obtained from a Kac-Siegert-Stratonovich-Hubbard-Edwards (KSSHE) [17][18][19][20][21][22] transformation devised to rewrite the grand canonical partition function (GCPF) of the inhomogeneous fluid as the partition function of a statistical field theory involving the stochastic real field φ. The method is an alternative to the so-called collective variables method developed notably by I. Mryglod and coworkers [23,24], that was mainly used for homogeneous fluids in [23][24][25] and is considered here in studying a planar interface. It is important to mention that the present work is not based on the determination of an effective surface hamiltonian, conversely to [26,27].…”

“…In order to get a canonical field theory we perform the translation φ = ϕ + ∆ν with ∆ν = ν − ν 0 where ν 0 is some arbitrary uniform reference chemical potential to be conveniently chosen later. The Jacobian of the transformation is obviously equal to one and then, performing a Taylor functional expansion of the grand potential ln {Ξ HS [ν 0 + φ]} about ν 0 we obtain a standard scalar field theory [23][24][25], characterized by the partition function…”