The collective variables representation of simple fluids from the point of view of statistical field theory
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 th...
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| Cite this: | The collective variables representation of simple fluids from the point of view of statistical field theory / J.-M. Caillol, O. Patsahan, I. Mryglod // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 665–684. — Бібліогр.: 29 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1205132025-06-03T16:27:04Z The collective variables representation of simple fluids from the point of view of statistical field theory Представлення колективних змінних для простих плинів з точки зору статистико-польової теорії Caillol, J.-M. Patsahan, O. Mryglod, I. 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. Започаткований в роботах професора Ігоря Юхновського метод колективних змінних (КЗ) був успішно розвинутий до опису класичних статистичних систем українською школою. В даній роботі основи і структура представлення КЗ для рідин вивчається з точки зору статистико-польового підходу і порівнюється з іншими точними теоріями, що використовують перетворення Габбарда-Стратоновича. Для випадку простого плину отримано вираз для вільної енергії в обох версіях теорії і показано, що отримані результати співпадають в кожному порядку петлевого розвинення. Результати, отримані в однопетлевому наближенні є ідентичними до отриманих в наближенні хаотичних фаз. Проте, двопетлеве наближення дає новий вираз для тиску і вільної енергії і є новим типом наближення. 2005 Article The collective variables representation of simple fluids from the point of view of statistical field theory / J.-M. Caillol, O. Patsahan, I. Mryglod // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 665–684. — Бібліогр.: 29 назв. — англ. 1607-324X PACS: 05.20.Jj, 05.70.Ce DOI:10.5488/CMP.8.4.665 https://nasplib.isofts.kiev.ua/handle/123456789/120513 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
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. |
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Article |
| author |
Caillol, J.-M. Patsahan, O. Mryglod, I. |
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Caillol, J.-M. Patsahan, O. Mryglod, I. The collective variables representation of simple fluids from the point of view of statistical field theory Condensed Matter Physics |
| author_facet |
Caillol, J.-M. Patsahan, O. Mryglod, I. |
| author_sort |
Caillol, J.-M. |
| title |
The collective variables representation of simple fluids from the point of view of statistical field theory |
| title_short |
The collective variables representation of simple fluids from the point of view of statistical field theory |
| title_full |
The collective variables representation of simple fluids from the point of view of statistical field theory |
| title_fullStr |
The collective variables representation of simple fluids from the point of view of statistical field theory |
| title_full_unstemmed |
The collective variables representation of simple fluids from the point of view of statistical field theory |
| title_sort |
collective variables representation of simple fluids from the point of view of statistical field theory |
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Інститут фізики конденсованих систем НАН України |
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2005 |
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https://nasplib.isofts.kiev.ua/handle/123456789/120513 |
| citation_txt |
The collective variables representation of simple fluids from the point of view of statistical field theory / J.-M. Caillol, O. Patsahan, I. Mryglod // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 665–684. — Бібліогр.: 29 назв. — англ. |
| series |
Condensed Matter Physics |
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2025-12-01T04:37:25Z |
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2025-12-01T04:37:25Z |
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1850279301857935360 |
| fulltext |
Condensed Matter Physics, 2005, Vol. 8, No. 4(44), pp. 665–684
The collective variables representation
of simple fluids from the point of view
of statistical field theory ∗
J.-M.Caillol 1 , O.Patsahan 2 , I.Mryglod 2
1 Laboratoire de Physique théorique, CNRS UMR 8627,
Bât. 210, Université de Paris-Sud,
91405 Orsay Cedex France
2 Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
Received July 18, 2005, in final form October 14, 2005
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 sta-
tistical 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 ex-
pansion. 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.
Key words: statistical field theory, loop expansion, collective variables
PACS: 05.20.Jj, 05.70.Ce
1. Introduction
Functional methods in modern statistical physics represent one of the most pow-
erful tools for the study of both equilibrium and dynamical properties (see, e.g. [1,2]).
In many cases the partition function of simple models (Ising and Heisenberg spins
or classical particles in interaction) can be re-expressed as a functional integral af-
ter performing a Hubbard-Stratonovich transformation, a simple device proposed in
∗This paper is dedicated to Professor Ihor Yukhnovskii on the occasion of his 80th birthday.
c© J.-M.Caillol, O.Patsahan, I.Mryglod 665
J.-M.Caillol, O.Patsahan, I.Mryglod
the 50ies [3,4]. Nearly at the same time another method – the so-called collective
variables (CV) method – that allows us to explicitly construct a functional repre-
sentation for many-particle interacting systems was developed [5,6] and applied to
the description of charged particle systems, in particular, to the calculation of the
configurational integral of the Coulomb systems. The idea of this method is based
on: (i) the concept of collective coordinates being appropriate for the physics of
the system considered (see, for instance, [7]), and (ii) an integral giving an exact
functional representation for the configurational Boltzmann factor. Later the CV
methods were successfully developed for the description of classical many-particle
systems [8] and for the theory of second order phase transitions [9]. The method
has a wide range of applications and, to paraphrase Prof. Yukhnovskii in his book
[9] “The collective variables is a common name for a class of variables which are
specific for each kind of physical system. For magnetic systems CV are variables
corresponding to the modes of vibrations of spin moments, for ferroelectrics they
are cluster vibrational modes, for systems of charged particles they are associated
with the modes of generalized charge vibrations, for binary alloys they are modes
of one-particle distribution function, the CV of the liquid-gas system are modes of
some deviations from the critical density and so on.”
One of the goals of this paper is to reconsider the CV method from the point of
view of statistical field theory and to compare the results obtained with those found
recently by one of us by means of the KSSHE (Kac-Siegert-Stratonovich-Hubbard-
Edwards) theory [10].
We formulate the method of CV in real space and consider a one-component
continuous model consisting of hard spheres interacting through additive pair po-
tentials. The expression for the functional of the grand partition function is derived
and the CV action that depends upon two scalar fields – field ρ connected to the
number density of particles and field ω conjugate to ρ is calculated. We study the
correlations between these fields as well as their relations to the density and energy
correlations of the fluid. The grand partition function of the model is evaluated in
a systematic way using a well-known method of statistical field theory, namely the
so-called loop expansion. It consists in functionally expanding the action H around
a saddle point, so that the lowest order (zero loop) approximation defines the mean-
field (MF) level of the theory and the first order loop expressions correspond to the
random phase approximation (RPA). Recently [10] this technique was applied to
the action obtained within the framework of the KSSHE theory. In this paper we
perform a two-loop expansion of the pressure and the free energy of the homoge-
neous fluid which yields a new type of approximation which we plan to test in our
future work.We note that another perturbation scheme for the free energy within the
framework of the CV method (a group expansion), based on the Gaussian measure,
was proposed in the late 70ies [11].
The paper is organized as follows. In section 2, starting from the Hamiltonian, we
introduce the two different functional representations of the grand partition function
based on the KSSHE and CV methods. Here we also enter several types of statistical
field averages that are important in the further part of the paper. In section 3 we
666
The collective variables representation of simple fluids
introduce the CV and KSSHE field correlation functions, establish links between
them as well as their relation to the density correlation functions of the fluid. The
MF level of the KSSHE and CV field theories is formulated in section 4. Section 5
is devoted to the loop expansion of the grand potential. The pressure and the free
energy of the homogeneous fluid are obtained in the two-loop approximation in
section 5. We conclude with some final remarks in section 6.
2. Prolegomena
2.1. The model
We consider the case of a simple three dimensional fluid made of identical hard
spheres of diameter σ with additional isotropic pair interactions v(rij) (rij = |xi−xj|,
xi is the position of particle “i”). Since v(r) is an arbitrary function of r in the core,
i.e. for r 6 σ, we will assume that v(r) has been regularized in such a way that its
Fourier transform ṽq is a well behaved function of q and that v(0) is a finite quantity.
We denote by Ω the domain of volume V occupied by the molecules of the fluid.
The fluid is at equilibrium in the grand canonical (GC) ensemble, β = 1/kBT is
the inverse temperature (kB Boltzmann constant), and µ is the chemical potential.
Moreover, the particles are subject to an external potential ψ(x) and we will denote
the dimensionless local chemical potential by ν(x) = β(µ − ψ(x)). We will stick to
notations usually adopted in standard textbooks devoted to the theory of liquids
(see e.g. [12]) and will denote the dimensionless pair interaction by w(r) = −βv(r)
minus . Quite arbitrarily we will say that the interaction is attractive if the Fourier
transform w̃(q) is positive for all q; in the converse case it will be said to be repulsive.
In a given GC configuration C = (N ;x1 . . . xN) the microscopic density of parti-
cles reads
ρ̂(x|C) =
N∑
i=1
δ(3)(x− xi) , (1)
and the GC partition function Ξ [ν] can thus be written as
Ξ [ν] = Tr
[
exp
(
−βVHS(C) +
1
2
〈ρ̂|w|ρ̂〉 + 〈ν|ρ̂〉
)]
,
Tr [. . .] =
∞∑
N=0
1
N !
∫
Ω
d1 . . . dn . . . , (2)
where i ≡ xi and di ≡ d3xi. For a given volume V , Ξ [ν] is a function of β and a
convex functional of the local chemical potential ν(x) [13] which we have strength-
ened by using a bracket. In equation (2), exp(−βVHS(C)) denotes the hard sphere
contribution to the Boltzmann factor and ν = ν+νS where νS = −w(0)/2 is β times
the self-energy of a particle. It follows from our assumptions on w(r) that νS is a
finite quantity which depends, however, on the regularization of the potential in the
core. In the right hand side of equation (2) we have introduced convenient bracket
667
J.-M.Caillol, O.Patsahan, I.Mryglod
notations
〈ν|ρ̂〉 ≡
∫
Ω
d1 ν(1)ρ̂(1), (3)
〈ρ̂|w|ρ̂〉 ≡
∫
Ω
d1d2 ρ̂(1|C)w(1, 2)ρ̂(2|C) . (4)
Here we show how to re-express Ξ [ν] as a functional integral.
2.2. The Hubbard-Stratonovitch transform
The Hubbard-Stratonovich transformation [3,4] which was proposed in the 50ies
is based on simple properties of Gaussian integrals. Within this framework the GC
partition function Ξ [ν] can be formally re-expressed as a functional integral [3,4,14–
17]. Moreover Ξ [ν], when re-written in this manner, can be interpreted as the GC
partition function of a fluid of bare hard spheres in the presence of an external
random Gaussian field ϕ with a covariance given by the pair potential [10,18,19]. It
will be convenient to distinguish the case of attractive interactions (w̃(q) > 0) from
that of repulsive ones (w̃(q) < 0).
i) Attractive interactions (w̃(q) > 0)
We start from the well-known property of Gaussian integrals
exp
(
1
2
〈ρ̂|w|ρ̂〉
)
=
∫
Dϕ exp
(
−1
2
〈ϕ|w−1|ϕ〉 + 〈ρ̂|ϕ〉
)
Nw ≡
∫
Dϕ exp
(
−1
2
〈ϕ|w−1|ϕ〉
)
≡ 〈exp (〈ρ̂|ϕ〉)〉w , (5)
where ϕ is a real random field and the inverse w−1 must be understood in
the operator sense, i.e.
∫
Ω
d3 w(1, 3)w−1(3, 2) = δ(1, 2). The functional inte-
grals which enter the above equation can be given a precise meaning in the
case where the domain Ω is a cube of side L with periodic boundary conditi-
ons (PBC) which will be implicitly assumed henceforth. Additional technical
details are to be found in the appendix. We now insert equation (5) in the
definition (2) of the GC partition function Ξ [ν] which yields
Ξ [ν] = 〈Tr [ exp (−βVHS(C) + 〈ν + ϕ|ρ̂〉)]〉w
= 〈ΞHS [ν + ϕ]〉w . (6)
Ξ [ν], when re-written in this manner, can thus be re-interpreted as the GC
partition function of a fluid of bare hard spheres in the presence of a random
Gaussian field ϕ with a covariance given by the pair potential [10,18,19].
ii) Repulsive interactions (w̃(q) < 0)
This time we make use of
exp
(
−1
2
〈ρ̂|w|ρ̂〉
)
=
∫
Dϕ exp
(
1
2
〈ϕ|w−1|ϕ〉 + i 〈ρ̂|ϕ〉
)
N−w ≡
∫
Dϕ exp
(
1
2
〈ϕ|w−1|ϕ〉
)
≡ 〈exp (i 〈ρ̂|ϕ〉)〉w , (7)
668
The collective variables representation of simple fluids
which yields
Ξ [ν] = 〈ΞHS [ν + iϕ]〉(−w) . (8)
Note that since ϕ is a real scalar field ΞHS [ν + iϕ] has to be evaluated for
imaginary chemical potentials which may cause some troubles since log ΞHS [ν]
has singularities (i.e. branch cuts) in the complex plane [20]. In the repulsive
case, the hard core part of the interaction is in fact not compulsory in order to
ensure the existence of a thermodynamic limit [21] and the reference system
can be chosen as the ideal gas [20,22].
Some comments are in order. Firstly, equations (6) and (8) are easily generalized to
the case of mixtures [19,23] or molecular fluids. Secondly, when the pair interaction
w is neither attractive nor repulsive, it is necessary to introduce two real scalar
fields ϕ+ and ϕ− if some rigor is aimed at [10]. Alternatively, in a more sloppy way,
equation (6) can be considered to hold in any cases having in mind that ϕ will be a
complex scalar field in the general case. We will thus formally write in all cases
Ξ [ν] = N−1
w
∫
Dϕ exp (−HK [ν, ϕ]) , (9)
where the action (or effective Hamiltonian) of the KSSHE field theory reads as
HK [ν, ϕ] =
1
2
〈
ϕ|w−1|ϕ
〉
− ln ΞHS [ν + ϕ] . (10)
2.3. The collective variables representation
Now we briefly introduce the CV representation of Ξ [ν] and refer the reader to a
vast literature for a more detailed presentation (see, e.g. [6,8,9,24,25]). The starting
point is the formally trivial identity
exp
(
1
2
〈ρ̂|w|ρ̂〉
)
=
∫
Dρ δF [ρ− ρ̂] exp
(
1
2
〈ρ|w|ρ〉
)
, (11)
where δF [ρ] denotes the functional “delta” [2]. Making use of its functional integral
representation (see the appendix)
δF [λ] ≡
∫
Dω exp (i 〈ω|λ〉) , (12)
one finds for the GC Boltzmann factor
exp
(
1
2
〈ρ̂|w|ρ̂〉
)
=
∫
DρDω exp
(
1
2
〈ρ|w|ρ〉 + i 〈ω| {ρ− ρ̂}〉
)
. (13)
Inserting equation (13) in the definition (2) of the GC partition function Ξ [ν] one
obtains
Ξ [ν] =
∫
Dρ exp
(
1
2
〈ρ|w|ρ〉
)
J [ρ, ν] , (14)
669
J.-M.Caillol, O.Patsahan, I.Mryglod
where the Jacobian
J [ρ, ν] =
∫
Dω exp (i 〈ω|ρ〉) ΞHS [ν − iω] (15)
allows for the passage from the microscopic variables xi to the collective ones ρ. We
note that J [ρ, ν] does not depend on the pair interactions w(1, 2) but only on the
GC partition function of the reference HS system ΞHS [ν] which is supposed to be
known.
Equation (14) can also easily be recast in the form of a standard statistical field
theory, i.e. as
Ξ [ν] =
∫
DρDω exp (−HCV [ν, ρ, ω]) , (16)
where the action of the CV field theory reads as
HCV [ν, ρ, ω] = −1
2
〈ρ|w|ρ〉 − i 〈ω|ρ〉 − ln ΞHS [ν − iω] . (17)
We stress that ω and ρ are two real scalar fields and that equations (16) and (17)
are valid for repulsive, attractive as well as arbitrary pair interactions. Moreover,
with the clever normalization of Wegner [26] for the functional measures there are
no unspecified multiplicative constants involved in equation (16) (see the appendix
for more details).
The CV transformation is clearly more general than the KSSHE transformation
since it can be used for a pair interaction w(1, 2) which does not possess an inverse
and is easily generalized for n-body interactions (n > 2). The equivalence of the
CV and KSSHE representations (9) and (16) of Ξ [ν] is readily established in the
repulsive case ( w̃(q) < 0) by making use of the properties of Gaussian integrals
(cf. equation (7)). In the attractive or in the general case we cannot propose a
convincing way (i.e. a non-formal one) in order to establish this equivalence.
2.4. Statistical average
In the sequel it will be important to distinguish carefully, besides the usual GC
average 〈A(C)〉GC of a dynamic variable A(C), between two types of statistical field
averages. At first the KSSHE averages are defined as
〈A [ϕ]〉K = Ξ [ν]−1
∫
Dϕ A [ϕ] exp (−HK [ν, ϕ]) , (18)
where A [ϕ] is some functional of the KSSHE fields ϕ and, secondly the CV averages
are defined in a similar way as
〈A [ρ, ω]〉CV = Ξ [ν]−1
∫
DρDω A [ρ, ω] exp (−HCV [ν, ρ, ω]) , (19)
where A [ρ, ω] is an arbitrary functional of the two CV fields ρ an ω.
670
The collective variables representation of simple fluids
3. Correlation functions
To establish the link between the usual theory of liquids and the two statistical
field theories which were introduced in sections (2.2) and (2.3) means to find the
relations between density correlation functions on the one hand and field (either
KSSHE or CV) correlation functions on the other hand. This is the purpose of the
present section.
3.1. Density correlations
The ordinary and truncated (or connected) density correlation functions of the
fluid will be defined in this paper as [12,27,28]
G(n)[ν](1, . . . , n) =
〈
n∏
1=1
ρ̂(xi|C)
〉
GC
=
1
Ξ[ν]
δn Ξ[ν]
δν(1) . . . δν(n)
,
G(n),T [ν](1, . . . , n) =
δn log Ξ[ν]
δν(1) . . . δν(n)
. (20)
Our notation emphasizes the fact that the G(n) (connected and not connected) are
functionals of the local chemical potential ν(x) and ordinary functions of the space
coordinates (1, . . . , n) ≡ (x1, . . . , xn). We know from the theory of liquids that [27,28]
G(n),T [ν](1, . . . , n) = G(n)[ν](1, . . . , n) −
∑∏
m<n
G(m),T [ν](i1, . . . , im) , (21)
where the sum of products is carried out over all possible partitions of the set
(1, . . . , n) into subsets of cardinal m < n. Of course ρ[ν](x) ≡ G(n=1)[ν](x) =
G(n=1),T [ν](x) is the local density of the fluid.
It follows from the definition (20) of the G(n)[ν](1, . . . , n) that they can be reex-
pressed as KSSHE or CV statistical averages, i.e.
G(n)[ν](1, . . . , n) =
〈
G
(n)
HS [ν + ϕ](1, . . . , n)
〉
K
, (22)
G(n)[ν](1, . . . , n) =
〈
G
(n)
HS [ν − iω](1, . . . , n)
〉
CV
. (23)
However, the enlightening of these relations is not of a great use except for the
special case n = 1 which reads explicitly as
ρ [ν] (x) = 〈ρHS[ν + ϕ](x)〉K , (24)
ρ [ν] (x) = 〈ρHS[ν − iω](x)〉CV , (25)
where ρHS[ξ](x) is the local density of the hard sphere fluid at point x in the presence
of the local chemical potential ξ(x).
671
J.-M.Caillol, O.Patsahan, I.Mryglod
3.2. Field correlations
The correlation functions of the KSSHE field ϕ and the CV fields ρ and ω will
be defined as
G(n)
ϕ [ν](1, . . . , n) =
〈
n∏
1=1
ϕ (xi)
〉
K
,
G(n)
ρ [ν](1, . . . , n) =
〈
n∏
1=1
ρ (xi)
〉
CV
,
G(n)
ω [ν](1, . . . , n) =
〈
n∏
1=1
ω (xi)
〉
CV
, (26)
and their truncated counterparts will be defined in the same way as in equation (21).
The relations between the field correlation functions (26) and the density correlati-
on functions G(n)[ν](1, . . . , n) are easily obtained by introducing ad hoc generating
functionals and we quote here only the results referring the reader to the refer-
ences [10,29] for a detailed discussion.
i) relations between G
(n)
ϕ and G(n)
One has
〈ϕ(1)〉K = w(1, 1
′
)ρ [ν] (1
′
) ,
G(2),T
ϕ [ν] (1, 2) = w(1, 2) + w(1, 1
′
)w(2, 2
′
)G(2),T [ν] (1
′
, 2
′
) ,
G(n),T
ϕ [ν] (1, . . . , n) = w(1, 1
′
) . . . w(n, n
′
)G(n),T [ν] (1′, . . . , n′) for n > 3 , (27)
where we have adopted Einstein’s convention, i.e. space integration of variables
labeled by the same dummy indices is meant. It will be quite enlighting to
consider the GC correlations of the microscopic potential
V̂ (x, C) ≡
N∑
i=1
w(x, xi) =
∫
Ω
d3x
′
ρ̂(x′, C))w(x, x′) . (28)
We readily deduce from equations (27) that
〈ϕ(1)〉K =
〈
V̂ (1)
〉
GC
,
G(2),T
ϕ [ν] (1, 2) =
〈
V̂ (1)V̂ (2)
〉T
GC
+ w(1, 2) ,
G(n),T
ϕ [ν] (1, . . . , n) =
〈
V̂ (1) . . . V̂ (n)
〉T
GC
for n > 3 . (29)
Therefore the KSSHE field ϕ(x) identifies itself “nearly” with the microscopic
field V̂ (x) at a neutral point.
672
The collective variables representation of simple fluids
ii) relations between G
(n)
ρ , G
(n)
ω and G(n)
One easily finds that
G(n),T
ρ [ν] (1, . . . , n) = G(n),T [ν] (1, . . . , n) , (30)
which was expected and that
G(n),T
ϕ [ν] (1, . . . , n) = (−i)n G(n),T
ω [ν] (1, . . . , n) , (31)
which was suspected. We stress that the above relations are valid for all n. In
particular, for n = 1 note that 〈ρ(x)〉CV = ρ(x) and 〈ω(x)〉CV = i
〈
V̂ (x)
〉
GC
.
The CV ρ and ω are thus connected to the density of particles and the micro-
scopic potential respectively. We have therefore the correspondence ϕ ↔ −iω
which will be a guideline in the sequel.
4. Mean field theory
Contrary to the situation which prevails in the theory of liquid, the term “mean-
field” approximation is defined unambiguously in statistical field theory. It consists
in neglecting field fluctuations and in approximating the grand potential (i.e. the
logarithm of the partition function) by minus the action at its saddle point [1]. In this
section the KSSHE and CV theories will be considered at this level of approximation.
4.1. KSSHE representation
The mean-field (MF) or saddle point approximation of the KSSHE theory will
therefore be defined by the set of relations [10,19]:
ΞMF [ν] = exp (−HK [ν, ϕ0]) , (32)
where, for ϕ = ϕ0, the action is stationary, i.e.
δ HK [ν, ϕ]
δϕ
∣∣∣∣
ϕ0
= 0 . (33)
Replacing the KSSHE action by its expression (10) in equation (33) leads to an
implicit equation for ϕ0:
ϕ0(1) = w(1, 1
′
) ρHS [ν + ϕ0] (1
′
) , (34)
which reduces to
ϕ0 = w̃(0) ρHS [ν + ϕ0] (35)
for a homogeneous system. It follows from the stationary condition (33) that the
MF density is given by
ρMF [ν] (1) =
δ ln ΞMF [ν]
δν(1)
= ρHS [ν + ϕ0] (1) , (36)
673
J.-M.Caillol, O.Patsahan, I.Mryglod
and that the MF grand potential reads
ln ΞMF [ν] = ln ΞHS [ν + ϕ0] −
1
2
〈ρMF|w|ρMF〉 . (37)
Moreover, the MF Kohn-Scham free energy defined as the Legendre transform
βAMF [ρ] = sup
ν
{〈ρ|ν〉 − ln ΞMF [ν]} (38)
is found to be the convex envelop of
βAMF [ρ] = βAHS [ρ] − 1
2
〈ρ|w|ρ〉 +
1
2
∫
Ω
dx w(0)ρ(x) . (39)
It can be shown [10] that AMF [ρ] constitutes a rigorous upper bound for the exact
free energy A [ρ] if the interaction is attractive (w̃(q) > 0) and a lower bound in the
converse case (w̃(q) < 0).
Finally, the pair correlation and vertex (or direct correlation) functions at the
zero-loop order which are defined respectively as
G
(2)
MF [ν] (1, 2) =
δ2 ln ΞMF [ν]
δν(1) δν(2)
, (40)
C
(2)
MF [ρ] (1, 2) = − δ2βAMF [ρ]
δρ(1) δρ(2)
, (41)
are given by
G
(2)
MF(1, 2) =
(
1 − w ? G
(2)
HS [ν + ϕ0]
)−1
? G
(2)
HS [ν + ϕ0] (1, 2) , (42)
C
(2)
MF(1, 2) = −G(2)−1
MF (1, 2) = C
(2)
HS(1, 2) + w(1, 2) . (43)
We note that these relations coincide with the old random phase approximation
(RPA) of the theory of liquids [12]. Taking advantage of the arbitrariness of w(r) in
the core on can choose a regularization potential such that the pair correlation func-
tion g(r) = 0 for r < σ, then we recover the ORPA (optimized RPA) of the theory
of liquids [12]. To be complete let us point out that it follows from equation (27)
that the free propagator of the KSSHE field theory is given by
G
(2)
ϕ, MF(1, 2) =
(
1 − w ? G
(2)
HS [ν + ϕ0] (1, 2)
)−1
? w(1, 2) . (44)
4.2. CV representation
An analysis similar to that of section (4.1) can be made in the CV representation.
The MF level of the CV field theory will be defined by
ΞMF [ν] = exp (−HCV [ν, ρ0, ω0]) , (45)
674
The collective variables representation of simple fluids
where, for ρ = ρ0 and ω = ω0 the CV action is stationary, i.e.
δ HCV [ν, ρ, ω]
δρ
∣∣∣∣
(ρ0,ω0)
=
δ HCV [ν, ρ, ω]
δω
∣∣∣∣
(ρ0,ω0)
= 0 . (46)
Replacing the CV action by its expression (17) in equation (46) yields a set of two
coupled implicit equations for ρ0 and ω0:
0 = w(1, 2)ρ0(2) + iω0(1) ,
0 = ρ0(1) − ρHS [ν − iω0] (1) . (47)
If we define ϕ0 = −iω0, then the two previous equations can be rewritten
ϕ0 = w(1, 2)ρ0(2) ,
ρ0(1) = ρHS [ν + ϕ0] (1) , (48)
which shows that, as expected, ϕ0 coincides with the saddle point of the KSSHE
field theory (cf Sec 4.1). Moreover a direct calculation will show that
ln ΞMF [ν] = −HK [ν, ϕ0] = −HCV [ν, ρ0, ω0] . (49)
Therefore the local density, the grand potential, the free energy, the correlation and
the direct correlation functions coincide at the MF level in the CV and KSSHE field
theories.
5. Loop expansion
The loop expansion is a useful tool of statistical field theory [1]. Away from the
critical point it yields approximations for the free energy beyond the MF level.
5.1. Loop expansion of the grand potential
The loop expansion of the logarithm of the partition function of a scalar field
theory can be found in any standard textbook, see e.g. that of Zinn-Justin [1]. In
the case of the KSSHE field theory we have just to reproduce the known results.
One proceeds as follows. A small dimensionless parameter λ is introduced and the
loop expansion is obtained by the set of transformations
ϕ = ϕ0 + λ1/2χ ,
ln Ξ [ν] = λ ln
{
N−1
w
∫
Dχ exp
(
−HK [ν, ϕ]
λ
)}
= ln Ξ(0) [ν] + λ ln Ξ(1) [ν] + λ2 ln Ξ(2) [ν] + O(λ3) , (50)
where ϕ0 is the saddle point of the KSSHE action. The expansion of ln Ξ [ν] in powers
of λ is obtained by performing a cumulant expansion and by making a repeated use
of Wick’s theorem. At the end of the calculation one set λ = 1.
675
J.-M.Caillol, O.Patsahan, I.Mryglod
In order to obtain a loop expansion of ln Ξ in the CV representation a little
homework is necessary. In this case we are led to introduce the following set of
transformations
ρ = ρ0 + λ1/2δρ , (51)
ω = ω0 + λ1/2δω , (52)
ln Ξ [ν] = λ ln
{∫
Dδρ Dδω exp
(
−HCV [ν, ρ, ω]
λ
)}
, (53)
where (ρ0, ω0) is the saddle point of the CV action. The form retained in equati-
ons (51) and (52) is imposed by the exact relation 〈ω〉CV (1) = iw(1, 1
′
) 〈ρ〉CV (1
′
)
(cf. equations (30) and (31) for n = 1). Then one performs a cumulant expansi-
on of (53) and use the appropriate version of Wick’s theorem (i.e. for two coupled
Gaussian fields ρ and ω). The derivation is detailed in reference [29] and will not
be reproduced here. One finds that the KSSHE and CV loop expansions of ln Ξ [ν]
coincide at each order of the expansion (i.e. the ln Ξ(n) [ν] are the same) which was
expected.
Of course Ξ(0) [ν] ≡ ΞMF [ν]. At the one-loop order one finds [10,29]
Ξ(1) [ν] =
N∆ϕ0
Nw
=
∫
Dϕ exp
(
−1
2
〈ϕ|∆ϕ0
|ϕ〉
)
∫
Dϕ exp
(
−1
2
〈ϕ|w|ϕ〉
) , (54)
where we have adopted Zinn-Justin’s notations for the free propagator [1]
∆−1
ϕ0
(1, 2) = w−1(1, 2) −G
(2), T
HS [ν + ϕ0] (1, 2) . (55)
Of course ∆ϕ0
(1, 2) coincides with G
(2)
ϕ, MF(1, 2) as can be seen by comparing equa-
cD D Da b
Figure 1. Diagrams which contribute to lnΞ(2) [ν]. Da and Dc are irreducible
while Db is reducible.
tions (44) and (55). If ν is uniform then the system is homogeneous and ∆ϕ0
(1, 2)
takes on a simple form in Fourier space, i.e.
∆̃ϕ0
(q) =
w̃(q)
1 − w̃(q)G̃
(2), T
HS [ν + ϕ0] (q)
, (56)
and, in this case, the Gaussian integrals in equation (54) can be performed explicitly
(cf the appendix for more details). One finds that
ln Ξ(1) [ν] = −V
2
∫
q
ln
(
1 − w̃(q)G̃
(2), T
HS [ν + ϕ0] (q)
)
. (57)
676
The collective variables representation of simple fluids
The second-loop order contribution ln Ξ(2) to the grand potential has a complicated
expression involving the sum of three diagrams sketched in figure 1
ln Ξ(2) [ν] = Da +Db +Dc . (58)
More explicitly one has [1]
Da =
1
8
∫
Ω
d1 . . . d4 ∆ϕ0
(1, 2)∆ϕ0
(3, 4)G
(4), T
HS [ν + ϕ0] (1, 2, 3, 4),
Db =
1
8
∫
Ω
d1 . . . d3 d1
′
. . . d3
′
∆ϕ0
(1, 2)∆ϕ0
(1
′
, 2
′
)∆ϕ0
(3, 3
′
)
×G
(3), T
HS [ν + ϕ0] (1, 2, 3)G
(3), T
HS [ν + ϕ0] (1
′
, 2
′
, 3
′
),
Dc =
1
12
∫
Ω
d1 . . . d3 d1
′
. . . d3
′
∆ϕ0
(1, 1
′
)∆ϕ0
(2, 2
′
)∆ϕ0
(3, 3
′
)
×G
(3), T
HS [ν + ϕ0] (1, 2, 3)G
(3), T
HS [ν + ϕ0] (1
′
, 2
′
, 3
′
) . (59)
As they stand, the above relations are not particularly useful for practical applica-
tions (even in the homogeneous case) since they involve the three and four body
correlation functions of the reference HS fluid. We will introduce some reasonable
approximation in section (5.2) to tackle with this horrible expression. Quite remark-
ably it has been shown recently that for a symmetric mixture of charged hard spheres
ln Ξ(2) has a much more simple expression which involves only the pair correlation
functions of the HS fluid as a consequence of local charge neutrality [19,23].
5.2. Loop expansion of the pressure and the free energy
In this section we restrict ourselves to the homogeneous case, therefore ln Ξ [ν] =
V βP (ν), where P denotes the pressure and βA [ρ] = V βf(ρ) where f is the
Helmholtz free energy per unit volume. The two-loop expression that we derived
for P in section (5.1) is too complicated to be of any practical use since it involves
the 3 and 4 body density correlation functions of the HS fluid which are unknown,
whereas G
(2), T
HS is known approximatively, for instance in the Percus-Yevick (PY)
approximation [12]. A simple but coherent approximations for G
(3), T
HS and G
(4), T
HS
will be now proposed.
Recall first that it follows from their definitions [27,28] (see e.g. equations (20))
that the G
(n), T
HS [ν] satisfy the following relations
δ
δν(n+ 1)
G
(n), T
HS [ν] (1, . . . , n) = G
(n+1), T
HS [ν] (1, . . . , n, n+ 1) . (60)
For a homogeneous system (in which case ν is a constant) one infers from this
equation that
∫
Ω
d1 . . . dn G
(n+1), T
HS [ν] (1, . . . , n, n+ 1) =
∂n
∂νn
ρHS (ν) ≡ ρ
(n)
HS (ν) , (61)
677
J.-M.Caillol, O.Patsahan, I.Mryglod
where ρHS (ν) is the number density of hard spheres at the chemical potential ν. In
the rest of the section we will adopt the following approximation
G
(n), T
HS [ν] (1, . . . , n) = ρ
(n+1)
HS (ν) δ(n, 1) . . . δ(2, 1) for n > 3 . (62)
Note that this hypothesis is coherent with the exact relations (60) and (61). How-
ever, we keep the full G
(2), T
HS [ν] (1, 2) ≡ G
(2), T
HS [ν] (x12) with the assumption that it
is a known function of x12 (in the PY approximation for instance). Thus the free
propagator ∆ϕ0
(x12) has a well defined expression as well. It is not difficult to con-
vince oneself that the set of approximations that we have introduced is reasonable
as long as the range of the KSSHE field correlation functions is (much) larger than
the range of the HS density correlation functions. This will be true if w is a long
range pair interaction (near the Kacs limit for instance).
With the hypothesis (62) it is easy to obtain the two-loop order approximation
for the pressure. One finds
βP (ν) = βP (0)(ν) + λβP (1)(ν) + λ2βP (2)(ν) + O(λ3) ,
βP (0)(ν) = βPMF(ν) = PHS(ν + ϕ0) −
ϕ2
0
2w̃(0)
,
βP (1)(ν) = −1
2
∫
q
ln
(
1 − w̃(q)G̃
(2), T
HS [ν + ϕ0] (q)
)
,
βP (2)(ν) =
ρ
(3)
0
8
∆2
ϕ0
(0) +
[
ρ
(2)
0
]2
8
∆̃ϕ0
(0)∆2
ϕ0
(0) +
[
ρ
(2)
0
]2
12
∫
d3x ∆3
ϕ0
(x) , (63)
where ρ
(n)
0 ≡ ρ
(n)
HS (ν + ϕ0) and
∫
q
≡
∫
d3q/(2π)3.
The free energy βf(ρ) is obtained as the Legendre transform of βP (ν). The
calculation is lengthy and tricky and we must contend ourselves to give the final
result (see reference [29] for the complete proof). Setting λ = 1 one finds
βf(ρ) = βfHS(ρ) −
w̃ (0)
2
ρ2 +
1
2
∫
q
{
ln
(
1 − w̃ (q) G̃T
HS, ρ (q)
)
+ ρw̃ (q)
}
− ρ(3)
8
∆2
ρ(0) +
1
8
∆2
ρ(0)
[
ρ(2)
]2
ρ(1)
−
[
ρ(2)
]2
12
∫
d3x ∆3
ρ(x) , (64)
where the HS pair correlation function GT
HS, ρ must be evaluated at the density ρ
and, similarly,
∆̃ρ(q) =
w̃ (q)
1 − G̃T
HS, ρ (q) w̃ (q)
. (65)
Finally, in equation (64)
ρ(1) =
1
ν
(1)
HS (ρ)
, ρ(2) =
∂ρ
(1)
0
∂ν
=
−ν(2)
HS(ρ)
[
ν
(1)
HS(ρ)
]3 ,
678
The collective variables representation of simple fluids
ρ(3) =
∂ρ
(2)
0
∂ν
=
3
[
ν
(2)
HS(ρ)
]2
− ν
(3)
HS(ρ)ν
(1)
HS(ρ)
[
ν
(1)
HS(ρ)
]5 , (66)
where ν
(n)
HS (ρ) denotes the nth derivative of the HS chemical potential with respect
to the density (it can be computed within the framework of the PY or Carnahan-
Starling approximations, for instance [12]). Some remarks are in order.
i) It must be pointed out that, quite unexpectedly, the reducible diagramDb does
not have to be cancelled by the Legendre transform. Usually, in statistical field
theory it is the case (cf. [1]). The reason is that the chemical potential ν is
not the field conjugate to the order parameter m = 〈ϕ〉K of the KSSHE field
theory. However one of us have shown elsewhere [19,23] that for the symmetric
mixtures of charged hard spheres only irreducible diagrams contribute to βf (2).
ii) The two first terms of equation (64) (i.e. the one-loop order result) supplement-
ed by equation (44) for the pair correlation function are exactly constitutive of
the RPA theory of liquid. The full result including the two-loop contribution
is a new approximation as far as the authors know.
iii) All the quantities which enter equation (64) can be computed numerically
(for instance in the PY approximation). However we stress once again that
the validity of equation (64) is restricted to long range pair potentials (for
instance w(r) = γ3Φ(r/γ) with γ → 0) and we are not aware of the exact
results (i.e. obtained by Monte Carlo simulations) which would allow us to
test the validity of our expression for the free energy for such kind of fluids.
6. Conclusion
Using the CV method we have reconsidered the basic relations of the statistical
field theory of simple fluids that follow from this approach. Contrary to the KSSHE
theory [10] the corresponding CV action depends on two scalar fields, the field ρ
connected to the number density of particles and the field ω conjugate to ρ, con-
nected to the local potential. Explicit relations between field and density correlation
functions were obtained.
For a one-component continuous model of fluid, consisting of hard spheres inter-
acting through an attractive pair potential, we have calculated the grand partition
function for both versions of statistical field theory using the loop expansion tech-
nique. As it was expected, at all the orders of loop expansion considered, both
versions of the theory produced indeed the same analytical results. The expressions
for the pressure as well as for the free energy were derived at the two-loop level. It
yields a new type of approximation and Monte Carlo simulations are wanted to test
its validity.
From our analysis of the CV and KSSHE transformations we can also conclude
that the former has some important advantages which could be very useful for more
679
J.-M.Caillol, O.Patsahan, I.Mryglod
complicated models of fluids. In particular, it is valid for an arbitrary pair potential
(including a pair interaction w(1, 2) which does not possess an inverse) and is easily
generalized for the case of n-body interparticle interactions with n > 2.
The statistical field theory of liquids is still in its infancy and many advances
including the implementation of Wilson’s ideas regarding the renormalization group
for the study of gas-liquid critical points are likely to be achieved in the future, this
being at least the hope of the authors.
Acknowledgements
This work was carried out within the framework of the cooperation project be-
tween the CNRS and the NASU (ref. CNRS 17110). OP and IM also thank the
Fundamental Research State Fund of the Ministry of Education and Science of
Ukraine for partial support (Proj. No. 02.07/00303).
680
The collective variables representation of simple fluids
Functional measures and integrals
In this appendix we give some details concerning functional measures and in-
tegrals. Let us consider a real scalar field ϕ(x) defined in a cube C3 of side L and
volume V = L3. We assume periodic boundary conditions, i.e. we restrict ourselves
to the fields which can be written as a Fourier series,
ϕ(x) =
1
L3
∑
q∈Λ
ϕ̃q eiqx , (67)
where Λ = (2π/L) Z
3 is the reciprocal cubic lattice. The reality of ϕ implies that, for
q 6= 0 ϕ̃q = ϕ̃?
−q, where the asterisk means complex conjugation. Following Wegner
[26] we define the normalized functional measure Dϕ as
Dϕ ≡
∏
q∈Λ
dϕ̃q√
2πV
, (68)
dϕ̃qdϕ̃−q = 2 dReϕ̃q dImϕ̃q for q 6= 0 . (69)
Equation (6) can thus be conveniently rewritten as
Dϕ =
dϕ0√
2πV
∏
q∈Λ?
dReϕ̃q dImϕ̃q
πV
, (70)
where the sum in the right hand side runs over only the half Λ∗ of all the vectors of
the reciprocal lattice Λ (for instance those with qx > 0). The normalization constant
of equation (6) can thus be explicitly evaluated as
Nw =
∫
Dϕ exp
(
−1
2
〈
ϕ|w−1|ϕ
〉)
= exp
(
1
2
∑
q∈Λ
ln w̃(q)
)
∼exp
(
V
2
∫
q
ln w̃(q)
)
,
(71)
where w(1, 2) is positive and satisfies w(1, 2) = w(2, 1) ≡ w(x12) and (2π)3
∫
q
≡∫
d3q. It is worth noting that in this case we have trivially Nw−1 = 1/Nw. More
generally and with the same hypothesis we have the two useful identities
〈exp ((i) 〈ϕ|ω〉)〉w ≡ N−1
w
∫
Dϕ exp
(
−1
2
〈
ϕ|w−1|ϕ
〉
+ (i) 〈ϕ|ω〉
)
= exp
(
+(−)
1
2
〈ω|w|ω〉
)
, (72)
where ω is a real scalar field.
Now we define the “functional delta” distribution δF [λ] as
δF [λ] ≡
∫
Dω exp (i 〈ω|λ〉) , (73)
681
J.-M.Caillol, O.Patsahan, I.Mryglod
where both ω and λ are real scalar fields defined on C. Since
〈ω|λ〉 =
∫
C
d3x ω(x)λ(x) =
1
V
∑
q∈Λ
ω̃−qλ̃q , (74)
it follows from equation (6) that we have more explicitly
δF [λ] =
√
2πV δ
(
λ̃0
) ∏
q∈Λ?
[
πV δ
(
Reλ̃q
)
δ
(
Imλ̃q
)]
. (75)
Therefore ∫
Dλ δF [λ] = 1 , (76)
and, more generally ∫
Dλ F [λ] δF [λ− λ0] = F [λ0] , (77)
where F [λ] is some arbitrary functional of the field λ(x).
682
The collective variables representation of simple fluids
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J.-M.Caillol, O.Patsahan, I.Mryglod
Представлення колективних змінних для простих
плинів з точки зору статистико-польової теорії
Ж.-М.Кайоль 1 , О.Пацаган 2 , І.Мриглод 2
1 Лабораторія теоретичної фізики університету Парі-Сюд, 91405,
Орсей, Франція
2 Інститут фізики конденсованих систем НАН України,
79011 Львів, вул. Свєнціцького, 1
Отримано 18 липня 2005 р., в остаточному вигляді –
14 жовтня 2005 р.
Започаткований в роботах професора Ігоря Юхновського метод ко-
лективних змінних (КЗ) був успішно розвинутий до опису класичних
статистичних систем українською школою. В даній роботі основи
і структура представлення КЗ для рідин вивчається з точки зору
статистико-польового підходу і порівнюється з іншими точними
теоріями, що використовують перетворення Габбарда-Стратоновича.
Для випадку простого плину отримано вираз для вільної енергії в
обох версіях теорії і показано, що отримані результати співпадають
в кожному порядку петлевого розвинення. Результати, отримані в
однопетлевому наближенні є ідентичними до отриманих в набли-
женні хаотичних фаз. Проте, двопетлеве наближення дає новий
вираз для тиску і вільної енергії і є новим типом наближення.
Ключові слова: статистико-польова теорія, петлеве розвинення,
колективні змінні
PACS: 05.20.Jj, 05.70.Ce
684
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