STATISTICAL FIELD THEORY FOR A MULTICOMPONENT FLUID: THE COLLECTIVE VARIABLES APPROACH


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1 ЖУРНАЛ ФIЗИЧНИХ ДОСЛIДЖЕНЬ т., (007) с JOURNAL OF PHYSICAL STUDIES v., No. (007) p STATISTICAL FIELD THEORY FOR A MULTICOMPONENT FLUID: THE COLLECTIVE VARIABLES APPROACH O. Patsahan, I. Mryglod, J.M. Caillol Institute for Condensed Matter Physics, National Academy of Sciences of Uraine, Svientsitsii St., Lviv, UA 790, Uraine Laboratoire de Physique Théorique CNRS UMR 867, Bât. 0 Université de ParisSud 9405 Orsay Cedex, France (Received January 5, 007) Using the collective variables (CV) method the basic relations of statistical field theory of a multicomponent nonhomogeneous fluids are reconsidered. The corresponding CV action depends on two sets of scalar fields: the fields ρ α connected to the local density fluctuations of the αth species of particles and those ω α conjugated to ρ α. Explicit expressions for the CV field correlations and their relation to the density correlation functions are found. The perturbation theory is formulated and a mean field level (MF) of this theory is considered in detail. Key words: theory of liquids, mixtures, statistical field theory, functional methods, collective variables. PACS number(s): Fh, 0.70.Rr I. INTRODUCTION In recent years much attention has been focused on an issue of the phase transitions in multicomponent fluid mixtures, especially in ionic fluids. In spite of significant progress in this field, such systems are far from being completely understood. The investigation of complex models is of great importance in understanding the nature of critical and phase behavior of real ionic fluids which demonstrate both the charge and size asymmetry. The powerful tools for the study of multicomponent continuous systems are those based on the functional methods. In many cases the partition function of multicomponent models (see, e. g. []) can be reexpressed as a functional integral after performing the Hubbard Stratonovich transformation [, 3], a simple device proposed in the 950s. Nearly at the same time another method, the method of collective variables (CVs), that allows in an explicit way to construct a functional representation for manyparticle interacting systems was developed [4,5]. The method proposed initially in the 950s [4 6] for the description of the classical charged many particle systems and developed later for the needs of the phase transition theory [7 0] was in fact one of the first successful attempts to address the problems of statistical physics using the functional integral representation. Recently, the rigorous scalar field KSSHE (Kac Siegert Stratonovich Hubbard Edwards) theory [, ], which uses the Stratonovich Hubbard transformation, was developed to describe the phase equilibria in simple and ionic fluids. As was shown [3, 4], both groups of theories are in close relation. In [3] the CV representation of simple (onecomponent) fluids was reexamined from the point of view of statistical field theory. Our goal here is to derive the exact functional representation for the grand canonical partition function of a nonhomogeneous multicomponent fluid. We reformulate the method of CV in real space and derive the CV action that depends on two sets of scalar fields: the fields {ρ α } connected to the densities of the αth species and the fields {ω α } conjugate to {ρ α }. We study the correlations between these fields as well as their relations to the density correlations of the fluid. The CV method is based on: (i) the concept of collective coordinates being appropriate for the physics of the system considered (see, for instance, [5]) and (ii) the integral identity allowing to derive an exact functional representation for the configurational Boltzmann factor. Being applied to the continuous system the CV method uses the idea of the reference system (RS), one of the basic ideas of the liquid state theory [6]. The idea consists in the splitting of an interparticle interaction potential into two parts: the potential of a shortrange repulsion which describes the mutual impenetrability of the particles and the potential describing mainly the behaviour at moderate and large distances. The equilibrium properties of the system interacting via a shortrange repulsion are assumed to be nown. Therefore, this system can be regarded as the reference system. The remainder of the interaction is described in the phase space of CVs (collective coordinates). The fluid of hard spheres is most frequently used as the RS in the liquid state theory since its thermodynamic and structural properties are well nown. In this paper we derive the functional representation for the grand canonical partition function of a multicomponent fluid which includes both shortrange and longrange interactions. This paper is dedicated to Professor Ivan Vaarchu on the occasion of his 60th birthday 33
2 O. PATSAHAN, I. MRYGLOD, J.M. CAILLOL The paper is organized as follows. In Section II we obtain the exact expression for the functional of the grand partition function of a multicomponent nonhomogeneous mixture. Section III is devoted to the study of the correlations of CVs fields and their relation to the density correlation functions of a multicomponent fluid. In Section IV we formulate the perturbation theory. The MF level of the theory is considered in detail. II. THE FUNCTIONAL REPRESENTATION OF THE GRAND PARTITION FUNCTION A. The model Let us consider a classical mcomponent system consisting of N particles among which there exist N particles of species, N particles of species,... and N m particles of species m. The potential energy of the system is assumed to be of the form U N...N m = m N m U αβ (r α i, r β j ) + N ψ α (r α i ), () α,β i j α= i= where U αβ (r α i, rβ j ) denotes the interaction potential of two particles and the second term is the potential energy due to external forces. We present the pair interaction potential U αβ (r α i, rβ j ) as U αβ (r α i, r β j ) = v0 αβ(r α i, r β j ) + w αβ(r α i, r β j ), () where vαβ 0 (rα i, rβ j ) is a potential of a shortrange repulsion that can be chosen as an interaction between two hard spheres of respective diameters σ α and σ β, respectively we call the mcomponent system with the interaction vαβ 0 (rα i, rβ j ) a reference system (RS). The thermodynamic and structural properties of the RS are assumed to be nown. w αβ (r α i, rβ j ) is some potential which can describe both repulsion (e. g. soft repulsion) and attractive interactions. In general,w αβ (r α i, rβ j ) can be presented in the form w αβ (r α i, r β j ) = wr αβ(r α i, r β j ) + wa αβ(r α i, r β j ), where wαβ R (rα i, rβ j ) and wa αβ (rα i, rβ j ) are repulsive and attractive parts of the interaction potential w αβ (r α i, rβ j ). Since w αβ (r α i, rβ j ) are arbitrary in the core, i. e. for r σ αβ (σ α +σ β )/, we assume that the w αβ (r α i, rβ j ) have been regularized in such a way that their Fourier transforms w αβ () are wellbehaved functions of i and that w αβ (0) are finite quantities. We denote by Ω the domain of volume V occupied by particles. We present the potential energy U N...N m as follows U N...N m = V RS N...N m + ρ α w αβ ρ β + ψ α ρ α N α ν S α, (3) where N α ρ α (r) = δ(r r α i ) (4) i= is the microscopic density of the αth species in a given configuration and ψ α (r) is some external onebody potential acting on particles of species α. The following is the contribution from an mcomponent RS, να S is the selfenergy of the αth species notations are introduced in (3): V RS N...N m ν S α = w αα(0). (5) In (3) we have also introduced Dirac s bracets notations ρ α w αβ ρ β = dr α dr β ρ α(r α )w αβ (r α, r β ) ρ β(r β ), Ω ψ α ρ α = Ω dr α ψ α (r α ) ρ α (r α ). In the above formulas summation over repeated indices is meant. The system under consideration is at equilibrium in the grand canonical (GC) ensemble, β = / B T is the inverse temperature ( B Boltzmann constant), µ α is the chemical potential of the αth species. Then, the GC partition function can be written as Ξ[{ν α }] = N! N 0 exp N!... (dγ) (6) N m! N 0 N m 0 [ βvn RS...N m β ] ρ α w αβ ρ β + ν α ρ α, where ν α (r) = ν α +να S βψ α (r) is the local chemical potential of the αth species and ν α = βµ α 3 ln Λ α, Λ α = (πm α β /h ) / is the inverse de Broglie thermal wavelength. (dγ) = α dγ N α, dγ Nα = dr α dr α... dr α N α is the element of the configurational space of N particles. For a given volume V, Ξ[ν α ] is a function of the temperature T and a logconvex functional of the local chemical potentials ν α (r α ). B. The collective variables representation We introduce the collective variable ρ α (r) which describes the field of the number particle density of the αth species. To this end we use the identity ( ) exp ρ w ρ = Dρ δ F [ρ ρ ] ( ) exp ρ w ρ. (7) In (7) the functional delta function δ F [ρ] is defined as [7] 34
3 STATISTICAL FIELD THEORY FOR A MULTICOMPONENT FLUID:... δ F [ρ] Dω exp (i ω ρ), (8) Using (8) we can present the Boltzmann factor which does not include the RS interaction in the form ( ) exp ρ α w αβ ρ β = DρDω ( ) exp ρ α w αβ ρ β + i ω α {ρ α ρ α }. (9) Inserting equation (9) in definition (7) of the GC partition function Ξ[ν α ] one obtains ( Ξ [{ν α }] = Dρ exp β ) ρ α w αβ ρ β J [{ρ α, ν α }], (0) where the Jacobian J [{ρ α, ν α }] = Dω exp (i ω α ρ α ) Ξ RS [{ν α iω α }] () allows one for the passage from the microscopic variables r α (the Cartesian coordinates of particles) to the collective variables ρ α (r) (fields of the number density of particles). In () Ξ RS [{ν α iω α }] = Ξ RS [Z ] is the GC partition function of an mcomponent RS Ξ RS [Z ] = N 0 N! N 0 N!... N m 0 N m! (dγ) exp ( βvn RS )...N m Z (r i ), () where Z (r) = α Z α(r) = exp(να(r)). Zα is the activity of the species α associated with the dimensionless local chemical potential να(r) = ν α (r) iω α (r). It should be noted that J [{ρ α, ν α }] does not depend on the pair interaction w αβ (r i, r j ) but only on the GC partition function of the RS Ξ RS [Z ] which is supposed to be nown. Equation (0) can also easily be recast in the form of a standard statistical field theory, i. e. as Ξ [{ν α }] = DρDω exp ( H[{ν α, ρ α, ω α }]), (3) where the action H[{ν α, ρ α, ω α }] of the CV field theory reads as H [{ν α, ρ α, ω α }] = β ρ α w αβ ρ β i ω α ρ α i ln Ξ RS [{ν α iω α }]. (4) Functional integrals which enter the abovementioned formulas can be given a precise meaning in the case where the domain Ω is a cube of side L ( V = L 3 ) with periodic boundary conditions. This means that we restrict ourselves to the fields ρ α (r) and ω α (r) which can be written as Fourier series and ρ α (r) = L 3 ω α (r) = L 3 ρ,α e ir, (5) Λ ω,α e ir, (6) Λ where Λ = (π/l) Z 3 is the reciprocal cubic lattice. The reality of ρ α (and ω α ) implies that, for 0 ρ,α = ρ,α (ω,α = ω,α ), where the star means complex conjugation. Then the normalized functional measure Dρ (and Dω) is defined as [8] Dρ α Λ dρ,α πv (7) dρ,α dρ,α = drρ,α diρ,α, 0. (8) Equation (7) can be rewritten as Dρ = α dρ 0,α πv drρ q,α diρ q,α, (9) πv q Λ where the sum in the r. h. s. runs over only the half Λ of all the vectors of the reciprocal lattice Λ. We have for Dω, respectively Dω = α dω 0,α πv drω q,α diω q,α. (0) πv q Λ Now let us present action (4) for the isotropic interaction potential w αβ (r) as follows H[{ν α, ρ α, ω α }] = Φ αβ ()ρ,α ρ,β i α α,β ω,α ρ,α ln Ξ RS [{ ν α iω α }], () Here CV ρ,α describes the th mode of number density fluctuations of the αth species. Φ αβ () = β V w αβ(), where w αβ () is the Fourier transform of the interaction potential w αβ (r). In order to obtain another equivalent representation of the action H[{ν α, ρ α, ω α }] we first distinguish the chemical potential ν 0 α of the particle of the species α in the RS. To this end we put and obtain ν α iω α (r) = ν 0 α iω α (r) () iω α (r) = ν α + iω α (r) with ν α = ν α να. 0 As a result, the action (or the Hamiltonian) () can be written as 35
4 O. PATSAHAN, I. MRYGLOD, J.M. CAILLOL H[{ν α, ρ α, ω α }] = α ν α ρ =0,α + Φ αβ ()ρ,α ρ,β i α α,β ω,α ρ,α ln Ξ RS [{να 0 iω α }]. (3) We have obtained the exact representations of the grand partition function of a multicomponent system (eqs. (3) (4), () and (3)) in terms of CVs ρ α (r), which are number density fields of the αth species particles (or fluctuation modes of the αth species number density) We also stress that ρ α (r) and ω α (r) are two real scalar fields and that eqs. (3) (4) (as well as eqs. () and (3)) are valid for repulsive, attractive as well as arbitrary pair interactions. Besides the usual GC average A GC of a dynamic variable we introduce statistical field averages of the type A [{ρ α, ω α }] CV = Ξ [{ν α }] DρDω A [{ρ α, ω α }] exp ( H [{ν α, ρ α, ω α }]), (4) where A [{ρ α, ω α }] is a functional of the two CV fields ρ α and ω α. III. CORRELATION FUNCTIONS A. General relations Let us write some important relations. First, according to [0, ] we introduce the ordinary and truncated (or connected) density correlation functions n α...α n [{ν α }](,..., n) = ρ αi (i) = δ n Ξ[{ν α }] Ξ[ν α ] δν α ()... δν αn (n), = GC,T α...α n [{ν α }](,..., n) = δn log Ξ[{ν α }] δν α ()... δν αn (n). (5) Our notation emphasizes the fact that the correlation functions (connected and not connected) are functionals of the local chemical potential ν α (r) and functions of the coordinates (,,..., n) (r, r,..., r n ). For the sae of simplicity, we omit below the notations which indicate the functional dependence of the correlation functions of ν α (r). In standard textboos of liquid theory [6] the nparticle correlation functions are more frequently defined as functional derivatives of Ξ or log Ξ with respect to the activities Z α = exp(ν α ) rather than with respect to the local chemical potentials ρ α...α n (,,..., n) Zα ()Zα ()... Zα n (n) = δ n Ξ Ξ δzα ()δzα ()... δzα n (n), (6) ρ T α...α n (,,..., n) Zα ()Zα ()... Zα n (n) = δ n ln Ξ Ξ δzα ()δzα ()... δzα n (n). (7) We also define the partial distribution function g α...α n (,,..., n) and the partial correlation functions h α...α n (,,..., n) [0] and g α...α n (,,..., n) = ρ α...α n (,,..., n) ρ α ()ρ α ()... ρ αn (n) (8) h α...α n (,,..., n) = ρt α...α n (,,..., n) ρ α ()ρ α ()... ρ αn (n). (9) Expressions (5) and (8) (9) differ by the terms involving products of delta functions. For instance, for n = and for a homogeneous system one has G () αβ (, ) = ρ αρ β g αβ (, ) + ρ α δ αβ δ(, ), αβ (, ) = ρ α ρ β h αβ (, ) + ρ α δ αβ δ(, ), (30) where ρ α is the equilibrium density of the species α. Besides, functions h αβ (r) and g αβ (r) are connected by the relation h αβ (r) = g αβ (r) for a homogeneous case. 36
5 STATISTICAL FIELD THEORY FOR A MULTICOMPONENT FLUID:... B. CV field correlations Let us consider the correlations of fields ρ α and ω α. We start with the definitions of the correlation functions n [{ν α }](,..., n) = ρ αi (i), [{ν α }](,..., n) = i= n ω αi (i) i= and their truncated (connected) parts CV CV, (3),T (,..., n) = (,..., n) m<n G (m),t ρ α...ρ αm (i,..., i m ),,T (,..., n) = (,..., n) m<n G (m),t ω α...ω αm (i,..., i m ), (3) where the sum of products is carried out over all possible partitions of the set (,..., n) into subsets of cardinality m < n. a. Correlation functions. Now we introduce the modified partition function Ξ [{ν α, J α }] = DρDω exp ( H CV [{ν α, ρ α, ω α }] + J α ρ α ), (33) where J α is a real scalar field and Ξ [{ν α, J α }] is the generator of field correlation functions result, we have [{ν α }](,..., n) = Ξ [{ν α }],T [{ν α }](,..., n) = δn log Ξ [{ν α, J α }] [9]. As a δ n Ξ [{ν α, J α }] δj α ()... δj αn (n), Jαi =0. (34) δj α ()... δj αn (n) Jαi =0 The simplest way to obtain the relations between the G ρ (n) α...ρ αn [{ν α }](,..., n) and the density correlation functions is to start from definition (5). One has α...α n (,..., n) = δ n Ξ[{ν α }] Ξ[{ν α }] δν α ()... δν αn (n) = Ξ[{ν α }] DρDω exp +i ω α ρ α ) δn Ξ RS [{ν α iω α }] δν α ()... δν αn (n) = Ξ[{ν α }] where we introduce the notation w αβ = βw αβ and use the equality ( DρDω exp ρα w αβ ρ β ) ( ρα w αβ ρ β + i ωα ρ α (i) n δn Ξ RS [{ν α iω α }] δω α ()... δω αn (n), δ n Ξ RS [{ν α iω α }] δν α ()... δν αn (n) = (i)n δn Ξ RS [{ν α iω α }] δω α ()... δω αn (n). (35) Performing now n integral by parts yields ( α...α n (,..., n) = ( i)n DρDω exp ρα w δ Ξ[{ν α }] αβ ρ β + ln ΞRS [{ν α iω α }]) n exp (i ω α ρ α ) δω α ()... δω αn (n) n = ρ αi (i). i= We have just proved the expected result CV α...α n [{ν α }] (,..., n) = [{ν α }] (,..., n). (36) Obviously, the following relation is also valid for the truncated (connected) correlation functions,t α...α n [{ν α }] (,..., n) =,T [{ν α }] (,..., n). (37) 37
6 O. PATSAHAN, I. MRYGLOD, J.M. CAILLOL b. Correlation functions. Let us define the modified partition function Ξ [{ν α, J α }] = DρDω exp ( H CV [{ν α, ρ α, ω α }] + J α ω α ), (38) where J α is a real scalar field. Ξ [{ν α, J α }] is the generator of the functions and we thus have δ n Ξ [{ν α, J α }] [{ν α }](,..., n) = Ξ [{ν α ]} δj α ()... δj αn (n), Jαi =0,T [{ν α }](,..., n) = δn log Ξ [{ν α, J α }]. (39) δj α ()... δj αn(n) In order to relate the correlation functions J αi =0 and α...α n (,..., n) we perform the change of variables ρ α ρ α + ij α in Eq. (38). The functional Jacobian of the transformation is of course equal to unity and one obtains the relation ln Ξ [{ν α, J α }] = Jα w αβ J β + ln Ξ [ {ν α, iw αβ J β } ], (40) where the star means space convolution and Ξ is defined in (33). The idea is to perform now n successive functional derivatives of both sides of Eq. (40) with respect to J α. Since it follows from expression (34) that δ n log Ξ [{ν α, iwαβ J β}] = i n wα δj α ()... δj αn (n) (, α )... wα (n, nα n n )G ρ (n),t [{ν α α}](,..., n ),...ρα n one obtains Jαi =0 ω α () CV = i w α α (, ) ρ α ( ) CV, (4) ω α ω α [{ν α }] (, ) = w α α (, ) w α α (, )w α α (, ), ρ α ρα [{ν α}] (, ),,T [{ν α }] (,..., n) = i n w α α (, )... w α nα n (n, n ),T ρ α...ρα n [{ν α}] (,..., n ), n 3. IV. THE PERTURBATION THEORY A. Meanfield theory Let us consider the functional of the GC partition function (3) with the action given by Eq. () for the case of an isotropic interaction potential w αβ (r). At the MF level one has [9] Ξ MF [{ν α }] = exp( H[{ν α, ρ α, ω α }]), (4) where, for ρ α and ω α, the action is stationary, i. e. δ H [{ν α, ρ α, ω α }] = δ H [{ν α, ρ α, ω α }] δρ α δω α (ρα,ω α) (ρα,ω α) = 0. (43) Replacing the CV action by its expression () in Eq. (43) leads to implicit equations for ρ α and ω α : ρ α () = ρ MF α () = ρ RS α [{ ν α iω α }](), iω α () = Φ αβ (, )ρ β (), (44) where ρ RS α [{ ν α iω α }](i) denotes the αth species number density of the RS fluid with the chemical potentials { ν α iω α }. For a homogeneous system (44) can be rewritten in the form ρ α = ρ MF α = ρ RS α [{ ν α iω α }], iω α = ρ β Φαβ (0), (45) It follows from the stationary condition (43) that the MF density is given by ρ MF α [{ν α }] () = δ ln Ξ MF [{ν α }] δν α () = ρ RS α [{ν α iω α }] (), (46) 38
7 STATISTICAL FIELD THEORY FOR A MULTICOMPONENT FLUID:... and that the MF grand potential reads ln Ξ MF [{ν α }] = ln Ξ RS [{ν α iω α }] + β ρ MF α w αβ ρ MF β. (47) The MF KohnScham free energy of a multicomponent system defined as the Legendre transform βa MF [{ρ α }] = sup { ρ α ν α ln Ξ MF [{ν α }]}, (48) ν α has the following form in the MF approximation βa MF [{ρ α }] = βa RS [{ρ α }] (49) + β ρ α w αβ ρ β β dr w αα (0)ρ α (r). Using the formulas MF,α α [{ν α }] (, ) = δ ln Ξ MF [{ν α }] δν α () δν α (), C () MF,α α [{ρ α }] (, ) = δ βa MF [{ρ α }] δρ α () δρ α (), where (,,..., n) (r, r,..., r n ), one can get the wellnown expressions for the partial pair correlation and vertex (or direct correlation) functions in the MF approximation. C () MF,α α [{ρ α }] (, ) is obtained readily from expression (49) C () Ω (, ) = G(),T (, ) = C () RS,αβ (, ) w αβ(, ), where C () RS,αβ (, ) means the exact twopoint proper vertex of the RS fluid at the mean field density ρ MF α. The twopoint vertex function C () is connected to the usual direct correlation function of the theory of liquids c (, ) C () (, ) = c (, ) ρ α () δ αβδ(, ). In order to calculate (, ) we start with equation ρ RS α ρmf α [{ ν α iω α }]() (, ) = ν β () = ρrs α [{ ν α iω α }](). (50) ν β () [{ν α }] depends on ν α directly but also through the mean field ω α. Therefore, one has ρrs α [{ ν α iω α }]() (, ) = ν β () ω + ρrs α [{ ν α iω α }]() ω γ (3) ω γ (3) (5) νγ ν β (). Taing into account (44) and (50) we obtain finally (, ) = G(),T RS,αβ (, ) β MF,αγ (, 3)w γδ(3, 4)G T RS,δβ(4, ). (5) (5) can be rewritten in a matricial form as [] MF (, ) = G(),T (, ) RS MF (, 3)w(3, 4)G(),T (4, ), (53) where MF(RS) (i, j) denotes the matrix of elements MF(RS),αβ (i, j) and w(i, j) that of elements βw αβ(i, j). The formal solution of Eq. (53) is then MF (, ) = ( + w RS RS ) G (),T RS (, ), (54) where = δ αβ δ(, ) is the unit operator and the denotes a convolution in space. B. Beyond the MF approximation In order to tae into account fluctuations we present CVs ρ α and ω α in the form: ρ α () = ρ α + δρ α (), ω α () = ω α + δω α (), where the quantities with a bar are given by (44). The function ln Ξ RS [{ν α ; iω α }] in (4) can be presented in the form of the cumulant expansion ln Ξ RS [{ν α iω α }] = ( i) n d... dn M α...α n! n (,..., n)δω α ()... δω αn (n), (55) n α,...,α n where M α...α n (,..., n) is the nth cumulant defined by M α...α n (,..., n) = n ln Ξ RS [{ν α iω α }] ( i) n δω α ()... δω αn (n) δωαi =0. (56) 39
8 O. PATSAHAN, I. MRYGLOD, J.M. CAILLOL As is seen from (56) and (5), the nth cumulant is equal to the nparticle partial truncated (connected) correlation function at ω α = ω α. The expressions for the several cumulants given in the Cartesian coordinate phase space are as follows M α () = ρ α (), (57) M αα (, ) = ρ α ()ρ α ()h αα (, ) + ρ α ()δ αα δ(, )], (58) M αα α 3 (,, 3) = ρ α ()ρ α ()ρ α3 (3)h αα α 3 (,, 3) + ρ α ()ρ α ()h αα (, )δ αα 3 δ(, 3) +ρ α ()ρ α3 (3)h αα 3 (, 3)δ αα δ(, ) + ρ α ()ρ α3 (3)h αα 3 (, 3)δ αα δ(, ) +ρ α ()δ αα δ αα 3 δ(, )δ(, 3), (59) M αα α 3α 4 (,, 3, 4) = ρ α ()ρ α ()ρ α3 (3)ρ α4 (4)h αα α 3α 4 (,, 3, 4) + δ αiα l δ(i, l) + i,j,,l i,j,,l ρ αi (i)ρ αj (j)ρ α ()h αiα jα (i, j, ) ρ αi (i)ρ α ()h αiα (i, )δ αiα j δ α α l δ(i, j)δ(, l) + i,j,,l ρ αi (i)ρ αl (l) h αiα l (i, l)δ αiα j δ αiα δ(i, j)δ(i, ) + ρ α ()δ αα δ αα 3 δ αα 4 δ(, )δ(, 3)δ(, 4). (60) In the above formulas ρ αi (i) is the local density of the α i th species in the RS and h α...α n (,..., n) is the nparticle partial correlation function of an mcomponent RS, defined in the GC ensemble (see Eqs. (8) (9)): h αα (, ) = g αα (, ) g α ()g α (), h αα α 3 (,, 3) = g αα α 3 (,, 3) g αα (, )g α3 (3) g αα 3 (, 3)g α () g αα 3 (, 3)g α () + g α ()g α ()g α3 (3) h αα α 3α 4 (,, 3, 4) = g αα α 3α 4 (,, 3, 4).... (6) In the case of a homogeneous system a Fourier image of the nth cumulant can be presented in the form M α...α n (,..., n ) = ( N α,... N αn ) /n S α...α n (,..., n )δ n, (6) where S α...α n (,..., n ) is the nparticle partial structure factor of the RS. Substituting (55) in (3) one can obtain Ξ [{ν α }] = Ξ RS [{ν α iω α }] DδρDδω exp β δρ α w αβ δρ β + i δω α δρ α + ( i) n n! n } M α...α n (,..., n)δω α ()... δω αn (n) α,...,α n d... dn. (63) Integrating in (63) over δω αi (i) we have in the homogeneous case { Ξ[{ν α }] = Ξ MF Ξ (dδρ) exp L αβ ()δρ,α δρ,β + } H n (δρ α ). (64)! n 3 α,β 40
9 STATISTICAL FIELD THEORY FOR A MULTICOMPONENT FLUID:... c. Gaussian approximation. In the Gaussian approximation, which corresponds to taing into account in (64) only the terms with n (H n 0), we have L αβ () = C αβ (), where C αβ () are the Fourier transforms of the partial direct correlation functions. After integrating in (64) we arrive at the GPF of an m component system in the random phase approximation (RPA). Using the Gaussian averages one can develop a loop expansion of Ξ[{ν α }] in the CV representation as it was done recently for a onecomponent fluid [3]. V. CONCLUSION Using the CV method we have reconsidered the basic relations of statistical field theory of a multicomponent nonhomogeneous fluid that follow from this approach. Contrary to the KSSHE theory [] the corresponding CV action depends on two sets scalar fields: the field ρ α connected to the number density of the αth species particles and the field ω α conjugated to ρ α. We derive the explicit expressions for the CV field correlations and obtain their relation to the density correlation functions of a multicomponent system. Contrary to the theories based on the Stratonovich Hubbard transformation [,3], the CV representation has some important advantages which could be very useful for more complicated models of fluids. In particular, it is valid for an arbitrary pair potential (including the pair interaction w αβ (, ) which does not possess an inverse) and is easily generalized for the case of nbody interparticle interactions with n >. [] I. A. Vaarchu, I. F. Margolych, Teor. Mat. Fiz. 7, 46 (987). [] R. L. Stratonovich, Sov. Phys. Solid State, 84 (958). [3] J. Hubbard, Phys. Rev. Lett., 77 (959). [4] D. N. Zubarev, Dol. Acad. Nau SSSR 95, 757 (954). [5] I. R. Yuhnovsy, Zh. Esp. Teor. Fiz. 34, 379 (958). [6] D. Bohm, D. Pines, Phys. Rev. 8, 65 (95). [7] I. R. Yuhnovsii, Phase Transitions of the Second Order: Collective Variables Method (Singapore, World Scientific, 987). [8] I. R. Yuhnovsii, Proc. Stelov Inst. Math., 3 (99). [9] O. V. Patsagan, I. R. Yuhnovsii, Teor. Mat. Fiz. 83, 7 (990). [0] I. R. Yuhnovsii, O. V. Patsahan, J. Stat. Phys. 8, 647 (995). [] J.M. Caillol, Mol. Phys. 0, 67 (003). [] J.M. Caillol, J. Stat. Phys. 5, 46 (004). [3] J.M. Caillol, O. Patsahan, I. Mryglod, Physica A 368, 36 (006). [4] O. Patsahan, I. Mryglod, Condens. Matter Phys. 9, 659 (006). [5] I. R. Yuhnovsii, M. F. Holovo, Statistical Theory of Classical Equilibrium Systems (Nauova Duma, Kiev, 980). [6] J. P. Hansen, I. R. McDonald, Theory of simple liquids (Academic Press, 986). [7] J. W. Negele, H. Orland, Quantum ManyParticle Systems (Frontiers in Physics, AddisonWesley, 988). [8] F. J. Wegner, in Phase Transitions and Critical Phenomena, 6, edited by C. Domb, M. S. Green (Academic Press, New Yor, 976), p. 7. [9] J. ZinnJustin, Quantum Field Theory and Critical Phenomena (Clarendon Press, Oxford, 989). [0] G. Stell, in Phase Transitionss and Critical Phenomena, 5b, edited by C. Domb, M. S. Green (Academic Press, New Yor, 973), p. 05. [] G. Stell, in Equilibrium theory of Classical Fluids, edited by H. L. Frisch, J. L. Lebowitz (NorthHolland, Amsterdam, 98), p. 7. СТАТИСТИКОПОЛЬОВА ТЕОРIЯ ДЛЯ БАГАТОКОМПОНЕНТНОГО ПЛИНУ: МЕТОД КОЛЕКТИВНИХ ЗМIННИХ O. Пацаган, I. Мриглод, Ж.М. Кайоль Iнститут фiзики конденсованих систем НАН України вул. Свєнцiцького,, Львiв, 790, Україна Лабораторiя теоретичної фiзики, Унiверситет ПарiСюд, 9405, Орсей, Францiя Використовуючи метод колективних змiнних (КЗ), розглянуто основнi спiввiдношення статистикопольової теорiї багатокомпонентної неоднорiдної системи. Показано, що дiя, отримана в представленнi КЗ, залежить вiд двох наборiв скалярних полiв, а саме, полiв ρ α, якi описують флюктуацiї локальної густини частинок сорту α i полiв ω α, спряжених до ρ α. Знайдено явнi вирази для кореляцiйних функцiй полiв КЗ, а також їх зв язок iз кореляцiйними функцiями густини. Сформульовано теорiю збурень i детально розглянуто наближення середнього поля. 4
arxiv:0707.0192v1 [condmat.soft] 2 Jul 2007
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