Ginzburg-Landau-Wilson Hamiltonian for a multi-component continuous system: a microscopic description

Ginzburg-Landau-Wilson Hamiltonian for a multi-component continuous system: a microscopic description

Recently we proposed the microscopic approach to the description of the phase behaviour and critical phenomena in binary fluid mixtures. It was based on the method of collective variables (CV) with a reference system. The approach allowed us to obtain the functional of the Ginzburg-LandauWilson...

Повний опис

Збережено в:
Бібліографічні деталі
Дата:2002
Автор: Patsahan, O.V.
Формат: Стаття
Мова:English
Опубліковано: Інститут фізики конденсованих систем НАН України 2002
Назва видання:Condensed Matter Physics
Онлайн доступ:http://dspace.nbuv.gov.ua/handle/123456789/120661
Теги: Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
Назва журналу:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:Ginzburg-Landau-Wilson Hamiltonian for a multi-component continuous system: a microscopic description / O.V. Patsahan // Condensed Matter Physics. — 2002. — Т. 5, № 3(31). — С. 413-428. — Бібліогр.: 29 назв. — англ.

Репозитарії

Digital Library of Periodicals of National Academy of Sciences of Ukraine
id irk-123456789-120661
record_format dspace
spelling irk-123456789-1206612017-06-13T03:06:27Z Ginzburg-Landau-Wilson Hamiltonian for a multi-component continuous system: a microscopic description Patsahan, O.V. Recently we proposed the microscopic approach to the description of the phase behaviour and critical phenomena in binary fluid mixtures. It was based on the method of collective variables (CV) with a reference system. The approach allowed us to obtain the functional of the Ginzburg-LandauWilson (GLW) Hamiltonian expressed in terms of the collective variables (“density” variables). The corresponding set of collective variables included the variable connected with the order parameter. In this paper, based on the previous results, we construct the GLW Hamiltonian in the phase space of the “field” variables φˆ ~k (fluctuating fields) conjugate to the “density” variables. We apply the obtained GLW functional to the study of both the binary symmetrical mixture and the restricted primitive model. In the former case we consider the Gaussian approximation only and show that the obtained results are the same as those found previously using the CV method. In the latter case we calculate the phase diagram taking into account the powers of φˆ ~k higher than the second one Недавно ми запропонували мікроскопічний підхід до опису фазових переходів і критичних явищ в бінарних флюїдних сумішах, який базується на методі колективних змінних (КЗ) з виділеною системою відліку. Цей підхід дозволив нам отримати функціонал гамільтоніана Гінзбурга-Ландау-Вільсона (ГЛВ), який виражений в термінах колективних змінних (“густинних” змінних). Відповідний набір колективних змінних включав змінну, пов’язану з параметром порядку. В цій статті, базуючись на попередніх результатах, ми будуємо гамільтоніан ГЛВ у фазовому просторі “польових” змінних φˆ ~k (флуктуюючих полів), спряжених до “густинних” змінних. Ми застосовуємо отриманий функціонал ГЛВ до вивчення бінарної симетричної суміші і найпростішої іонної моделі. В першому випадку ми розглядаємо тільки гауссо-ве наближення і показуємо, що отримані результати є такі ж як і отримані раніше в рамках методу КЗ. В другому випадку ми обчислюємо фазову діаграму, враховуючи вищі степені φˆ ~k ніж друга. 2002 Article Ginzburg-Landau-Wilson Hamiltonian for a multi-component continuous system: a microscopic description / O.V. Patsahan // Condensed Matter Physics. — 2002. — Т. 5, № 3(31). — С. 413-428. — Бібліогр.: 29 назв. — англ. 1607-324X PACS: 05.70.Fh, 05.70.Jk, 65.10.+h DOI:10.5488/CMP.5.3.413 http://dspace.nbuv.gov.ua/handle/123456789/120661 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description Recently we proposed the microscopic approach to the description of the phase behaviour and critical phenomena in binary fluid mixtures. It was based on the method of collective variables (CV) with a reference system. The approach allowed us to obtain the functional of the Ginzburg-LandauWilson (GLW) Hamiltonian expressed in terms of the collective variables (“density” variables). The corresponding set of collective variables included the variable connected with the order parameter. In this paper, based on the previous results, we construct the GLW Hamiltonian in the phase space of the “field” variables φˆ ~k (fluctuating fields) conjugate to the “density” variables. We apply the obtained GLW functional to the study of both the binary symmetrical mixture and the restricted primitive model. In the former case we consider the Gaussian approximation only and show that the obtained results are the same as those found previously using the CV method. In the latter case we calculate the phase diagram taking into account the powers of φˆ ~k higher than the second one
format Article
author Patsahan, O.V.
spellingShingle Patsahan, O.V.
Ginzburg-Landau-Wilson Hamiltonian for a multi-component continuous system: a microscopic description
Condensed Matter Physics
author_facet Patsahan, O.V.
author_sort Patsahan, O.V.
title Ginzburg-Landau-Wilson Hamiltonian for a multi-component continuous system: a microscopic description
title_short Ginzburg-Landau-Wilson Hamiltonian for a multi-component continuous system: a microscopic description
title_full Ginzburg-Landau-Wilson Hamiltonian for a multi-component continuous system: a microscopic description
title_fullStr Ginzburg-Landau-Wilson Hamiltonian for a multi-component continuous system: a microscopic description
title_full_unstemmed Ginzburg-Landau-Wilson Hamiltonian for a multi-component continuous system: a microscopic description
title_sort ginzburg-landau-wilson hamiltonian for a multi-component continuous system: a microscopic description
publisher Інститут фізики конденсованих систем НАН України
publishDate 2002
url http://dspace.nbuv.gov.ua/handle/123456789/120661
citation_txt Ginzburg-Landau-Wilson Hamiltonian for a multi-component continuous system: a microscopic description / O.V. Patsahan // Condensed Matter Physics. — 2002. — Т. 5, № 3(31). — С. 413-428. — Бібліогр.: 29 назв. — англ.
series Condensed Matter Physics
work_keys_str_mv AT patsahanov ginzburglandauwilsonhamiltonianforamulticomponentcontinuoussystemamicroscopicdescription
first_indexed 2025-07-08T18:17:59Z
last_indexed 2025-07-08T18:17:59Z
_version_ 1837103768764678144
fulltext Condensed Matter Physics, 2002, Vol. 5, No. 3(31), pp. 413–428 Ginzburg-Landau-Wilson Hamiltonian for a multi-component continuous system: a microscopic description O.V.Patsahan Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii Str., 79011 Lviv, Ukraine Received April 13, 2002 Recently we proposed the microscopic approach to the description of the phase behaviour and critical phenomena in binary fluid mixtures. It was based on the method of collective variables (CV) with a reference system. The approach allowed us to obtain the functional of the Ginzburg-Landau- Wilson (GLW) Hamiltonian expressed in terms of the collective variables (“density” variables). The corresponding set of collective variables included the variable connected with the order parameter. In this paper, based on the previous results, we construct the GLW Hamiltonian in the phase space of the “field” variables φ̂~k (fluctuating fields) conjugate to the “density” vari- ables. We apply the obtained GLW functional to the study of both the binary symmetrical mixture and the restricted primitive model. In the former case we consider the Gaussian approximation only and show that the obtained results are the same as those found previously using the CV method. In the latter case we calculate the phase diagram taking into account the powers of φ̂~k higher than the second one. Key words: phase transition, a two-component continuous system, order parameter, fluctuating field PACS: 05.70.Fh, 05.70.Jk, 65.10.+h 1. Introduction Nowadays the theory of phase transitions and critical phenomena is well devel- oped in general. It enables us to obtain both universal and non-universal properties for many model systems. However, a number of questions still remains open, for instance, a criticality of ionic fluids, a Yang-Yang anomaly and others. In order to answer these questions it is important to obtain the explicit form of an effective Ginzburg-Landau-Wilson (GLW) Hamiltonian with coefficients related to the mi- croscopic parameters of the system. While for lattice models this task does not pose c© O.V.Patsahan 413 O.V.Patsahan special difficulties, the case of continuous systems is more complicated from a math- ematical point of view. Particularly, the latter case was studied in [1–5]. However, these considerations either did not go beyond the mean field approximation [2,3] or just took into account Gaussian fluctuations [1,4,5]. Another microscopic approach to the study of phase transitions was proposed in the late 80ies. First it was applied to a 3D Ising model [6] and then it was developed for a simple fluid near the gas-liquid critical point [7–9]. This theory has its origin in the approach based on a functional representation of a partition function in the collective variables (CV) space [10,11]. Its particular feature is a choice of the phase space in which the system is considered. Among the independent variables of this space there should be the ones connected with the order parameters. This phase space is formed by a set of CV. Each of them is a mode of density fluctuations cor- responding to the specificity of the model under consideration. In particular, for a magnetic system, the CVs are the variables connected with spin density fluctuation modes, while for a one-component fluid they are connected with particle density fluctuation modes. This approach allows one to determine, on microscopic grounds, the explicit form of the effective GLW Hamiltonian and then to integrate the parti- tion function in the neighbourhood of the phase transition point taking into account the renormalization group symmetry. As a result, non-classical critical exponents and analytical expressions for thermodynamic functions are obtained [6,9]. More recently this theory has been developed for a binary fluid mixture [12–17]. In contrast to lattice systems, the description of phase transitions in continuous systems has a number of the important peculiarities. On the one hand, as one usu- ally does in the liquid state theory, we should distinguish a reference system (RS) describing the behaviour at short distances. This will allow us to take into consider- ation the short-range and the long-range interactions simultaneously. On the other hand, the grand canonical ensemble (GCE) should be used in order to describe the processes relating to the phase transitions in multi-component fluids in which the composition fluctuations play a crucial role (e.g., the gas-gas and liquid-liquid equi- libria in binary fluid mixtures). The task of the development of the CV method for the case of the GCE is also caused by the problem of selecting the CV phase space which includes the variable connected with the order parameter. The set of CV can be called “density” variables. The proper choice of the CV phase space is very important for determining the order parameter appearing in complex systems below the phase transition point. When the “density” variable connected with the order parameter is known we can represent the GLW Hamil- tonian in the phase space of the “field” variables (fluctuating fields) conjugate to the “density” ones. This is the main purpose of the present paper. In section 2 us- ing the Hubburd-Schofield method [18] we obtain the functional representation of a grand partition function of a multi-component continuous system in the phase space of “field” variables. A two-component system is considered in section 3. The fields conjugate to the CV connected with the order parameters are determined here. The case of the binary symmetrical fluid is discussed. In section 4 we use this approach to the study of a restricted primitive model (RPM). The terms of the order higher 414 Ginzburg-Landau-Wilson Hamiltonian than the second one in the GLW Hamiltonian are taken into account. As a result, the phase diagram is obtained which demonstrates both the gas-liquid and charge ordering phase transitions. Section 5 contains some concluding remarks. 2. Functional representation of a grand partition function of a multi-component continuous system Let us consider a classical m-component continuous system of interacting parti- cles consisting of Na1 particles of species a1, Na2 particles of species a2, . . . and Nam particles of species am. The system is in volume V at temperature T . The interaction potential between particles has a pairwise additive character and may be presented as a sum of two terms: Uγδ(r) = ψγδ(r) + Φγδ(r), where ψγδ(r) is a potential of a short-range repulsion that characterizes the mutual impermeability of particles. In the simplest case it can be chosen as an interaction between the two hard spheres σγγ and σδδ. Φγδ(r) is an attractive part of the potential which dominates at large distances. Let us start with a grand partition function Ξ = ∞∑ Na1 =0 ∞∑ Na2 =0 . . . ∞∑ Nam=0 am∏ γ=a1 zNγ γ Nγ ! ∫ (dγ) exp  −β 2 ∑ γδ ∑ ij Uγδ(rij)   , (2.1) where (dΓ) = ∏ γ dΓNγ , dΓNγ = d~r γ 1 d~r γ 2 . . .d~r γ Nγ is an element of the configurational space of the γth species; zγ is the fugacity of the γth species: zγ = exp(βµ′ γ), µ′ γ = µγ + β−1 ln[(2πmγβ −1)3/2/h3]; β = 1/kBT , kB is the Boltzmann constant, T is temperature; mγ is mass of the γth species, h is the Planck constant. µ′ γ is determined from ∂ ln Ξ ∂βµ′ γ = 〈Nγ〉, where 〈Nγ〉 is the average number of the γth species. Let us introduce operators ρ̂~k,γ ρ̂~k,γ = Nγ ∑ j=1 exp(−i~k~r γ j ), which are Fourier transforms of the particle number density operators. Now we write the attractive part of the potential in the form ∑ i6=j Φγδ(rij) = 1 V ∑ ~k Φ̃γδ(k)(ρ̂~k,γ ρ̂−~k,δ − ρ̂0,γδγδ), where Φ̃γδ(k) is the Fourier transform of Φγδ(rij) Then we can represent Ξ as Ξ = ΞRSΞ1. (2.2) 415 O.V.Patsahan Here ΞRS is the grand partition function of the RS ΞRS = ∞∑ Na1 =0 ∞∑ Na2 =0 . . . ∞∑ Nam=0 am∏ γ=a1 exp (βµ(0) γ Nγ) Nγ! ∫ (dΓ) exp  −β 2 ∑ γδ ∑ ij ψγδ(rij)   , (2.3) where µ(0) γ is the chemical potential of the γth species in the RS. Ξ1 is as follows [19]: Ξ1 = 〈exp{β m∑ γ=1 µ(1) γ ρ̂0,γ − β 2V ∑ γ,δ ∑ ~k Φ̃γδ(k)ρ̂~k,γ ρ̂−~k,δ}〉RS . (2.4) Here 〈. . .〉RS means the average over RS. The chemical potential µ(1) γ = µ′ γ − µ(0) γ is determined from the equation ∂ ln Ξ1 ∂βµ (1) γ = 〈Nγ〉. We assume the thermodynamic and structural properties of the RS to be known. In the matrix representation (2.2)–(2.4) can be written in the form Ξ = ΞRS〈exp{βµ̂(1)ρ̂0 + 1 2 ∑ ~k α̂(k)ρ̂~kρ̂−~k}〉RS , (2.5) where ρ̂~k denotes a column-vector ρ̂~k =        ρ̂~k,1 ρ̂~k,2 ... ρ̂~k,m        , µ̂(1) denotes a row-vector µ̂(1) = ( µ (1) 1 , µ (1) 2 , . . . , µ(1) m ) and α̂(k) is a symmetrical (m×m) matrix with elements αγδ(k) = −(β/V )Φ̃γδ(k). We perform in (2.5) the Hubburd-Stratonovich transformation [20] ∞∫ −∞ N∏ i=1 dxi exp(−1 4 xiV −1 ij xj + yixi) = const × exp(yiVijyj), (2.6) where summation over repeated indices is implied, and V is any symmetric positive matrix. As a result, we obtain Ξ = ΞRS〈 ∏ ~k (2π)−1/2(det B̂(k))1/2 ∞∫ −∞ . . . ∞∫ −∞ m∏ γ=1 ∏ ~k dφ̂~k,γ × exp{−1 2 ∑ ~k B̂(k)φ̂~kφ̂−~k + ∑ ~k ρ̂~k(φ̂~k + δ~kβµ̂ (1))}〉RS , (2.7) 416 Ginzburg-Landau-Wilson Hamiltonian where φ̂~k is a row-vector with elements φ̂~k,γ. B̂(k) denotes an inverse matrix to α̂(k) with elements bij. After transformation φ̂′ ~k = φ̂~k + δ~kβµ̂ (1) in (2.7) we obtain Ξ = ΞRS ∏ ~k (2π)−1/2(det B̂(k))1/2 ∞∫ −∞ . . . ∞∫ −∞ s∏ m=1 ∏ ~k dφ̂′ ~k,γ × exp{−1 2 ∑ ~k B̂(k)φ̂′ ~k φ̂′ −~k + βµ̂(1)B̂(0)φ̂′ 0 − 1 2 B̂(0)(βµ̂(1))2}〈exp ∑ ~k ρ̂~kφ̂ ′ ~k 〉RS . (2.8) Here φ̂′ ~k has a meaning of a “field” variable conjugate to a “density” variable ρ̂~k. We can present the expression 〈exp ∑ ~k ρ̂~kφ̂ ′ ~k 〉RS in the form of a cumulant expan- sion [21,22]: 〈exp ∑ ~k ρ̂~kφ̂ ′ ~k 〉RS = exp ∑ n>1 D̂n(φ̂′), (2.9) where D̂n(φ̂′) = 1 n! ∑ ~k1,...,~kn M̂n(~k1, . . . , ~kn)φ̂ ′ ~k1 φ̂′ ~k2 . . . φ̂′ ~kn , (2.10) M̂n(~k1, . . . , ~kn) is a symmetrical m×m× . . .×m ︸ ︷︷ ︸ n matrix (see Appendix A). In general, the dependence of M γ1...γn n (~k1, . . . , ~kn) on wave vectors ~k1, . . . , ~kn is complicated. Since we are interested in the critical properties, the small-~k expansion of the cumulants can be considered. Hereafter we shall replace M γ1...γn n (~k1, . . . , ~kn) by their values in the long-wavelength limit M γ1...γn n (0, . . . , 0). The recurrence formulas for M γ1...γn n (0, . . . , 0) are obtained in [23]. As a result, we can present (2.8) in the form Ξ = ΞRS ∏ ~k (2π)−1/2(det B̂(k))1/2 ∞∫ −∞ . . . ∞∫ −∞ s∏ m=1 ∏ ~k dφ̂′ ~k,γ × exp{−1 2 ∑ ~k (B̂(k) − M̂2(0, 0))φ̂′ ~k φ̂′ −~k + (βµ̂(1)B̂(0) + M̂1(0))φ̂′ 0 − 1 2 B̂(0)(βµ̂(1))2} exp ∑ n>3 D̂n(φ̂′). (2.11) 3. A two-component system Let us consider a classical two-component system of interacting particles consist- ing of Na particles of a species and Nb particles of b species. We pass to new variables φ̂+ ~k and φ̂− ~k in (2.8)–(2.10) by means of the orthogonal linear transformation φ̂+ ~k = 1√ 2 (φ̂~k,a + φ̂~k,b), φ̂− ~k = 1√ 2 (φ̂~k,a − φ̂~k,b), 417 O.V.Patsahan As a result, we obtain for Ξ Ξ = ΞRS ∏ ~k (2π)−1/2(det B̂(k))1/2 ∫ (dφ̂+)(dφ̂−) × exp{−1 2 ∑ γ,δ bγδβµ (1) γ βµ (1) δ +M+φ̂+ 0 +M−φ̂− 0 − 1 2 ∑ ~k [Ṽ(k)φ̂+ ~k φ̂+ −~k + 2Ũ(k)φ̂+ ~k φ̂− −~k + W̃(k)φ̂− ~k φ̂− −~k ] + ∑ n>1 D̂n(φ̂+, φ̂−)]}. (3.1) Here the following notations are introduced: M+ = 1√ 2 [βµ(1) a (Ṽ(0) + Ũ(0)) + βµ (1) b (Ṽ(0) − Ũ(0))], (3.2) M− = 1√ 2 [βµ(1) a (W̃(0) + Ũ(0)) − βµ (1) b (W̃(0) − Ũ(0))], (3.3) Ṽ(k) = (b11(k) + b22(k) + 2b12(k))/2, W̃(k) = (b11(k) + b22(k) − 2b12(k))/2, Ũ(k) = (b11(k) − b22(k))/2, (3.4) D̂1(φ̂ +, φ̂−) = 1√ 2 ( M (0) 1 φ̂+ 0 + M (1) 1 φ̂− 0 ) , (3.5) D̂2(φ̂ +, φ̂−) = 1 2! ∑ ~k1,~k2 1 ( √ 2)2 ( M (0) 2 φ̂+ ~k1 φ̂+ ~k2 + 2M (1) 2 φ̂+ ~k1 φ̂− ~k2 + M (2) 2 φ̂− ~k1 φ̂− ~k2 ) δ~k1+~k2 , (3.6) D̂3(φ̂ +, φ̂−) = 1 3! ∑ ~k1...~k3 1 ( √ 2)3 ( M (0) 3 φ̂+ ~k1 φ̂+ ~k2 φ̂+ ~k3 + 3M (1) 3 φ̂+ ~k1 φ̂+ ~k2 φ̂− ~k3 + 3M (2) 3 φ̂+ ~k1 φ̂− ~k2 φ̂− ~k3 + M (3) 3 φ̂− ~k1 φ̂− ~k2 φ̂− ~k3 ) δ~k1+~k2+~k3 , (3.7) D̂4(φ̂ +, φ̂−) = 1 4! ∑ ~k1...~k4 1 ( √ 2)4 ( M (0) 4 φ̂+ ~k1 φ̂+ ~k2 φ̂+ ~k3 φ+ ~k4 + 4M (1) 4 φ̂+ ~k1 φ̂+ ~k2 φ̂+ ~k3 φ̂− ~k4 + 6M (2) 4 φ̂+ ~k1 φ̂+ ~k2 φ̂− ~k3 φ̂− ~k4 + 4M (3) 4 φ̂+ ~k1 φ̂− ~k2 φ̂− ~k3 φ̂− ~k4 + M (4) 4 φ̂− ~k1 φ̂− ~k2 φ̂− ~k3 φ̂− ~k4 ) δ~k1+~k2+~k3+~k4 . (3.8) The expressions for M (in) n are the same as those in [14]. Let us consider the Gaussian approximation of the functional of the grand par- tition function (n 6 2 in (3.1)): Ξ = ΞRS ∏ ~k (2π)−1/2(det B̂(k))1/2 ∫ (dφ̂+)(dφ̂−) 418 Ginzburg-Landau-Wilson Hamiltonian × exp{−1 2 ∑ γ,δ βµ(1) γ bγδβµ (1) δ + (M+ + M̄ (0) 1 )φ̂+ 0 + (M− + M̄ (1) 1 )φ̂− 0 − 1 2 ∑ ~k [A11(k)φ̂ + ~k φ̂+ −~k + 2A12(k)φ̂ + ~k φ̂− −~k + A22(k)φ̂ − ~k φ̂− −~k ]}, (3.9) where A11(k) = Ṽ(k) − M̄ (0) 2 , A22(k) = W̃(k) − M̄ (2) 2 , A12(k) = Ũ(k) − M̄ (1) 2 , (3.10) M̄ (in) n = 1√ 2 n M (in) n . In order to determine the “field” variables conjugate to the order parameters we diagonalize the square form in (3.9) by means of the orthogonal transformation: φ̂+ ~k = A(k)χ1,~k + B(k)χ2,~k , φ̂− ~k = C(k)χ1,~k + D(k)χ2,~k , (3.11) where A(k) = β1 √ 1 + β2 1 , B(k) = β2 √ 1 + β2 2 , C(k) = 1 √ 1 + β2 1 , D(k) = 1 √ 1 + β2 2 , β1,2 = − A22 − A11 ∓ √ (A11 − A22)2 + 4A2 12 2A12 . As a result, we get for the square form −1 2 2∑ s=1 ∑ ~k εs(k)χs,~kχs,−~k , where ε1,2(k) = A11(k) + A22(k) ∓ √ (A11(k) − A22(k))2 + 4A2 12(k). (3.12) In a general case only one of the quantities ε1(k) and ε2(k) is a critical one, no matter whether the system approaches the gas-liquid or mixing-demixing critical point [15,16]. 419 O.V.Patsahan We pass in (3.5)–(3.8) to new variables χs,~k (see Appendix B). As a result, the functional of the grand partition function (3.9) has the form Ξ = ΞRS ∏ ~k (2π)−1/2(det B̂(k))1/2 ∫ (dχ1)(dχ2) × exp{−1 2 ∑ γ,δ βµ(1) γ bγδβµ (1) δ + (AM+ + CM−)χ1,0 + (BM+ + DM−)χ2,0 − 1 2 2∑ s=1 ∑ ~k εs(k)χs,~kχs,−~k + ∑ n>3 D̂n(χ1, χ2)}, (3.13) In order to obtain the effective GLW Hamiltonian we can follow the program proposed in [13,16], namely: (1) to determine the critical branch εs(k) and the ordering fields χs,~k connected to it; (2) to integrate over the remaining χs,~k (irrelevant variables), using the Gaussian density measure as the basic one; (3) as a result of the integration performed in (2), to construct the functional including higher powers of the ordering fields χs,~k than the second power. As a result, we obtain the GLW Hamiltonian with coefficients which are the known functions of the microscopic parameters, temperature, concentration and density. A symmetrical mixture. We consider a symmetrical binary fluid mixture (SBFM), i.e. a system in which the two pure components “a” and “b” are identical and only interactions between the particles of dissimilar species differ: σaa = σbb = σ and Φaa(r) = Φbb(r) = Φ(r) 6= Φab(r). Notwithstanding its simplicity, the SBFM exhibits all the three types of two-phase equilibria which are observed in real binary fluids, namely: gas-liquid, liquid-liquid and gas-gas equilibria. In this case the functional of the grand partition function (3.13) is reduced to the form Ξ = ΞRS ∏ ~k (2π)−1/2(det B̂(k))1/2 ∫ (dφ̂+)(dφ̂−) × exp{−1 2 ∑ γ,δ βµ(1) γ bγδβµ (1) δ +M+φ̂+ 0 +M−φ̂− 0 − 1 2 ∑ ~k (ε1(k)φ̂ − ~k φ̂− −~k + ε2(k)φ̂ + ~k φ̂+ −~k ) + ∑ n>3 D̂n(φ̂+, φ̂−)}, (3.14) where ε1(k) = 1 α(k) − αab(k) − M̄ (2) 2 , ε2(k) = 1 α(k) + αab(k) − M̄ (0) 2 , (3.15) M+ = 1√ 2 ( 〈N〉 + 2βµ(1) a α(0) + αab(0) ) , M− ≡ 0 (3.16) and in (3.5)–(3.8) all cumulants M̄ (in) n ≡ 0 if in are odd [13]. 420 Ginzburg-Landau-Wilson Hamiltonian The nth cumulant M (in) n with in = 0 is connected with the nth structure factor of the one-component system Sn(0) [13]: M (0) n (0) = 〈N〉Sn(0). Structure factors Sn(0) with n > 2 can be obtained from S2(0) by means of a chain of equations for correlation functions [24]. Cumulants with in 6= 0 can be expressed in terms of M (0) n (0) (see formulae (4.8) in [13]): M̄ (2) n = M̄ (0) n−1√ 2 , M̄ (4) n = 3 2 M̄ (0) n−2 − 2√ 2 M̄ (0) n−3 . As a result, we can rewrite (3.14)–(3.16) in the form Ξ = CΞRS ∫ (dφ̂+)(dφ̂−) exp(−βH), (3.17) where C = ∏ ~k (2π)−1/2(det B̂(k))1/2 exp{−1 2 ∑ γ,δ βµ(1) γ bγδβµ (1) δ }. H is the Hamiltonian expressed in terms of the two fluctuating fields φ̂+ ~k and φ̂− ~k (in the approximation of the φ4 model): H = −M+φ̂+ 0 −M−φ̂− 0 + 1 2 ∑ ~k (ε1(k)φ̂ − ~k φ̂− −~k + ε2(k)φ̂ + ~k φ̂+ −~k ) − 1 3! ∑ ~k1...~k3 ( M̄ (0) 3 φ̂+ ~k1 φ̂+ ~k2 φ̂+ ~k3 + 3M̄ (2) 3 φ̂+ ~k1 φ̂− ~k2 φ̂− ~k3 ) δ~k1+~k2+~k3 − 1 4! ∑ ~k1...~k4 ( M̄ (0) 4 φ̂+ ~k1 φ̂+ ~k2 φ̂+ ~k3 φ+ ~k4 + 6M̄ (2) 4 φ̂+ ~k1 φ̂+ ~k2 φ̂− ~k3 φ̂− ~k4 + M̄ (4) 4 φ̂− ~k1 φ̂− ~k2 φ̂− ~k3 φ̂− ~k4 ) δ~k1+~k2+~k3+~k4 . (3.18) The fluctuating fields φ̂+ ~k and φ̂− ~k are conjugate to the fluctuating (or collective) densities, namely: ρ̂~k = ρ̂~k,a + ρ̂~k,b and ĉ~k = ρ̂~k,a − ρ̂~k,b . Lately [14], for the SBFM we obtained the expression for the effective GLW Hamiltonian expressed in terms of the collective variables (fluctuating densities) ρ~k and c~k. The two representations yield the same results in the Gaussian approximation (see [13]). As was shown, in such a system the two phase transitions can occur, namely, the gas-liquid and mixing- demixing phase transitions. The corresponding critical temperatures are determined from the equations (in the Gaussian approximation): ε1(k = 0) = 0, ε2(k = 0) = 0. In the above formulas an interaction in the system was not specified. The SBFM was studied in detail within the framework of the method of CV and the following model systems were considered: the hard-core Yukawa system [25], the hard-core Morse system [26] and the hard-core square-well system [14,17]. In the last case the non-universal critical properties were studied within the framework of the φ4 model. Below we shall use the approach proposed to the ionic fluid. 421 O.V.Patsahan 4. Restricted primitive model (RPM) We consider the simplest continuous model of an ionic fluid, namely, the restrict- ed primitive model (RPM). It consists of N = N+ +N− hard spheres of diameter σ with N+ carrying charges +q and N− (= N+) charges −q, in a medium of dielectric constant D. The interaction potential of the RPM has the form Uγδ(r) =    ∞, r < σ qγqδ Dr , r > σ , qi = ±q. We split the potential Uγδ(r) into short- and long-range parts using the Weeks- Chandler-Andersen partition [27]. As a result, we have ψγδ(r) = { ∞, r 6 σ 0, r > σ , Φγδ(r) =    qγqδ Dσ , r 6 σ qγqδ Dr , r > σ . The Fourier transform of Φγδ(r) has the form βρΦ̃γδ(k) = 24β∗η sin x x3 , where β∗ = βq2 Dσ , η = π 6 ρσ3 is fraction density, x = kσ. Let us start with formulas (3.14)–(3.16). One must remember that for the RPM the transformation (2.6) has a formal character only. Thus the transition from the SBFM to the RPM should be performed in the final formulas for thermodynam- ic functions. For the RPM the fluctuating fields φ̂+ ~k and φ̂− ~k are conjugate to the fluctuating densities ρ̂+ ~k and ρ̂−~k , the total number density and the charge density respectively. For this model the Gaussian approximation yields the only phase tran- sition, connected with the charge ordering. The corresponding boundary of stability is determined from the condition ε1(k) = 0 or from the equation T ∗ = −24η sin x x3 , T ∗ = 1 β∗ . (4.1) As a result, the equation of the charge ordering spinodal line is obtained T ∗ c (x∗, η) = −8η cos x∗ x∗2 , (4.2) where x∗ is determined from the condition tan x∗ = x∗ 3 , (4.3) which yields x∗ ' 4.0783. 422 Ginzburg-Landau-Wilson Hamiltonian In order to obtain the gas-liquid critical point we should take into consideration the order terms higher than the second order terms. As it was proposed in [13,16], we integrate in (3.17)–(3.18) (taking into account (3.15)–(3.16)) over φ̂− ~k with the Gaussian measure density as the basic one. We consider the integral I = ∫ (dφ̂−) exp  −1 2 ∑ ~k ε̃1(k)φ̂ − ~k φ̂− −~k    1 + 1 2! ∑ ~k φ̂− ~k φ̂− −~k A1 + . . .   , (4.4) where A1 = M̄ (2) 3 φ̂+ 0 + 1 2 M̄ (2) 4 ∑ ~k1 φ̂+ ~k1 φ̂+ − ~k1 . After the integration in (4.4)(taking into account the first term of the expansion) we obtain Ξ = CC ′ΞRS ∫ (dφ̂+) exp(−βH ′), (4.5) where the following notations are introduced C ′ = ∏ ~k √ 2π ε̃1(k) , (4.6) H ′ = −M̃+φ̂+ 0 + 1 2 ∑ ~k ε̃2(k)φ̂ + ~k φ̂+ −~k − 1 3! ∑ ~k1...~k3 M̄ (0) 3 φ̂+ ~k1 φ̂+ ~k2 φ̂+ ~k3 δ~k1+~k2+~k3 − 1 4! ∑ ~k1...~k4 M̄ (0) 4 φ̂+ ~k1 φ̂+ ~k2 φ̂+ ~k3 φ̂+ ~k4 δ~k1+~k2+~k3+~k4 , (4.7) M̃+ = M+ + M̄ (2) 3 ã(β), (4.8) ε̃2(k) = ε2(k) − M̄ (2) 4 ã(β), (4.9) where ã(β) = 1 2 ∑ ~k 1 ε1(k) . (4.10) Next, the shift is carried out in order to eliminate the cubic term in (4.7) φ̂+ ~k = φ̂′+ ~k + ∆δ~k , where ∆ = −M̄ (0) 3 /M̄ (0) 4 . Then (4.5)–(4.7) has the form Ξ = CC ′C ′′ΞRS ∫ (dφ̂′+) exp(−βH̄), (4.11) where C ′′ = −M̃+ M̄ (0) 3 M̄ (0) 4 − 1 2 ε̃2   M̄ (0) 3 M̄ (0) 4   2 − 1 8 (M̄ (0) 3 )4 (M̄ (0) 4 )3 . (4.12) 423 O.V.Patsahan H̄ = −M̄+φ̂+ 0 + 1 2 ∑ ~k ε̄2(k)φ̂ + ~k φ̂+ −~k − 1 4! ∑ ~k1...~k4 M̃ (0) 4 φ̂+ ~k1 φ̂+ ~k2 φ̂+ ~k3 φ+ ~k4 δ~k1+~k2+~k3+~k4 , (4.13) M̄+ = M̃+ + ε̃2 M̄ (0) 3 M̄ (0) 4 + 1 3 (M̄ (0) 3 )3 (M̄ (0) 4 )2 , (4.14) ε̄2(k) = ε̃2(k) + 1 2 (M̄ (0) 3 )2 M̄ (0) 4 . (4.15) The chemical potential µ (1) + is determined from the equation ∂ ln Ξ1 ∂βµ (1) + = 〈N〉. Provided the terms of the fourth order in (4.13) are neglected µ (1) + is equal to µ (1) + = − 1 2β M (0) 2 ã(β) − 1 3 (M (0) 3 )3/(M (0) 4 )2 M (0) 2 + 1 2 M (0) 3 ã(β) − 1 2 (M (0) 3 )2/M (0) 4 . (4.16) As a result, we obtain the equation for the gas-liquid spinodal curve ã(β) = −2 M (0) 2 M (0) 3 + M (0) 3 M (0) 4 , or 2 π ∫ ∞ 0 x2 sin x dx x3T ∗ + 24η sin x = 2 S2(0) S3(0) − S3(0) S4(0) , (4.17) where Sn(0) is the nth structure factor of the one-component hard-sphere system at k = 0. The phase diagram of the RPM (in the considered approximation) is shown in figure 1. The curve with the maximum is the gas-liquid spinodal calculated by (4.17).The Percus-Yevick approximation is used for S2(0). The straight line calculat- ed by (4.2)–(4.3) corresponds to the charge ordering phase transition. The gas-liquid critical point is located at T ∗ c = 0.0502 and ηc = 0.022. While the value for T ∗ c is in good agreement with the recent data of computer simulations [28,29] (T ∗ c ' 0.05), the critical density is underestimated (ηc ' 0.04). 5. Conclusions Based on our previous studies, we develop the approach which allows one to obtain the GLW Hamiltonian defined in the phase space of the fluctuating fields φ̂~k conjugate to the fluctuating densities connected with the order parameter. We use this approach to the study of both the SBFM and the RPM. In the former case we 424 Ginzburg-Landau-Wilson Hamiltonian 0,01 0,02 0,03 0,04 0,05 0,00 0,01 0,02 0,03 0,04 0,05 T * η Figure 1. The phase diagram of the RPM (see the text for the explanation). consider the Gaussian approximation only and show that the equations obtained for the phase instability boundaries are the same as those found in [13] within the framework of the CV method. For the RPM we calculate the phase diagram taking into account the powers of φ̂~k higher than the second one. The obtained value for the gas-liquid critical temperature correlates well with the MC simulation data. The proposed approach can also be used in the case when both long-range (Coulombic) and short-range (i.e., van der Waals) interactions are involved the mod- el simultaneously. This task will be considered elsewhere. Appendix A D̂1(φ̂) = ∑ ~k m∑ γ=1 M γ 1 φ̂~k,γ , D̂2(φ̂) = 1 2! ∑ ~k1 ~k2 ∑ γ1,γ2 M γ1,γ2 2 (k1, k2)φ~k1,γ1 φ~k2,γ2 , D̂3(φ̂) = 1 3! ∑ ~k1 ~k2 ~k3 ∑ γ1,γ2,γ3 M γ1,γ2,γ3 3 (k1, k2, k3)φ~k1,γ1 φ~k2,γ2 φ~k3,γ3 , D̂4(φ̂) = 1 4! ∑ ~k1...~k4 ∑ γ1,...,γ4 M γ1,...,γ4 4 (k1, . . . , k4)φ~k1,γ1 φ~k2,γ2 φ~k3,γ3 φ~k4γ4 . Here the nth cumulant M γ1,...,γn n (k1, . . . , kn) is connected with Sγ1...γn(k1, . . . , kn), the n-particle partial structure factor of the RS, by means of the relation M γ1...γn n (~k1, . . . , ~kn) = n √ Nγ1 . . . NγnS γ1...γn n (k1, . . . , kn)δ~k1+···+~kn , δ~k1+···+~kn is a Kronecker symbol. 425 O.V.Patsahan Appendix B D̂3(χ1, χ2) = 1 3! ∑ ~k1...~k3 1 ( √ 2)3 ( M̌ (0) 3 χ1,~k1 χ1,~k2 χ1,~k3 + 3M̌ (1) 3 χ1,~k1 χ1,~k2 χ2,~k3 + 3M̌ (2) 3 χ1,~k1 χ2,~k2 χ2,~k3 + M̌ (3) 3 χ2,~k1 χ2,~k2 χ2,~k3 ) δ~k1+~k2+~k3 , where M̌ (0) 3 = M (0) 3 A3 + 3M (1) 3 A2C + 3M (2) 3 AC2 + M (3) 3 C3, M̌ (1) 3 = M (0) 3 A2B + M (1) 3 A(AD + 2BC) + M (2) 3 C(2AD + BC) + M (3) 3 C2D, M̌ (2) 3 = M (0) 3 AB2 + M (1) 3 B(2AD + BC) + M (2) 3 D(AD + 2BC) + M (3) 3 CD2, M̌ (3) 3 = M (0) 3 B3 + 3M (1) 3 B2D + 3M (2) 3 BD2 + M (3) 3 D3. D̂4(χ1, χ2) = 1 4! ∑ ~k1...~k4 1 ( √ 2)4 ( M̌ (0) 4 χ1,~k1 χ1,~k2 χ1,~k3 χ1,~k4 + 4M̌ (1) 4 χ1,~k1 χ1,~k2 χ1,~k3 χ2,~k4 + 6M̌ (2) 4 χ1,~k1 χ1,~k2 χ2,~k3 χ2,~k4 + 4M̌ (3) 4 χ1,~k1 χ2,~k2 χ2,~k3 χ2,~k4 + M̌ (4) 4 χ2,~k1 χ2,~k2 χ2,~k3 χ2,~k4 ) δ~k1+~k2+~k3+~k4 , where M̌ (0) 4 = M (0) 4 A4 + 4M (1) 4 A3C + 6M (2) 4 A2C2 + 4M (3) 4 AC3 + M (4) 4 C4, M̌ (1) 4 = M (0) 4 A3B + M (1) 4 A2(AD + 3BC) + 3M (2) 4 AC(AD + BC) + M (3) 4 C2(3AD + BC) + M (4) 4 C3D, M̌ (2) 4 = M (0) 4 A2B2 + 2M (1) 4 AB(AD + BC) + M (2) 4 (A2D2 + B2C2 + 4ABCD) + 2M (3) 4 CD(AD + BC) + M (4) 4 C2D2, M̌ (3) 4 = M (0) 4 AB3 + M (1) 4 B2(3AD + BC) + 3M (2) 4 BD(AD + BC) + M (3) 4 D2(AD + 3BC) + M (4) 4 CD3, M̌ (4) 4 = M (0) 4 B4 + 4M (1) 4 B3D + 6M (2) 4 B2D2 + 4M (3) 4 BD3 + M (4) 4 D4. References 1. Chen X.S., Forstmann F. // J. Chem. Phys., 1992, vol. 97, p. 3696. 2. Ciach A., Stell G. // J Mol. Liq., 2002, vol. 87, p. 53. 3. Brilliantov N.V., Valleau J.P. // J. Chem. Phys., 1998, vol. 108, p. 1115. 4. Brilliantov N.V. // Phys. Rev. E, 1998, vol. 58, p. 2678. 5. Brilliantov N.V., Loskutov A.Yu., Malinin V.V. // Theoret. and Math. Phys., 2002, vol. 130, p. 145 (in Russian). 6. Yukhnovskii I.R. Phase Transitions of the Second Order: Collective Variables Method. Singapore, World Scientific, 1987. 426 Ginzburg-Landau-Wilson Hamiltonian 7. Yukhnovskii I.R. // Physica A, 1990, vol. 168, p. 999. 8. Yukhnovskii I.R. // Proceed. of the Steclov Inst. of Math., 1992, vol. 2, p. 223. 9. Yukhnovskii I.R., Idzyk I.M., Kolomietc V.O. // J. Stat. Phys., 1995, vol. 80, p. 405. 10. Zubarev D.N. // DAN USSR, 1964, vol. 95, p. 757 (in Russian). 11. Yukhnovskii I.R., Holovko M.F. Statistical Theory of Classical Equilibrium Systems. Kiev, Naukova Dumka, 1980 (in Russian). 12. Patsagan O.V., Yukhnovskii I.R. // Theoret. Math. Phys., 1990, vol. 83, p. 387. 13. Yukhnovskii I.R., Patsahan O.V.// J. Stat. Phys., 1995, vol. 81, p. 647. 14. Patsahan O.V., Kozlovskii M.P., Melnyk R.S. // J. Phys.: Condens. Matter, 2000, vol. 12, p. 1595. 15. Patsahan O.V., Patsahan T.M. // J. Stat. Phys., 2001, vol. 105, p. 287. 16. Patsahan O.V.// Physica A, 1999, vol. 272, p. 358. 17. Patsahan O.V., Melnyk R.S., Kozlovskii M.P. // Condens. Matter Phys., 2001, vol. 4, p. 235. 18. Hubburd J., Schofield P. // Phys. Lett. A, 1972, vol. 40, p. 245. 19. Patsahan O.V., Holovko M.F. On the theory of phase transitions of fluids in porous media. Preprint of the Institute for Condensed Matter Physics, ICMP–01–08U, Lviv, 2001, 24 p. (in Ukrainian). 20. Hubburd J. // Phys. Rev. Lett., 1959, vol 3, p. 77; Stratonovich R.L. // Sov. Phys. Dokl., 1957, vol. 2, p. 416. 21. Kubo R. // J. Phys. Soc. Jpn., 1962, vol. 17, p. 100. 22. Shiryajev A.N. Probability. Moscow, Nauka, 1989 (in Russian). 23. Patsahan O.V. // Ukr. Fiz. Zh., 1996, vol. 41, p. 877 (in Ukrainian). 24. Stell G. The Equilibrium Theory of Classical Fluids. New York, Benjamin, 1964. 25. Patsahan O.V. // Condens. Matter Phys., 1995, No. 5, p. 124. 26. Kozlovskii M.P., Patsahan O.V., Melnyk R.S. // Ukr. Fiz. Zh., 2000, vol. 45, p. 381 (in Ukrainian). 27. Weeks J.D., Chandler D., Andersen H.C. // J. Chem. Phys., 1971, vol. 54, p. 5237. 28. Orkoulas G., Panagiotopoulos A.Z. // J. Chem. Phys., 1999, vol. 110, p. 1581. 29. Caillol J.-M., Levesque D., Weis J.-J. Critical behavior of the restricted primitive model revisited. Preprint cond-mat/0201301 v1. 427 O.V.Patsahan ������� � �� �����������������������������! "���$#%�$����&'� � ��(����)��# �+* �������, ��-�����.)����/��, ,��0213��/�.)/���/���42��0215(�0�(, /��76 ����-$����(�-���.��98���0�:3.�� #;1�� # <>=@?A=@B+C2D,C2EFC2G H I�JLK�M�K�NOK�PRQ STMVUOMRU�WXIOY ZVI�J�WX[L\VI]M_^�J]M]JLKLZV`>acb�a%d UOe]\�f�I]Mcg hVikj$lkl2m�n_[�Q [kgL[FNOo pOq�[_rkI]sVQ sVn_U�WXtuW gTl v+wkx y�z�{�|V}>~������k� wk|k���c��� ��x�� a,Z_Y�\V[�I�W�`�M S_\V�kekWX�TWXILNT[L\�oXMA`�Q UOekWVJ]U�WX�kQ ��I]MV�'�kQ Y�^TQ Y%Y WAWX�kM]JLN�P+\TS�WX[�M_^ �TZVekZ_^�W�Y,Q [%QTUOe�M�K�M]��I]M_^A�k[�MV��['�XQ I�\Ve�I]M_^AP�oX��f Y,I]M_^%JLNT`�Q �+\_^Vgk�kUOMV���k\�� SON�r_K�nOJ���I�\�`�Z�KLW�Y,QTU�WToVZVU�K�MV[�I]M_^%ST`�Q I]I]M_^����V�2�$S�[�M_Y,Q oVZVI�WX��J]M]JLKLZV`�WX� [�Q YXoXQ U�N_p��2ZV�A�kQ Y�^TQ Y%Y WkST[_WToXMV[+I�\V`�WTK�e�MV`2\�K�M+P�NTI]UOsVQ WXI�\�o�t \V`�Q oXn¡KLWXI]Q \VI�\ ¢�Q I�S��TNTe�t \��@m)\VIOY�\�N]�¡£,Q oXnOJ�WXI�\'��¢¤m�£��¤g��kUOMV� [�MVe]\_¥�ZVI]MV�R[2KLZVe�`�Q I�\_^�U�WToVZVUL� K�MV[�I]M_^)ST`�Q I]I]M_^���¦ut�N�JLK�MVI]I]M_^]§VST`�Q I]I]M_^]�¤pO£,Q Y,�TWX[�Q Y,I]MV�+I�\T�XQ e�U�WToVZVU�K�MV[�I]M_^ ST`�Q I]I]M_^�[�U�oX���O\V[�ST`�Q I]ILN_gF�TWX[k¨ ��S_\VILN)S���\Ve]\V`�Z�K�ekWX`��TWXe]�LY,U�N_p�£�sVQ ��JLKO\�K�� K�Q g��k\TSONT����M]J]n�I�\"�TWX�TZVekZ_Y,I]Q ^©ekZTSONOoXn¡KO\�KO\_^Vg `�M��TNFYXN�rk`�W5t \V`�Q oXn¡KLWXI]Q \VI ¢¤m�£�NRP+\TS�WX[_WX`,NA�kekWVJLKLWXe�Q�¦u�TWToXnOWX[�M_^]§�ST`�Q I]I]M_^ φ̂~k ��P�oTNTU�K�NT������M_^��TWc� oXQ [T�¤g�J]�ke]�L¥�ZVI]M_^�Y W+¦ut�N�JLK�MVI]I]M_^]§kST`�Q I]I]M_^VpFª�M�S_\]JLKLWVJ�WX[FN�rk`�W�WTK�e�MV`2\VI]MV� P�NTI]UOsVQ WXI�\�o;¢¤m�£�Y W�[�MV[_�LZVI]I����XQ I�\Ve�I�W]f�J]MV`�Z�K�e�M]��I�W]f�JLNT`�Q �RQuQ I�\V�V�kekWVJLK�Q � �;W]f_Q WXI]I�W]fL`�W�Y Z�oXQ pF£%�TZVe��;WX`,N�[�MV��\_Y,U�N�`�M+ekWkSTt9ok�LY�\]rk`�W)K�Q oXn_UOM�t \�N�J�J�Wc� [_Z�I�\T�ToXM_¥�ZVI]I��+Qu�TWXU!\TSON�rk`�W g��RW�WTK�e�MV`2\VI]Q ekZTSONOoXn¡KO\�K�M�r2KO\VUOQ�¥��kU$Q WTK�e�M�� `2\VI]Q�e]\VI]Q �;Z'[�e]\V`�U!\_^�`�Z�KLW�YXN;�V�)p�£'Y,e�NTtuWX`,N+[�MV��\_Y,U�N+`�M;WV�V��M]JLoX��rk`�W P+\TS�WX[FN�Y,Q \Vt�e]\V`,N_gO[�e]\_^�WX[FNT����M [�MV�AQ�JLKLZV�TZVI]Q φ̂~k I]Q ¥�Y,e�NTt \cp «c¬,­'®]¯ °V±O²]¬ ¯ °k³$´kµ�¶]·�¸]¹�ºV» ¼�½_¾$½O¿�À Á2Â�Á�¹]¸]ÃL¸]Ä�¼�¸]Å�½TÅXÆ¡Åk¶�Å�½T¼�½_¾$½_¾�¹�Åk¶ Ç º Ç ÆȽTÄ�¶kÂ�¼k¶�¾�¶]Ä�½VƤ¾%¼�¸�¾XÉ�Á�Ã�ÊFÂ]µ�Ë�ÊLÃ_ÆuÊLÌ�ÌRÍF½ ¼�¸_Ë�½ ÎVÏ)Ð�Ñ�´TÒ]ÓVÔÖÕ�ÒkÔØ×�ÙVÂ�Ò]ÓVÔÖÕ�ÒkÔ Ú_Û�Â�ÜkÓVÔ!Ý!ÒkÔ�ÞTÙ 428

Схожі ресурси