Yukawa fluids: a new solution of the one component case

In recent work a solution of the Ornstein-Zernike equation for a general Yukawa closure for a single component fluid was found. Because of the complexity of the equations a simplifying assumption was made, namely that the main scaling matrix Γ had to be diagonal. While in principle this is mathe...

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Дата:	2003
Автори:	Blum, L., Hernando, J.A.
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Мова:	English
Опубліковано:	Інститут фізики конденсованих систем НАН України 2003
Назва видання:	Condensed Matter Physics
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/120754
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Цитувати:	Yukawa fluids: a new solution of the one component case / L. Blum, J.A. Hernando // Condensed Matter Physics. — 2003. — Т. 6, № 3(35). — С. 447-458. — Бібліогр.: 20 назв. — англ.

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spelling	irk-123456789-1207542017-06-13T03:02:47Z Yukawa fluids: a new solution of the one component case Blum, L. Hernando, J.A. In recent work a solution of the Ornstein-Zernike equation for a general Yukawa closure for a single component fluid was found. Because of the complexity of the equations a simplifying assumption was made, namely that the main scaling matrix Γ had to be diagonal. While in principle this is mathematically correct, it is not physical because it will violate symmetry conditions when different Yukawas are assigned to different components. In this work we show that by using the symmetry conditions the off diagonal elements of Γ can be computed explicitly for the case of two Yukawas solving a quadratic equation: There are two branches of the solution of this equation, and the physical one has the correct behavior at zero density. The non-physical branch corresponds to the solution of the diagonal approximation. Although the solution is different from the diagonal case, the excess entropy is formally the same as in the diagonal case. В сучасних роботах знайдено розв’язок рівняння Орнштейна-Церніке для простого однокомпонентного плину в рамках узагальнених умов замикання Юкави. У зв’язку зі складністю рівнянь було зроблено припущення про те, що головна скейлінгова матриця Γ має бути діагональною. Хоча це математично правильно, фізично це порушує умови симетрії при співставлянні різних потенціалів Юкави різним компонентам. В цій роботі ми показуємо, що використовуючи умови симетрії, недіагональні елементи матриці Γ можуть бути точно обчислені, розв’язуючи квадратне рівняння для двох потенціалів Юкави. Існують два розв’язки цього рівняння, але тільки один з них має фізично правильну поведінку при нульовій густині. Нефізичний розв’язок відповідає розв’язку з діагональною апроксимацією. I хоча наш розв’язок відрізняється від того, що в діагональному випадку, надлишкова ентропія формально залишається такою ж. 2003 Article Yukawa fluids: a new solution of the one component case / L. Blum, J.A. Hernando // Condensed Matter Physics. — 2003. — Т. 6, № 3(35). — С. 447-458. — Бібліогр.: 20 назв. — англ. 1607-324X PACS: 61.20.Gy DOI:10.5488/CMP.6.3.447 http://dspace.nbuv.gov.ua/handle/123456789/120754 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	In recent work a solution of the Ornstein-Zernike equation for a general Yukawa closure for a single component fluid was found. Because of the complexity of the equations a simplifying assumption was made, namely that the main scaling matrix Γ had to be diagonal. While in principle this is mathematically correct, it is not physical because it will violate symmetry conditions when different Yukawas are assigned to different components. In this work we show that by using the symmetry conditions the off diagonal elements of Γ can be computed explicitly for the case of two Yukawas solving a quadratic equation: There are two branches of the solution of this equation, and the physical one has the correct behavior at zero density. The non-physical branch corresponds to the solution of the diagonal approximation. Although the solution is different from the diagonal case, the excess entropy is formally the same as in the diagonal case.
format	Article
author	Blum, L. Hernando, J.A.
spellingShingle	Blum, L. Hernando, J.A. Yukawa fluids: a new solution of the one component case Condensed Matter Physics
author_facet	Blum, L. Hernando, J.A.
author_sort	Blum, L.
title	Yukawa fluids: a new solution of the one component case
title_short	Yukawa fluids: a new solution of the one component case
title_full	Yukawa fluids: a new solution of the one component case
title_fullStr	Yukawa fluids: a new solution of the one component case
title_full_unstemmed	Yukawa fluids: a new solution of the one component case
title_sort	yukawa fluids: a new solution of the one component case
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2003
url	http://dspace.nbuv.gov.ua/handle/123456789/120754
citation_txt	Yukawa fluids: a new solution of the one component case / L. Blum, J.A. Hernando // Condensed Matter Physics. — 2003. — Т. 6, № 3(35). — С. 447-458. — Бібліогр.: 20 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT bluml yukawafluidsanewsolutionoftheonecomponentcase AT hernandoja yukawafluidsanewsolutionoftheonecomponentcase
first_indexed	2025-07-08T18:31:25Z
last_indexed	2025-07-08T18:31:25Z
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fulltext	Condensed Matter Physics, 2003, Vol. 6, No. 3(35), pp. 447–458 Yukawa fluids: a new solution of the one component case L.Blum 1 , J.A.Hernando 2 1 Department of Physics P.O. Box 23343, University of Puerto Rico, Rio Piedras, PR 00931-3343, USA 2 Department of Physics, Comision Nacional de Energia Atomica, Av. del Libertador 8250, 1429 Buenos Aires, Argentina Received March 12, 2003, in final form June 23, 2003 In recent work a solution of the Ornstein-Zernike equation for a general Yukawa closure for a single component fluid was found. Because of the complexity of the equations a simplifying assumption was made, namely that the main scaling matrix Γ had to be diagonal. While in principle this is mathematically correct, it is not physical because it will violate symmetry conditions when different Yukawas are assigned to different components. In this work we show that by using the symmetry conditions the off diago- nal elements of Γ can be computed explicitly for the case of two Yukawas solving a quadratic equation: There are two branches of the solution of this equation, and the physical one has the correct behavior at zero density. The non-physical branch corresponds to the solution of the diagonal ap- proximation. Although the solution is different from the diagonal case, the excess entropy is formally the same as in the diagonal case. Key words: Yukawa fluids, mean spherical approximation, entropy, scaling approximations PACS: 61.20.Gy 1. Introduction There are many problems of practical and academic interest that can be formu- lated as closures of either a scalar or matrix Ornstein-Zernike (OZ) equation. These closures can always be expressed by a sum of exponentials, which do form a complete basis set if we allow for complex numbers [1–5]. Another interesting application is to fluids with non-spherical molecules, like water [6,7]. While the initial motivation was to study simple approximations like the Mean Spherical (MSA) [8] or Generalized Mean Spherical Approximation (GMSA) [9– 13], the availability of closed form scaling solutions [14–16], an equation makes it possible to write down analytical solutions for any given approximation that can be c© L.Blum, J.A.Hernando 447 L.Blum, J.A.Hernando formulated by writing the direct correlation function c(r) outside the hard core as c(r) = M ∑ n=1 K(n) e −zn(r−σ) r = M ∑ n=1 K(n) e −znr r . (1) In this equation K(n) is the interaction/closure constant used in the general solution first found by Blum and Hoye (which we will call BH78) [17], while K(n) is the definition used in the later general solution by Blum, Vericat and Herrera (BVH92 in what follows) [16]. In this work we will use the more common notation of BVH92. The case of factored interactions discussed by Blum, [14,18] was simplified by Ginoza [15,19,20] for the one Yukawa case. We have K(n) = K(n)δ(n)δ(n), K(n) = K(n)d(n)d(n), (2) where δ(n) = d(n)e−znσ/2. (3) The general solution of this problem was formulated by Blum, Vericat and Her- rera [16] in terms of a scaling matrix Γ. The full solution was recently given by Blum et al. [1,2,4]. For only one component the matrix Γ was assumed to be diagonal and explicit expressions for the closure relations for any arbitrary number of Yukawa exponents M were obtained. Then, the solution is remarkably simple in the MSA since explicit formulas for the thermodynamic properties are obtained. The diagonal assumption is however not correct for mixtures, even if they are of the same hard core diameter. It is also more natural to solve the full problem with no diagonal assumption and compute the nondiagonal terms of the Γ matrix using the symmetry relations: In this work we do precisely the following: the sym- metry relations are used to calculate explicitly the off diagonal terms of Γ in the 1 component, 2 Yukawa case. 2. Summary of the previous work We study the Ornstein-Zernike (OZ) equation hij(12) = cij(12) + ∑ k ∫ d3hik(13)ρkckj(32), (4) where hij(12) is the molecular total correlation function and cij(12) is the molecular direct correlation function, ρi is the number density of the molecules i, and i = 1, 2 is the position ~ri, r12 = \|~r1 − ~r2\| and σij is the distance of the closest approach of two particles (or species) i, j. The direct correlation function is cij(r) = M ∑ n=1 K (n) ij e−zn(r−σij) r , r > σij , (5) 448 Yukawa fluids: a new solution of the one component case and the pair correlation function is hij(r) = gij(r) − 1 = −1, r 6 σij . (6) We use the Baxter-Wertheim (BW) factorization of the OZ equation [ I + ρH̃(k) ] [ I − ρC̃(k) ] = I, (7) where I is the identity matrix, and we have used the notation H̃(k) = 2 ∫ ∞ 0 dr cos(kr)J(r), (8) C̃(k) = 2 ∫ ∞ 0 dr cos(kr)S(r). (9) The matrices J and S have matrix elements Jij(r) = 2π ∫ ∞ r ds shij(s), (10) Sij(r) = 2π ∫ ∞ r ds scij(s), (11) [ I − ρC̃(k) ] = [ I − ρQ̃(k) ] [ I − ρQ̃T (−k) ] , (12) where Q̃T (−k) is the complex conjugate and transpose of Q̃(k). The first matrix is non-singular in the upper half complex k-plane, while the second is non-singular in the lower half complex k-plane. It can be shown that the factored correlation functions must be of the form Q̃(k) = I − ρ ∫ ∞ λji dreikrQ̃(r), (13) where we used the following definitions σji = 1 2 (σj + σi), λji = 1 2 (σj − σi), (14) S(r) = Q(r) − ∫ dr1Q(r1)ρQ T(r1 − r). (15) Similarly, from equation (12) and equation (7) we get, using the analytical prop- erties of Q and Cauchy’s theorem J(r) = Q(r) + ∫ dr1J(r − r1)ρQ(r1). (16) The general solution is discussed in [3,15,18], and yields qij(r) = q0 ij(r) + M ∑ n=1 D (n) ij e−znr, λji < r, (17) 449 L.Blum, J.A.Hernando q0 ij(r) = (1/2)Aj[(r − σj/2)2 − (σi/2)2] + βj[(r − σj/2) − (σi/2)] + M ∑ n=1 C (n) ij e−znσj/2[e−zn(r−σj/2) − e−znσi/2], λji < r < σij. (18) From here X (n) i − σiφ0(znσi)Π (n) i = δ (n) i − 1 2 σiφ0(znσi) ∑ ` ρ`β 0 `X (n) ` − σ3 i z 2 nψ1(znσi)∆ (n), (19) or ∑ ` ρ` { −Ĵ (n) j` Π (n) ` + Î (n) j` X (n) ` } = δ (n) j . (20) Here we use the notation of our previous work [3] β0 ` = πσ` ∆ . 2.1. The Laplace transforms From equation 16 we obtain the Laplace transform of the pair correlation function 2π ∑ ` g̃i`(s)[δ Kr `j − ρ`q̃`j(is)] = q̃0 ij ′(is), (21) where q̃0 ij ′(is) = ∫ ∞ σij dre−sr[q0 ij(r)] ′ = [( 1 + sσi 2 ) Aj + sβj ] e−sσij s2 − ∑ m zm s+ zm e−(s+zm)σijC (m) ij . (22) The Laplace transform of equations (17) and (18) yields esλji q̃ij(is) = σ3 i ψ1(sσi)Aj + σ2 i φ1(sσi)βj + ∑ m 1 s + zm [(C (m) ij +D (m) ij )e−zmλji − C (m) ij e−zmσjizmσiφ0(sσi)C (m) ij e−zmσji ]. (23) This result will be used below. Another important relation deduced from equation [21]by setting s = zn is −Π (n) j = ∑ m M̃nma (m) j , (24) where M̃nm = 1 zn + zm ∑ ` ρ` [ X (n) ` (zmX (m) ` − Π (m) ` ) +X (m) ` Π (n) ` ] . (25) 450 Yukawa fluids: a new solution of the one component case 3. The general one component closure The closure relation (BVH92 [16] ) is, for only one component 2πK(n)δ(n) zn + a(n)I(n) − ∑ m 1 zn + zm ρa(n)a(m) [ J (n)[Π(m) − zmX (m)] − I(n)X(m) ] = 0. (26) This simplifies to 2πK(n)ρ[X (n)]2 + znρa (n)X(n) + ∑ m zn zn + zm [ρa(n)a(m)] [ ρX (m)X(n) ] = 0, (27) which is the desired expression. This equation is in a more compact form [1] 2πρK(n) [ X(n) ]2 + znβ (n) [ 1 + ∑ m 1 zn + zm β(m) ] = 0, (28) where β(n) is β(n) = ρX (n)a(n). (29) 4. Symmetry In this section we will summarize and extend our previous analysis of the most general scaling relation [16] for the multi Yukawa closure of the Ornstein Zernike equation. We have Π (n) i = − ∑ m ΓnmX (m) i (30) where Γmn is the M ×M matrix of scaling parameters. This matrix is not uniquely defined by the MSA closure relations and must be supplemented by M(M−1) equa- tions obtained from symmetry requirements for the correlations. From the symmetry of the direct correlation function at the origin, equation (15) qij(λji) = qji(λij), (31) we write a (n) i = ∑ m ΛnmX (m) i , (32) where, as was shown in reference [16], Λ must be a symmetric matrix. From the symmetry of the contact pair correlation function equation (16) we get {gij(σij) = gji(σij)} =⇒ {qij(σij) ′ = qji(σij) ′}, (33) which are ∑ n (Π (n) i − znX (n) i )a (n) j = ∑ n (Π (n) j − znX (n) j )a (n) i , (34) 451 L.Blum, J.A.Hernando from which we get the scaling relation Π (n) i − znX (n) i = ∑ m Υnma (m) i , (35) and a new set of M(M − 1)/2 symmetry relations Υmn = Υnm. (36) Furthermore, using the scaling relations we get M̃ · Λ = Γ, (37) where the matrix M̃ (see equation [25])has elements [M̃]nm = 1 zn + zm ∑ ` ρ` [ X (n) ` {zmX (m) ` − Π (m) ` } +X (m) ` Π (n) ` ] . (38) Solving these equations yields the relations M̃−1 · Γ = Λ (39) and − ( I + z · Γ−1 ) · M̃ = Υ. (40) Both Υ and Λ must be symmetric matrices. We have, therefore, a total of M(M−1) symmetry relations, which together with the M closure equations give the required equations for the M 2 elements of the matrix Γ. The symmetry requirements are more explicit Γ · M̃T = M̃ · ΓT M̃T · [ ΓT ] −1 = Γ−1 · M̃ SI (41) and ( I + z · Γ−1 ) · M̃ = M̃T · ( I + [Γ−1]T · z ) SII (42) the matrix M̃ as M̃ = 1 2 D̃ + 1 2 M̃A (43) and [M̃A]nm = −1 snm ∑ ` ρ` [ X (n) ` (zmX (m) ` − 2Π (m) ` ) − (znX (n) ` − 2Π (n) ` )X (m) ` ] = −1 zn + zm ∑ p ρ [ X(n)X(p)(zmδ Kr pm + 2Γmp) − (znδ Kr pn + 2Γnp)X (m)X(p) ] , (44) [M̃A]nm = −X (n)X(m) [γnm + αnm] , (45) where αnm = − 2ρ zn + zm ∑ p [ Γmp X(p) X(m) − Γnp X(p) X(n) ] , (46) and γnm = 2Γ(nn) + zn − 2Γ(mm) − zm zm + zn . (47) The second symmetry condition in equation (42) is M̃A = Γ−1 · M̃ · z − z · M̃T · [ΓT ]−1. (48) 452 Yukawa fluids: a new solution of the one component case 5. The two-Yukawa case: symmetric matrix results We write equation (24) in matrix form [1] −~Πi = M̃ · ~ai (49) where ~Xi = [ X (1) i X (2) i ] , ~Πi = [ Π (1) i Π (2) i ] , ~ai = [ a (1) i a (2) i ] . (50) Using the symmetry relation equation (41) we get ( Γ(12)X (2) X(1) − Γ(21)X (1) X(2) ) = s12 2 [χ12 − γ12] (51) with χ12 = z1 − z2 z1 + z2 + 2 Γ(11) + 2 Γ(22) , (52) γ12 = z1 − z2 + 2 Γ(11) − 2 Γ(22) z1 + z2 , (53) in equation (45) we can write M̃ = ρ 2 [ X(1) 0 0 X(2) ] [ 1 1 − χ12 1 + χ12 1 ] [ X(1) 0 0 X(2) ] . (54) We rewrite equation (49) as 2 [ G(1) G(2) ] = [ 1 1 − χ12 1 + χ12 1 ] [ β(1) β(2) ] . (55) Here we have defined G(1) = Γ(11) + X(2) X(1) Γ(12), G(2) = Γ(22) + X(1) X(2) Γ(21). (56) If we also define 2G(s) = G(1) + G(2), 2G(12) = G(1) − G(2), (57) then we can solve equation (55) 2G(s) = 2βs + β12χ12, 2G(12) = −βsχ12, (58) or βs = −2 G(12) χ12 , β12 = 2 χ12 [ G(s) + G(12) χ12 ] . (59) 453 L.Blum, J.A.Hernando From the second symmetry condition equation (60) we get X(2) X(1) z1Γ (12) − X(1) X(2) z2Γ (21) − 2Γ(12)Γ(21)χ12 + 2τ12 = 0, (60) where τ12 = ( z2Γ (11)(z1 + Γ(11)) − z1Γ (22)(z2 + Γ(22)) z1 + z2 + 2 Γ(11) + 2 Γ(22) ) . (61) We also remark that τ12 = 1 2 [z2Γ (11)(1 + χ12) − z1Γ (22)(1 − χ12)] + χ12Γ (11)Γ(22), (62) in equation (60) we get X(2) X(1) z1Γ (12) − X(1) X(2) z2Γ (21) + 2χ12DΓ + [z2Γ (11)(1 + χ12) − z1Γ (22)(1 − χ12)] = 0. (63) Using now equation (51) X2 2 X2 1 Γ2 12 − X2 X1 Γ12 [s12χ12 2 + z2 + 2Γ(22) ] + Γ(22)(z2 + Γ(22)) = 0, (64) and X2 1 X2 2 Γ2 21 − X1 X2 Γ21 [ − s12χ12 2 + z1 + 2Γ(11) ] + Γ(11)(z1 + Γ(11)) = 0, (65) from where G(1) = 1 2 [{ s12χ12 2 + z12 χ12 } − z1 − √ ∆ (2) Γ ] , (66) G(2) = 1 2 [{ − s12χ12 2 + z12 χ12 } − z2 − √ ∆ (1) Γ ] , (67) with ∆ (2) Γ = z2 2 + s12χ12 [s12χ12 4 + z2 + 2Γ(22) ] , (68) ∆ (1) Γ = z2 1 + s12χ12 [s12χ12 4 − z1 − 2Γ(11) ] , (69) [ 2G(1) − {s12 2 + z12 } χ12 + z1 ]2 − [ 2G(2) − { − s12 2 + z12 } χ12 + z2 ]2 = = ∆ (2) Γ − ∆ (1) Γ = 0. (70) From here we get the equation { χ12 − 2z12 s12 + 2Gs }{ χ12 − z12 + 2G(12) s12 } = 0, (71) 454 Yukawa fluids: a new solution of the one component case which yields the two solutions χ12 = 2z12 s12 + 2Gs (A), χ12 = z12 + 2G(12) s12 (B). (72) Notice first that in the zero density limit we get χ12 =⇒ 2z12 s12 , χ12 =⇒ z12 s12 , (73) and then in equations (66) and (67) we get the correct zero density limit only from the choice (B) G(1) ' 1 2 [{z12 2 + s12 } − z1 − s12 2 ] = 0, (74) G(2) ' 1 2 [{ − z12 2 + s12 } − z2 − s12 2 ] = 0. (75) Then βs = −2G(12) s12 (z12 + 2G(12)) , β12 = 2 s12 (z12 + 2G(12)) [ G(s) + 2G(12)s12 (z12 + 2G(12)) ] . (76) These expressions turn out to be identical to those derived by Blum and Ubriaco using the diagonal approximation [2]. 6. Thermodynamics by parameter integration We will use the notation and the results of Blum and Hernando [3]. We recall that J (n)Π(n) = I(n)X(n) − δ(n). (77) Remember that X(n) = γ(n) + Ĵ (n)B̂(zn). (78) Here Ĵ (n) = σφ0(znσ) − 2ρβ0σ3ψ1(zn), (79) and γ̂(n) = δ(n) − 2β0 z2 n ρδ(n)(1 + znσ 2 ). (80) The total excess internal energy is E(β) kTV = ∑ n K(n) { ρδ(n)B̂(n) } . (81) From equation (30) we show that −Π(n) = G(n)X(n), (82) 455 L.Blum, J.A.Hernando where G(n) is a (generally algebraic) function of the coefficients β ≡ {β1, β2, · · ·}. In fact in equation (77) δ(n) = ∑ m [Mnm]X(m) = ∑ m {I(n)δKr nm + J (n)Γ(nm)}X (m) = I(n)X(n) + J (n) ∑ m Γ(nm)}X (m) = {I(n) + J (n)G(n)}X (n) (83) with G(n) = ∑ m Γ(nm)X (m) X(n) . (84) For the 1 component case we get X(n) = δ(n) I(n) + G(n)J (n) . (85) Then, since the “charge” parameters are constants at constant temperature, the derivative of B̂(n) with respect to the scaling parameter G(n) is [ ∂B̂(n) ∂G(n) ] = [ J (n) ]−1 { ∂ ( X(n) ) ∂G(n) } = − [ J (n) ]−1 [ δ(n)J (n) (I(n) + G(n)J (n))2 ] , (86) where we use the fact that J (n) is independent of G(n). The desired energy derivative equation (81) is ∂E ∂G(n) = −ρ[X (n)]2 (87) or ∂E ∂G(s) = − ∑ n ρ[X (n)]2, ∂E ∂G(nm) = −ρ{[X (n)]2 − [X (m)]2}. (88) The integrability condition is satisfied since ∂2E ∂G(n)∂G(m) = ∂2E ∂G(m)∂G(n) = δKr nm [ 2ρ[X (n)]2 J (n) I(n) + G(n)J (n) ] . (89) Now we use equation (29) to obtain ∂E ∂G(s) = 1 2 [ β2 s + s12βs + z12β12 ] = s12z12 2(2G(12)+z12) [ 2G(s) + s12 − s12z12 (2G(12)+z12) ] (90) 456 Yukawa fluids: a new solution of the one component case and ∂E ∂G(12) = 1 2 [ βs(βs + s12) + z12βs + z12 2s12 {β2 s − β2 12} ] = s12z12 4(2G(12) + z12)2 [ s2 12 + z2 12 − { 2G(s) + s12 − 2s12z12 (2G(12) + z12) }2 ] − s12z12 4 . (91) Thermodynamic integration of these equations leads to − 2π k ∆S = ( 1 8 s12z12 (z12 + 2G(12)) )3 { 1 3 + ( 1 − (z12 + 2G(12))(s12 + 2G(s)) s12z12 )2 } − (s12z12 8 ) ( s2 12 + z2 12 (z12 + 2G(12)) − z12 + 2G(12) ) + s3 12 12 , (92) ∆S = − k 2π [ β3 s 6 + βs 4 [(βss12 + β12z12] − z2 12(β 2 s − β2 12) 8(βs + s12) ] . (93) An interesting point here is that this expression is correct in the zero density limit without any further adjustement. As was shown by Lin et al. [13], the result ob- tained directly from the diagonal assumption [2] does not automatically satisfy this requirement, and the reason for this is that the wrong branch of the solution is used in this approximation. Acknowledgements The authors thank Yurij Kalyuzhnij for his careful proofreading of the manuscript. References 1. Blum L., Herrera J.N. // Mol. Phys., 1999, vol. 96, p. 821. 2. Blum L., Ubriaco M. // Mol. Phys., 2000, vol. 98, p. 829. 3. Blum L., Hernando J.A. // J. Phys.: Cond. Matt., 2002, vol. 14, p. 10933. 4. Blum L., Herrera J.N. // Mol. Phys., 1998, vol. 95, p. 425. 5. Blum L., Hernando J.A. Condensed Matter Theories. (Hernandez S., Clark J.W. edi- tors). New York, Nova Publishers, 2001, vol. 16, p. 411. 6. Blum L., Vericat F., Degreve L. // Physica A, 1999, vol. 265, p. 396. 7. Blum L., Vericat F. // J. Phys. Chem., 1996, vol. 100, p. 1197. 8. Waisman E. // Mol. Phys., 1973, vol. 25, p. 45. 9. Waisman E., Stell G., Høye J.S. // Chem. Phys. Letters, 1976, vol. 40, p. 3253. 10. Arrieta E., Jedrzejek C., Marsh K.N. // J. Chem. Phys., 1987, vol. 86, p. 3606. 11. Herrera J.N., Blum L., Garćıa-Llanos E. // J. Chem. Phys., 1996, vol. 105, p. 9606. 12. Høye J.S., Stell G. // J. Chem. Phys., 1980, vol. 89, p. 461. 13. Lin Y.Z., Li Y.G., Lu Y.F., Liu Z.P. // Mol. Phys., 2002, vol. 100, p. 3251. 457 L.Blum, J.A.Hernando 14. Blum L. // Mol. Phys., 1975, vol. 30, p. 1529. 15. Ginoza M. // J. Phys. Soc. Japan, 1986, vol. 55, p. 95,1782. 16. Blum L., Vericat F., Herrera J.N. // J. Stat. Phys., 1992, vol. 66, p. 249. 17. Blum L., Høye J.S. // J. Stat. Phys., 1978, vol. 19, p. 317. 18. Blum L. // J. Stat. Phys., 1980, vol. 22, p. 661. 19. Ginoza M. // Mol. Phys., 1990, vol. 71, p. 145. 20. Ginoza M., Yasutomi M. // J. Stat. Phys., 1998, vol. 90, p. 1475. Плин Юкави: новий розв’язок однокомпонентного випадку Л.Блюм 1 , Дж.А.Ернандо 2 1 Фізичний факультет, Університет Пуерто-Рiко, Ріо Пієдрас, PR 00931-3343, США 2 Фізичний факультет, Національна Рада ядерної енергетики, просп. дель Лібертадор, 8250, 1429 Буенос-Айрес, Аргентина Отримано 12 березня 2003 р., в остаточному вигляді – 23 липня 2003 р. В сучасних роботах знайдено розв’язок рівняння Орнштейна-Церні- ке для простого однокомпонентного плину в рамках узагальнених умов замикання Юкави. У зв’язку зі складністю рівнянь було зроб- лено припущення про те, що головна скейлінгова матриця Γ має бути діагональною. Хоча це математично правильно, фізично це по- рушує умови симетрії при співставлянні різних потенціалів Юкави різним компонентам. В цій роботі ми показуємо, що використовую- чи умови симетрії, недіагональні елементи матриці Γ можуть бути точно обчислені, розв’язуючи квадратне рівняння для двох потенці- алів Юкави. Існують два розв’язки цього рівняння, але тільки один з них має фізично правильну поведінку при нульовій густині. Нефізич- ний розв’язок відповідає розв’язку з діагональною апроксимацією. I хоча наш розв’язок відрізняється від того, що в діагональному ви- падку, надлишкова ентропія формально залишається такою ж. Ключові слова: плин Юкави, середньо-сферичне наближення, ентропія, скейлінгове наближення PACS: 61.20.Gy 458

Yukawa fluids: a new solution of the one component case

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