The equation of state of a cell fluid model in the supercritical region

The analytic method for deriving the equation of state of a cell fluid model in the region above the critical temperature (T > Tc) is elaborated using the renormalization group transformation in the collective variables set. Mathematical description with allowance for non-Gaussian fluctuations...

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Дата:	2018
Автори:	Kozlovskii, M.P., Pylyuk, I.V., Dobush, O.A.
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Мова:	English
Опубліковано:	Інститут фізики конденсованих систем НАН України 2018
Назва видання:	Condensed Matter Physics
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Цитувати:	The equation of state of a cell fluid model in the supercritical region / M.P. Kozlovskii, I.V. Pylyuk, O.A. Dobush // Condensed Matter Physics. — 2018. — Т. 21, № 4. — С. 43502: 1–26. — Бібліогр.: 29 назв. — англ.

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spelling	irk-123456789-1574682019-06-21T01:28:52Z The equation of state of a cell fluid model in the supercritical region Kozlovskii, M.P. Pylyuk, I.V. Dobush, O.A. The analytic method for deriving the equation of state of a cell fluid model in the region above the critical temperature (T > Tc) is elaborated using the renormalization group transformation in the collective variables set. Mathematical description with allowance for non-Gaussian fluctuations of the order parameter is performed in the vicinity of the critical point on the basis of the ρ⁴ model. The proposed method of calculation of the grand partition function allows one to obtain the equation for the critical temperature of the fluid model in addition to universal quantities such as critical exponents of the correlation length. The isothermal compressibility is plotted as a function of density. The line of extrema of the compressibility in the supercritical region is also represented. Використовуючи множину колективних змiнних та перетворення ренормалiзацiйної групи, розвинено аналiтичний метод розрахунку рiвняння стану комiркової моделi плину в областi вище критичної температури (T > Tc). Математичний опис з урахуванням негаусових флуктуацiй параметра порядку виконано в околi критичної точки на основi моделi ρ⁴ . Запропонований метод розрахунку великої статистичної суми дозволяє отримати, крiм унiверсальних величин, зокрема, критичних показникiв кореляцiйної довжини, рiвняння для критичної температури моделi плину. Побудовано кривi iзотермiчної стисливостi як функцiї густини. Також зображено лiнiю екстремумiв стисливостi у надкритичнiй областi. 2018 Article The equation of state of a cell fluid model in the supercritical region / M.P. Kozlovskii, I.V. Pylyuk, O.A. Dobush // Condensed Matter Physics. — 2018. — Т. 21, № 4. — С. 43502: 1–26. — Бібліогр.: 29 назв. — англ. 1607-324X PACS: 51.30.+i, 64.60.fd DOI:10.5488/CMP.21.43502 arXiv:1712.07164 http://dspace.nbuv.gov.ua/handle/123456789/157468 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	The analytic method for deriving the equation of state of a cell fluid model in the region above the critical temperature (T > Tc) is elaborated using the renormalization group transformation in the collective variables set. Mathematical description with allowance for non-Gaussian fluctuations of the order parameter is performed in the vicinity of the critical point on the basis of the ρ⁴ model. The proposed method of calculation of the grand partition function allows one to obtain the equation for the critical temperature of the fluid model in addition to universal quantities such as critical exponents of the correlation length. The isothermal compressibility is plotted as a function of density. The line of extrema of the compressibility in the supercritical region is also represented.
format	Article
author	Kozlovskii, M.P. Pylyuk, I.V. Dobush, O.A.
spellingShingle	Kozlovskii, M.P. Pylyuk, I.V. Dobush, O.A. The equation of state of a cell fluid model in the supercritical region Condensed Matter Physics
author_facet	Kozlovskii, M.P. Pylyuk, I.V. Dobush, O.A.
author_sort	Kozlovskii, M.P.
title	The equation of state of a cell fluid model in the supercritical region
title_short	The equation of state of a cell fluid model in the supercritical region
title_full	The equation of state of a cell fluid model in the supercritical region
title_fullStr	The equation of state of a cell fluid model in the supercritical region
title_full_unstemmed	The equation of state of a cell fluid model in the supercritical region
title_sort	equation of state of a cell fluid model in the supercritical region
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2018
url	http://dspace.nbuv.gov.ua/handle/123456789/157468
citation_txt	The equation of state of a cell fluid model in the supercritical region / M.P. Kozlovskii, I.V. Pylyuk, O.A. Dobush // Condensed Matter Physics. — 2018. — Т. 21, № 4. — С. 43502: 1–26. — Бібліогр.: 29 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT kozlovskiimp theequationofstateofacellfluidmodelinthesupercriticalregion AT pylyukiv theequationofstateofacellfluidmodelinthesupercriticalregion AT dobushoa theequationofstateofacellfluidmodelinthesupercriticalregion AT kozlovskiimp equationofstateofacellfluidmodelinthesupercriticalregion AT pylyukiv equationofstateofacellfluidmodelinthesupercriticalregion AT dobushoa equationofstateofacellfluidmodelinthesupercriticalregion
first_indexed	2025-07-14T09:53:44Z
last_indexed	2025-07-14T09:53:44Z
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fulltext	Condensed Matter Physics, 2018, Vol. 21, No 4, 43502: 1–26 DOI: 10.5488/CMP.21.43502 http://www.icmp.lviv.ua/journal The equation of state of a cell fluid model in the supercritical region M.P. Kozlovskii, I.V. Pylyuk, O.A. Dobush Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii St., 79011 Lviv, Ukraine Received May 8, 2018, in final form October 26, 2018 The analytic method for deriving the equation of state of a cell fluid model in the region above the critical tem- perature (T > Tc) is elaborated using the renormalization group transformation in the collective variables set.Mathematical description with allowance for non-Gaussian fluctuations of the order parameter is performed in the vicinity of the critical point on the basis of the ρ4 model. The proposed method of calculation of the grand partition function allows one to obtain the equation for the critical temperature of the fluid model in addition to universal quantities such as critical exponents of the correlation length. The isothermal compressibility is plotted as a function of density. The line of extrema of the compressibility in the supercritical region is also represented. Key words: cell fluid model, critical exponent, equation of state, supercritical region PACS: 51.30.+i, 64.60.fd 1. Introduction In recent decades the interest in supercritical fluids is steadily increasing. They possess unique prop- erties which appear to be efficient for chemical, technological and industrial appliances [1]. The behavior of supercritical fluids is intensively investigated in experiments by methods of computer simulations and by development of theoretical approaches. These directions of research are well covered in recent reviews by Yoon, Lee [2] and Vega [3]. The theoretical approaches for description of the critical phenomena in fluids are represented by the statistical mechanical theories such as the theory of integral equations, which consists of homogeneous Ornstein-Zernike Equation [4, 5] and the closure relations such as the Percus-Yevick [6] and hypernetted chain closures [7], and the fluctuation theory of solutions [8]. The renormalization group (RG) theory [9] has also been very successful in describing the properties of sys- tems near their critical point. The conception of RG theory was used by Yukhnovskii [10] to make a total integration in the partition function of the Ising model and the grand partition function of a gas-liquid system in the phase-space of collective variables and to investigate the behavior of these systems in the vicinity of a critical point. For over a century, scientists are trying to theoretically describe the nature of phase transitions and critical phenomena in liquid systems. In the course of long years of strenuous work, various worthy approaches have been elaborated to solve this problem in the frames of canonical ensemble but the devel- opment of resemblant theories on the basis of the grand canonical distribution still remains topical.The latter direction is of particular importance since due to the presence of chemical potential within the framework of the grand canonical ensemble, the actual systems of atoms andmolecules can be adequately represented. Only this thermodynamic parameter is responsible for the exchange of constituents between different parts of the system and with the environment. Moreover, it quantitatively describes the tendency of the thermodynamic system to establish a composition equilibrium. In articles [11, 12], we calculated the grand partition function of a cell fluid model in the mean-field type approximation. In this way, it was possible to describe the first-order phase transition and, in general, This work is licensed under a Creative Commons Attribution 4.0 International License . Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. 43502-1 https://doi.org/10.5488/CMP.21.43502 http://www.icmp.lviv.ua/journal http://creativecommons.org/licenses/by/4.0/ M.P. Kozlovskii, I.V. Pylyuk, O.A. Dobush the behavior of the system in a wide range of temperatures above and below the critical point, except its vicinity. In particular, using the Morse interaction potential [13], which well describes the interaction in liquid alkali metals, we obtained an equation of state, coexistence curves, and values of critical density and critical temperature for sodium and potassium. However, in the mean-field approximation, it is impossible to describe the behavior of a three-dimensional system in a close vicinity of a critical point, where fluctuations play a significant role and collective effects turn out to be primary. The solution of the problem will be the calculation of the grand partition function using the renormalization group transformation. We intend to implement the idea of the method elaborated in order to calculate the partition function of the 3D Ising-like model in the external field [14–16] as well as to analyze the critical behavior of this system. This waymakes it possible to provide a thorough study of the critical properties of the correlation function and thermodynamic functions such as specific heat, isothermal compressibility and isobaric expansion, to estimate the values of critical amplitudes and critical parameters, to plot the Widom line [17, 18], which is necessary for investigating a fluid in the supercritical region. Note that unique properties of the matter appear during phase transitions in other systems. In particular, the articles [19, 20] represent the investigation of the phase transition in a Fermi fluid from a normal state to a state in which the rotational symmetry breaking takes place in the momentum space. The authors obtained analytic expressions for the phase transition temperature as well as the order parameter near the critical point using the equation of self-consistency. In this article, we propose a theoretical description of the behavior of a cell fluidmodel near the critical point at temperatures above the critical. Section 2 is devoted to a staged calculation of the grand partition function of the cell fluid model within an approach of collective variables. Furthermore, the recurrence relations between the coefficients of effective non-Gaussian measures of density, the solutions of these relations and the equation for a phase transition temperature are presented. The thermodynamic potential is considered in section 3. The total expression of the thermodynamic potential in case of temperatures T > Tc is obtained as a compilation of terms derived for each fluctuation regime. A technique of calculating the equation of state with fluctuational effects taken into account is elaborated in section 4. The corresponding expressions are derived for the cases ofT > Tc. Plots of the isothermal compressibility and the line based on the points of maxima of isothermal compressibility are also presented in this section. Discussions and conclusions are presented in section 5. 2. Basic expressions As in our previous articles [11, 12], the object of the present investigation is a cell fluid model. The cell fluid is an approximation of a continuous system. Under continuous system one should mean the system of volume V composed of N interacting particles. Similarly to the case of a cell gas model [21], the idea of the cell fluid [22] consists in a fixed partition of the volume V of a system, where N particles reside, on Nv mutually disjoint cubes ∆®l = (−c/2, c/2]3 ⊂ R3, each of the volume v = c3 = V/Nv (c is the side of a cell). Instead of the distance between particles, the distance between the centers of cells is introduced. It is possible to theoretically describe the behavior of any system having its equation of state. One of the ways to obtain the latter is an analytical calculation of the grand partition function (GPF) of the system. As it had been already shown in [11, 12], the GPF of the cell fluid model within the framework of the grand canonical ensemble is of the form Ξ = ∞∑ N=0 (z)N N! ∫ V (dx)Nexp [ − β 2 ∑ ®l1,®l2∈Λ Ũl12 ρ®l1 (η)ρ®l2 (η) ] . (2.1) Here and forth, z = eβµ is the activity, β is the inverse temperature, µ is the chemical potential. Integration over the coordinates of all the particles in the system is noted as ∫ V (dx)N = ∫ V dx1 . . . ∫ V dxN , xi = ( x(1)i , x(2)i , x(3)i ) , η = {x1, . . . , xN } is the set of coordinates. Ũl12 is the potential of interaction, l12 = ��®l1−®l2�� is the difference between two cell vectors. Each ®li belongs 43502-2 The EoS of a supercritical cell fluid model to a set Λ, defined as Λ = { ®l = (l1, l2, l3)\|li = cmi; mi = 1, 2, . . . , Na; i = 1, 2, 3; Nv = N3 a } . Here, Na is the number of cells along each axis, ρ®l(η) is the occupation number of a cell ρ®l(η) = ∑ x∈η I∆®l (x) . (2.2) Here, I∆®l (x) is the indicator of ∆®l , that is, I∆®l (x) = 1 if x ∈ ∆` and I∆®l (x) = 0 otherwise. Interaction in the system is expressed as Ũl12 which is a function of distance between cells. We choose Ũl12 = Ψl12 −Ul12 in the form of a lattice analogue of the Morse potential Ψl12 = De−2(l12−1)/αR ; Ul12 = 2De−(l12−1)/αR, (2.3) here, αR = α/R0 where α is the effective interaction radius. The parameter R0 corresponds to the minimum of the function Ũl12 [ Ũ(l12 = 1) = −D determines the depth of the potential well ] . Note that in terms of convenience, the R0-units are used for length measuring. In the next sections we present some quantitative results using the following parameters of the interaction potential: R0 = 5.3678 Å, αR = 0.3385, D = 0.9241 × 10−20 J [23]. The latter parameters are used to describe the interaction in sodium by theMorse potential [13]. The value of themodel’s parameter c in this case is c = 1.3424, which is the same as we used in the mean-field description of the vapor-liquid transition in alkali metals [12]. The procedure of self-consistent determination of v = c3, as well as of other parameters of the model, is presented in [11, 12]. In our recent publication [22] we made an accurate calculation of the grand partition function of a single-sort cell model with Curie-Weiss potential and found that this model has a sequence of first order phase transitions at temperatures below the critical one Tc. The Curie-Weiss potential was chosen in the form of a function Φl12 = { −J1/Nv , x ∈ ∆®l1 , y ∈ ∆®l2 , ®l1 , ®l2 , J2δ®l1®l2 , x, y ∈ ∆®l1 , ®l1 = ®l2 . By virtue of the Curie-Weiss approach, it fails to be a function of distance between constituents of the system. The first term in Φl12 with J1 > 0 describes attraction. It is taken to be equal for all particles. The second termwith J2 > 0 describes the repulsion between two particles contained in one and the same cell. In the present research, the potential of interaction is a function of distance. This manoeuvre allows us to take account of the influence of long-range fluctuations. The analogue of interaction −J1 = const is an exponentially decreasing function of distance Ũl12(l12 , 0) < 0, the analogue of J2 is Ũl12(l12 = 0) > 0. From this point of view, the constant interaction Ũl12(l12 = 0) = De1/αR (e1/αR − 2) > 0 reflects the interaction energy within a cell. Clearly, this energy is of a repulsive nature and corresponds to the interaction between particles located in the same cell. In [11, 12] we obtained a general functional representation of the grand partition function of the cell fluid model in the set of collective variables in the following form Ξ = ∫ (dρ)Nv exp [ βµρ0 + β 2 ∑ ®k∈Bc W(k)ρ®k ρ−®k ] Nv∏ l=1 [ ∞∑ m=0 vm m! e−pm 2 δ(ρ®l − m) ] , (2.4) from which one can see that the occupation numbers of cells ρ®l(η), which are connected with the variables ρ®l , can take values m = 0, 1, 2, . . . . Due to the term e−pm2 , the more m increases the less m-th term contributes to the sum in (2.4). Therefore, the probability of hosting many particles in a single cell is very small. Note that (dρ)Nv = ∏ ®k∈Bc dρ®k . The variable ρ®k is the representation of ρ®l in reciprocal space. 43502-3 M.P. Kozlovskii, I.V. Pylyuk, O.A. Dobush Here and forth, the vector ®k belongs to the set Bc= { ®k= (k1, k2, k3) �� ki=− π c + 2π c ni Na , ni=1, 2, . . . , Na; i = 1, 2, 3; Nv = N3 a } . The Brillouin zone Bc corresponds to the volume of periodicity Λ with cyclic boundary conditions. The parameter p is expressed as follows: p = 1 2 βχΨ(0) , and W(k) is the Fourier transform of the effective potential of interaction W(k) = U(k) − Ψ(k) + χΨ(0) . (2.5) The Fourier transforms of the attractive and of the repulsive parts of the Morse potential (2.3) are, respectively, as follows: U(k) = U(0) ( 1 + α2 Rk2)−2 , Ψ(k) = Ψ(0) ( 1 + α2 Rk2/4 )−2 , (2.6) where U(0) = 16Dπ α3 R v eR0/α and Ψ(0) = Dπ α3 R v e2R0/α . (2.7) The positive parameter χ forms the Jacobian of transition from individual coordinates to collective variables. There are at least three possible ways of calculating (2.4), in order to derive the equation of state. The first option is to average the potential of interaction in the direct space or to replace it by the one that fails to be a function of the distance between constituents of the system. In this case, one gets a diagonal form in the integrand exponent. We did this in the article [22], using a Curie-Weiss potential to mimic the interaction in a cell model, and rigorously calculated the grand partition function of this system. However, such interaction potential has a non-physical property since the force of interaction is a function of the number of cells. Nevertheless, it allows one to determine sufficient thermodynamic properties. The second option is to use an approximation of the ρ4-model which consists of the cutting off terms proportional to the fifth power of the variable ρ®k and higher, and to use a type of the mean-field approximation considering only variables ρ®k with ®k = 0 (see [11, 12]). Both of these options make it possible to explicitly calculate the grand partition function and derive an equation of state. However, using them, one would describe the behavior of the model in a wide range of temperatures excluding a close vicinity of the critical point. The third option is presented in the present manuscript. It consists in using the idea of the method of calculating the partition function elaborated for a magnet [24, 25]. In this way, it is possible to take into account the influence of long-range fluctuations, which is necessary for the description of critical behavior. To follow this purpose, we start from the expression of the grand partition function of the cell fluid in ρ4-model approximation, which is derived from (2.4) (see [11, 12]) Ξ = gW eNv (Eµ−a0) ∫ (dρ)Nv exp [ MN1/2 v ρ0− 1 2 ∑ k∈B d(k)ρ®k ρ−®k− a4 24 1 Nv ∑ k1, . . .,k4 ®ki ∈B ρ®k1 ...ρ®k4 δ®k1+...+®k4 ] . (2.8) Here, gW = ∏ ®k∈B [ 2πβW(k) ]−1/2 , a34 = −a3/a4 , Eµ = − βW(0) 2 (M + ã1) 2 + Ma34 + 1 2 d(0)a2 34 − a4 24 a4 34 , M = µ W(0) − ã1 , ã1 = a1 + d(0)a34 + a4 6 a3 34 . (2.9) 43502-4 The EoS of a supercritical cell fluid model The following expression corresponds to the coefficient d(k) d(k) = 1 βW(k) − ã2 , ã2 = a4 2 a2 34 − a2 . (2.10) The effective potential of interaction W(k) from (2.5) is definitely positive when k = 0 for all χ > 1, but for 0 < χ < 1 it is either positive or negative depending on the ratio R0/α. If R0/α = 2.9544 [which is typical of Na (sodium)], χ = 1.1243 (p = 1.8100) and v = 2.4191 [12] we have the following: a0 = −0.3350, ã1 = −0.1891, ã2 = 0.3242, a4 = 0.0376, a34 = 2.4925. (2.11) Let us represent d(k) in the form of series in power k2 d(k) = d(0) + 2bk2/βW(0) + 0(k4). (2.12) Here, b = α2 R [ U(0) − Ψ(0)/4 ] /W(0). (2.13) As can be seen from the look of expressions of d(0) (2.10) and b (2.13), it is expedient to make a change of variables in the expression (2.8) ρk = ρ ′ k [ βW(0) ]1/2 . (2.14) As a result, we have Ξ = gW [ βW(0) ]Nv/2eNv (Eµ−a0) ∫ (dρ)Nv × exp [ w0N1/2 v ρ0 − 1 2 ∑ ®k∈B ( r0 + 2bk2)ρ®k ρ−®k − u0 24 1 Nv ∑ ®ki ∈B ρ®k1 ...ρ®k4 δ®k1+...+®k4 ] , (2.15) moreover, (the sign of stress near ρ®k is omitted) w0 = M [ βW(0) ]1/2 , r0 = 1 − βW(0)ã2 , u0 = a4 [ βW(0) ]2 . (2.16) Let us make in (2.15) a staged integration keeping with the technique elaborated in [26]. One should start from the variables ρ®k with large value of the wave vector and end with integration over the variable ρ0. The latter is the basis of a type of mean-field approximation. The contribution of ρ0 is the main one, in case of temperatures far from Tc (Tc is the critical temperature). It is essential to take account of the fluctuations in the vicinity of Tc; thus, one should consider terms ρ®k with k , 0 in calculations. According to [14], let us write the following:∑ ®k∈B ( r0 + 2bk2)ρ®k ρ−®k = ∑ ®k∈B1 ( r0 + 2bk2)ρ®k ρ−®k + ∑ ®k∈B/B1 (r0 + q)ρ®k ρ−®k , (2.17) here, q = 2bq′ and q′ = 〈k2〉B1,B is the average of k2 on the interval ®k ∈ (B1,B), q′ = 3 2 π2 c2 ( 1 + s−2) . (2.18) The range of values ®k ∈ B1 are of the form: B1 = { ®k = (k1, k2, k3)\|ki = − π c1 + 2π c1 ni N1i ; ni = 1, 2, ... , N1i; i = x, y, z; N1 = N3 1x } . (2.19) Here, c1 = sc, moreover, the way of dividing the space of collective variables into intervals (s > 1) is determined by the parameter s = B/B1. Rewrite (2.15) as Ξ = Gµ ∫ (dρ)N1e − 1 2 ∑ ®k∈B1 [r0+2bk2−(r0+q)]ρ ®kρ− ®k ew0 √ Nvρ0 ∏ ®k∈B1 δ(η®k − ρ®k) × ∫ (dη)Nv exp [ − 1 2 ∑ ®k∈B (r0 + q)η®kη−®k − u0 24 1 Nv ∑ ®ki ∈B η®k1 ...η®k4 δ®k1+...+®k4 ] , (2.20) 43502-5 M.P. Kozlovskii, I.V. Pylyuk, O.A. Dobush where Gµ = gW [ βW(0) ]Nv/2eNv (Eµ−a0), N1 = Nvs−3. Use the integral representation δ(η®k − ρ®k) = ∫ (dν)N1 exp [ 2πi ∑ ®k∈B1 ν®k ( η®k − ρ®k ) ] and integrate over Nv variables η®k . Thus, we have Ξ = Gµ [ Q(r0) ]Nv j1 ∫ (dρ)N1 N1∏ l=1 ∫ (dν)N1ew0 √ Nvρ0 exp − 1 2 ∑ ®k∈B1 2b ( k2 − q′ ) ρ®k ρ−®k − 2πiν®lρ®l − m0∑ m=1 (2π)2m P2m (2m)! ν2m ®l ] . (2.21) Here, ν®l = 1√ N1 ∑ ®k∈B1 ν®ke i®k®l is a site representation of the variable ν®k , j1 = 2(N1−1)/2 is the Jacobian of transition from variables ν®k to ν®l , Q(r0) = ∞∫ −∞ f (η)dη , f (η) = exp [ − 1 2 (r0 + q)η2 ®l − m0∑ m=2 u2m (2m)! η2m ®l ] . (2.22) Here, m0 = 2 is the approximation correspondent to the ρ4 model. However, any particular difficulties would arise if one uses higher-order approximations such as: ρ6, which corresponds to m0 = 3 etc. Keeping with these higher-order approximations, one should use in (2.8) the coefficients a2m where m > 2. The coefficients P2m are derived in the way it was done by [27] P2 = I2 , P4 = s−d ( − I4 + 3I2 2 ) , P6 = s−2d ( I6 − 15I4I2 + 30I3 2 ) (2.23) etc., where I2l = ∞∫ −∞ η2l f (η)dη ∞∫ −∞ f (η)dη . (2.24) The term s−nd in (2.23) is connected with the necessity of transition from the set of values ®k ∈ B to the set ®k ∈ B1 (see [27]), where d = 3 is the dimension of space. In terms of parabolic cylinder functions, we have the following: Q(r0) = (2π)1/2 ( 3 u0 )1/4 ex 2/4U(0, x) , P2 = ( 3 u0 )1/2 U(x) , P4 = s−3 ( 3 u0 ) ϕ(x). (2.25) The special functions U(x) and ϕ(x) U(x) = U(1, x) / U(0, x) , ϕ(x) = 3U2(x) + 2xU(x) − 2 43502-6 The EoS of a supercritical cell fluid model are expressed by the functions of parabolic cylinder U(a, x) U(a, x) = 2 Γ(a + 1 2 ) e−x 2/4 ∞∫ 0 t2a exp ( − xt2 − 1 2 t4 ) dt. The following expression is the result of integration over the variables ν®l in (2.21) Ξ = Gµ [ Q(r0) ]Nv [Q(P)]N1 j1 ∫ (dρ)N1 exp [ a(1)1 √ N1ρ0 − 1 2 ∑ ®k∈B1 g1(k)ρ®k ρ−®k ] × exp [ − a(1)4 4! 1 N1 ∑ ®ki ∈B1 ρ®k1 ...ρ®k4 δ®k1+...+®k4 ] , (2.26) moreover, Q(P) = ∞∫ −∞ dνg(ν), g(ν) = exp [ − ∞∑ m=1 (2π)2m P2m (2m)! ν2m ®l ] , a(1)1 = sd/2w0 . (2.27) For g1(k) and a(1)4 , we have the formulae: g1(k) = g1(0) + 2bk2, a(1)4 = 3 P4 ϕ(y) , g1(0) = ( 3 P4 )1/2 U(y) − q. (2.28) q is expressed in (2.18), y = s3/2U(x) [ 3 ϕ(x) ]1/2 , x = (r0 + q) (3/u0) 1/2 . (2.29) The following explicit form of the recurrence relations appears between the coefficients of the exponent in the grand partition function after the first stage of integration. Taking into account the formulae for P4 [see (2.25)], y and x [see (2.29)] one has g1(0) = N(x)(r0 + q) − q , (2.30) where N(x) = yU(y) xU(x) . (2.31) For a(1)4 , one derives a(1)4 = s−3u0E(x) , (2.32) where E(x) = s2dϕ(y)/ϕ(x). (2.33) Use the notion w1 = sa(1)1 , r1 = s2g1(0) , u1 = s4a(1)4 43502-7 M.P. Kozlovskii, I.V. Pylyuk, O.A. Dobush and write the expressions for a(1)1 from (2.27), g1(0) from (2.30) and a(1)4 from (2.32) in the following form: w1 = s d+2 2 w0 , r1 = s2 [ − q + (r0 + q)N(x) ] , u1 = s4−du0E(x). (2.34) The special functions N(x) and E(x) are defined in the above. According to [14, 27], after making n stages of integration in Ξ, we have Ξ = 2(Nn+1−1)/2Gµ [ Q(r0) ]NvQ1...Qn [ Q(Pn) ]Nn+1 ∫ Wn+1(ρ)(dρ)Nn+1, (2.35) where Q(dn) = (2π)1/2 [ 3 a(n)4 ]1/4 exp ( x2 n 4 ) U(0, xn) , Q(Pn) = (2π)−1/2 [ a(n)4 ϕ(xn) ]1/4 s3/4 exp ( y2 n 4 ) U(0, yn) , (2.36) Wnp+1(ρ) = exp [ a(n+1) 1 √ Nn+1ρ0 − 1 2 ∑ ®k∈Bn+1 gn+1(k)ρ®k ρ−®k − a(n+1) 4 4! 1 Nn+1 ∑ ®ki ∈Bn+1 ρ®k1 ...ρ®k4 δ®k1+...+®k4 ] , and also Qn = [ Q(Pn−1)Q(dn) ]Nn . (2.37) Moreover,1 xn = (rn + q)(3/un)1/2, yn = s3/2U(xn) [ 3/ϕ(xn) ]1/2 . (2.38) If a(n+1) 1 = s−(n+1)wn+1, gn+1(0) = s−2(n+1)rn+1, a(n+1) 4 = s−4(n+1)un+1, the recurrence relations can be represented as follows: wn+1 = s d+2 2 wn , rn+1 = s2 [ − q + (rn + q)N(xn) ] , un+1 = s4−dunE(xn) (2.39) with the initial condition (2.16). Using the following conditions, one finds the coordinates of the fixed point w∗, r∗, u∗ wn = wn+1 = w ∗, rn = rn+1 = r∗, un = un+1 = u∗. For w∗, there is w∗ = 0, since s > 1. The equation for un+1 yields sE(x∗) = 1, (2.40) which juxtaposes own x∗ to each s. The value s∗ = 3.5977 falls in with x∗ = 0 [14]. From the second equation, we have (2.39) (u∗)1/2 = q ( 1 − s−2)√3U(x∗) / [ y∗U(y∗) ] . (2.41) Therefore, the coordinates of the fixed point of the recurrence relations (2.39) are w∗ = 0, r∗ = −q, however, u∗ should be determined from (2.41). Note that the values of yn from (2.38) are large. Taking this into account, for (2.39) and (2.41) one obtains the following: wn+1 = s d+2 2 wn , 1Note that xn = gn(Bn+1, Bn) [ 3 a (n) 4 ] 1 2 , where gn(Bn+1, Bn) = gn(0) + qs−2n = s−2n(rn + q). 43502-8 The EoS of a supercritical cell fluid model rn+1 = s2 [ −q + √ un √ 3 1 U(xn) − 1 2s3 √ un √ 3 ϕ(xn) U3(xn) ] , un+1 = sun ϕ(xn) 3U4(xn) [ 1 − 7 2 s−3 ϕ(xn) U2(xn) ] and (u∗)1/2 = q(1 − s−2) √ 3U(x∗) [ 1 + 3 2 (y∗)−2 ] . The quantity q from (2.18) fails to be a function of temperature, so r∗ and u∗ fail to be functions of temperature as well. They depend on s and αR = α/R0. Besides, the formula (2.13) contains the function[ U(0) − Ψ(0)/4 ] / [ U(0) + Ψ(0)(χ − 1) ] , which turns into unity in case of χ = 3/4. Using eigenvalues of the transformation matrix R ©« wn+1 − w ∗ rn+1 − r∗ un+1 − u∗ ª®¬ = R ©« wn − w ∗ rn − r∗ un − u∗ ª®¬ , (2.42) which, in case of s = s∗, are equal to E1 = s d+2 2 = 24.551, E2 = 8.308, E3 = 0.374, (2.43) and also using eigenvectors of R, one has wn = w0En 1 , rn = r∗ + c1En 2 + c2REn 3 , un = u∗ + c1R1En 2 + c2En 3 , (2.44) where R = R(0)/(u∗)1/2, R1 = R(0)1 (u ∗)1/2, moreover, R(0) = −0.530, R(0)1 = 0.162. The expressions derived in [14] are valid for the coefficients c1 and c2 c1 = [ r0 − r∗ + (u∗ − u0)R ] D−1, c2 = [ u0 − u∗ + (r∗ − r0)R1 ] D−1, (2.45) and D = (E2 − E3)/(R22 − E3) ≈ 1.086. For R both with R1 we have R = R(0) /√ u∗, R(0) = R(0)23 E3 − R22 , R1 = R(0)1 √ u∗, R(0)1 = E2 − R22 R(0)23 . (2.46) The equation for Tc c1(Tc) = 0 (2.47) is of the following form: 1 − ã2βcW(0) − r∗ − R { a4 [ βW(0) ]2 − u∗ } = 0. Since r∗ = −q, we obtain the equation 1 + q + R(0) √ u∗ − ã2βcW(0) − R(0) a4 √ u∗ [ βcW(0) ]2 = 0 , (2.48) where q is defined in (2.18), ã2 and a4 are the coefficients of the expression (2.8). This equation allows us to find the critical temperature of the fluid model as a function of microscopic parameters of the interaction potential and coordinates of the fixed point of the recurrence relations. The 43502-9 M.P. Kozlovskii, I.V. Pylyuk, O.A. Dobush physical condition which yields this equation is the existence of critical regime of fluctuations (see [27]) near the critical point. The quantity βcW(0) = 4.412 is the solution of the equation (2.48) for the values in (2.11) and coordinates of the fixed point which are obtained above. Let us represent c1(T) and c2(T) from (2.45) as an expansion in powers of τ = (T − Tc)/Tc, which is used in section 3 to derive one of the terms of the thermodynamic potential of the model. Applying the equality (2.16) for the expressions of r0 and u0 and taking into account that the coordinates of the fixed point of the recurrence relations (2.39) are not functions of temperature, one has the following: c1 = c10 + c11τ + c12τ 2, c2 = c20 + c21τ + c22τ 2. (2.49) Here, c10 = 0, because of the equation (2.48), which, actually, is used to determine the critical temperature. For other coefficients, we have the following: c11 = βcW(0)D−1 [ ã2 + 2R(0)βcW(0)a4(u∗)−1/2 ] , c12 = −βcW(0)D−1 [ ã2 + 3R(0)βcW(0)a4(u∗)−1/2 ] . (2.50) For the coefficients c2l(l = 0, 1, 2), we get: c20 = D−1 { − u∗ − R(0)1 √ u∗(1 + q) + R(0)1 √ u∗ã2βcW(0) + a4 [ βcW(0) ]2 } , c21 = −D−1 { R(0)1 √ u∗ã2βcW(0) + 2a4 [ βcW(0) ]2 } , c22 = D−1 { R(0)1 √ u∗ã2βcW(0) + 3a4 [ βcW(0) ]2 } . (2.51) 3. A thermodynamic potential of the model The later calculation of (2.35) is based on the method proposed in [14]. In case of T > Tc, the thermodynamic potential Ω = −kBT lnΞ (3.1) is represented as follows: Ω = Ωµ +Ωr +Ω (+) CR +ΩLGR , (3.2) where Ωµ = −kBT ln Gµ = −kT { ln gW + 1 2 Nv ln [ βW(0) ] + Nv(Eµ − a0) } , Ωr = −kBT ln Q(r0) = −kT Nv [ 1 2 ln(2π) + 1 4 ln ( 3 u0 ) + x2 4 + ln U(0, x) ] . (3.3) In the critical region of fluctuations, one has Ω (+) CR = −kBT Nv ( γ01 + γ02τ + γ03τ 2) +Ω(s)CR . (3.4) The coefficients γ0l meet the expressions γ01 = s−3 f (0)CR (1 − s−3) , γ02 = s−3 c11d1E2 1 − E2s−3 , γ03 = s−3 ( c12d1E2 1 − E2s−3 + c2 11d3E2 2 1 − E2 2 s−3 ) , (3.5) 43502-10 The EoS of a supercritical cell fluid model which coincide with similar relations for the Ising-like system in an external field [14]. The quantities f (0)CR , dl [14] are expressed by the coordinates of the fixed point. The term Ω (s) CR = kBT Nv γ̄(+)s−3(np+1) (3.6) contains a nonanalytic function of temperature τ and chemical potential µ. Here, γ̄(+) = γ̄1 + γ̄2Hc + γ̄3H2 c , (3.7) where γ̄l are constants [14] γ̄1 = f (0)CR ( 1 − s−3)−1 , γ̄2 = d1q ( 1 − E2s−3)−1 , γ̄3 = d3q2 (1 − E2 2 s−3)−1 , (3.8) which have the following values at s = s∗ γ̄1 = 1.529, γ̄2 = −0.635, γ̄3 = −0.058. (3.9) For np, Hc and s−3(np+1), we have np = − ln ( h̃2 + h2 c ) 2 ln E1 − 1 , Hc = τ̃ ( h̃2 + h2 c )−1/(2p0), s−3(np+1) = ( h̃2 + h2 c )3/5 . (3.10) Here, τ̃ = τ(c11/q) is the renormalized relative temperature, h̃ = M [ βW(0) ]1/2 , hc = τ̃p0 . (3.11) Express the exponent p0 = ln E1 ln E2 (3.12) in the following form p0 = ν/µ. (3.13) Here, ν is the exponent of the correlation length ξ = ξ± \|τ \|−ν at M = 0. It is represented by the formula ν = ln s∗ ln E2 . (3.14) The behavior of the correlation length at T = Tc [ ξ = ξ(c)M−µ ] is described by the critical exponent µ µ = 2 d + 2 . (3.15) The values ν = 0.605 and p0 = 1.512 are derived for the model ρ4 at s = s∗. Note that the quantities Ωµ and Ωr are contained in the expression of the thermodynamic potential Ω [see (3.2) and (3.3)]. A temperature dependence can be singled out from these terms, but this would cause a renormalization of coefficients of the analytic part of the thermodynamic potential. Of particular interest are the terms of the thermodynamic potential which are the nonanalytic functions of both temperature and chemical potential. The part of the thermodynamic potentialΩLGR from (3.2) can be expressed as a sum of two terms [24] ΩLGR = Ω (+) TR +Ω ′. (3.16) 43502-11 M.P. Kozlovskii, I.V. Pylyuk, O.A. Dobush The former term Ω(+)TR is a contribution to the thermodynamic potential of the transitional region of fluctuations (from non-Gaussian to Gaussian fluctuations of the order parameter). The latter term Ω′ can be derived using the Gaussian distribution of fluctuations. The term Ω(+)TR holds for the formula (see [16]) Ω (+) TR = −kBT Nv fnp+1s−3(np+1). (3.17) The expression for the coefficient fnp+1 is fnp+1 = 1 2 ln ynp + 9 4 y−2 np + 1 4 x2 np+1 + ln U(0, xnp+1). (3.18) It is easy to derive xnp+1 from the relation xnp+m = x̄Em−1 2 Hc ( 1 + ΦqEm−1 2 Hc )−1/2 , (3.19) in case of m = 1. Here, x̄ = q(u∗)−1/2√3 , Φq = q(u∗)−1/2R(0)1 . Hc is defined in (3.10), and ynp should be derived from formula for yn (2.38) in case of n = np. In order to calculate the term Ω′ [see (3.16)], we have to reckon the correspondent expression of the grand partition function Ξ′ Ξ ′ = 2(Nnp+2−1)/2 [Q(Pnp+1) ]Nnp+2 Ξnp+2 , (3.20) where Ξnp+2 = ∫ (dρ)Nnp+2 exp [ h̃ √ Nvρ0 − 1 2 ∑ ®k∈Bnp+2 gnp+2(k)ρ®k ρ−®k − a(np+2) 4 4! N−1 np+2 ∑ ®k1, . . ., ®k4 ®ki ∈Bnp+2 ρ®k1 . . . ρ®k4 δ®k1+...+®k4 ] . (3.21) The expressions for the coefficients gnp+2(k) and a(np+2) 4 are as follows: gnp+2(k) = gnp+2(0) + 2bk2, gnp+2(0) = s−2(np+2)rnp+2 , a(np+2) 4 = s−4(np+2)unp+2 , (3.22) both rnp+2 and unp+2 meet the expressions rnp+2 = q(−1 + E2Hc) , unp+2 = u∗(1 + ΦqE2Hc). (3.23) The integral in (3.21) is convergent since the coefficient unp+2 is positive in case of any values of temperature and chemical potential. The coefficient rnp+2 is positive and much larger than unp+2 in case of small values of the chemical potential (hc � h̃). However, in case of large chemical potential (hc � h̃), the coefficient rnp+2 is small and turns into negative if temperature τ continues to decrease. It is possible to calculate the expression (3.21) by using the Gaussian distribution and applying the following substitution of variables: ρ®k = η®k + √ Nvσ+δ®k . (3.24) 43502-12 The EoS of a supercritical cell fluid model As a result, Ξnp+2 = eNvE0(σ+) ∫ (dη)Nnp+2 exp [ A0 √ Nvη0 − 1 2 ∑ ®k∈Bnp+2 ḡ(k)η®kη−®k − b̄ 6 N−1/2 np+2 ∑ ®k1, . . ., ®k3 ®ki ∈Bnp+2 η®k1 . . . η®k3 δ®k1+...+®k3 − ā4 24 N−1 np+2 ∑ ®k1, . . ., ®k4 ®ki ∈Bnp+2 η®k1 . . . η®k4 δ®k1+...+®k4 ] . (3.25) Here, E0(σ+) = h̃σ+ − rnp+2 2 s−2(np+2)σ2 + − unp+2 24 s−(np+2)σ4 + , A0 = h̃ − rnp+2s−2(np+2)σ+ − unp+2 6 s−(np+2)σ3 + , ḡ(k) = ḡ(0) + 2bk2, ḡ(0) = rnp+2s−2(np+2) + unp+2 2 s−(np+2)σ2 + , b̄ = unp+2s−5(np+2)/2σ+ , ā4 = unp+2s−4(np+2). (3.26) As in [16], the magnitude of the shift σ+ is found from the condition ∂E0(σ+) ∂σ+ = 0. (3.27) Recall the expression of A0 from (3.26) to write the equation A0 = 0 , (3.28) with a solution chosen in the form of σ+ = σ0s−(np+2)/2. (3.29) The solutions of the gotten cubic equation for σ0 σ3 0 + pσ0 + q = 0 , (3.30) where p = 6 rnp+2 unp+2 , q = −6 s5/2 unp+2 h̃( h̃2 + h2 c )1/2 , is analyzed by [14, 16]. In general case (T , Tc, M , 0), the solution of the equation (3.30) is a function of both chemical potential and temperature. For all τ > 0, M , 0 the real root of this equation is found using the Cardano solution σ0 = A + B (see [27, 28]), where A = ( − q/2 +Q1/2)1/3 , B = ( − q/2 −Q1/2)1/3 , Q = (p/3)3 + (q/2)2. Direct calculation shows that Q is positive in case of T > Tc. The plot of real solution σ0 as a function of the chemical potential M for various values of τ is shown in figure 1. The Gaussian distribution of fluctuations is a basis (a zero-order approximation at k , 0) for integration in (3.25) over the variables η®k with k , 0. Singling out in (3.25) the terms with k = 0 and integrating over η®k with k , 0, we obtain Ξnp+2 = eNvE0(σ+) ∏ ®k,0 ®k∈Bnp+2 [ π/ḡ(k) ]1/2 Ξ (0) np+2 . (3.31) 43502-13 M.P. Kozlovskii, I.V. Pylyuk, O.A. Dobush Figure 1. The behavior of σ0 as a function of the chemical potential M for various values of the relative temperature τ. The arrow points the correspondence between τ and curves of σ0 at the transition from down to up in the first quadrant of the coordinate plane. Here, Ξ (0) np+2 = +∞∫ −∞ dη0 exp [ A0 √ Nvη0 − 1 2 ḡ(0)η2 0 − b̄ 6 N−1/2 np+2η 3 0 − ā4 24 N−1 np+2η 4 0 ] . (3.32) The next stage of calculations is a return to the variable ρ0 using the “reverse” to (3.24) change of variables η0 = ρ0 − √ Nvσ+ (3.33) in (3.32). The following expression is the result of integration over ρ0 in the term (3.31) of the total thermodynamic potential using the Laplace method Ωnp+2 = −kBT NvE0(σ+) − 1 2 kBT Nnp+2 lnπ + 1 2 kBT ∑ ®k,0 ®k∈Bnp+2 ln ḡ(k). (3.34) Here, E0(σ+) = e(+)0 h̃s−(np+1)/2 − e(+)2 s−3(np+1), (3.35) where the following notation is used e(+)0 = σ0s−1/2, e(+)2 = σ2 0 2 s−3 ( rnp+2 + unp+2 12 σ2 0 ) . (3.36) Note that both the change of variables (3.33) and the substitution ρ0 = √ Nvρ in (3.32) cause an appearance of a sharp maximum of the integrand expression at the point ρ̄. Both the extremum condition of the integrand, from which it is possible to define ρ̄, and the representation ρ̄ = ρ̄′s−(np+2)/2 lead to the same equation (with the same coefficients) as (3.30), where the role of σ0 is played by ρ̄′. The expressions of ρ̄′ and σ0 coincide. The quantities of ρ̄ and σ+ (3.29) are also similar. Thus, the integrand E0(ρ̄) at the point ρ̄ is represented as E0(σ+) [see (3.35)] with coefficients (3.36), where ρ̄′ is changed to σ0. The sum over ®k ∈ Bnp+2 in (3.34) is calculated using the transition to the spherical Brillouin zone. Integration 43502-14 The EoS of a supercritical cell fluid model over k: 1 2 ∑ ®k∈Bnp+2 ln ḡ(k) = Nnp+2 [ 1 2 ln ( 1 + a2) − (np + 2) ln s + 1 2 ln rR − 1 3 + 1 a2 − 1 a3 arctan a ] , a = π c ( 2b rR )1/2 , rR = rnp+2 + unp+2 2 σ2 0 . (3.37) The expressions derived for Q(Pnp+1) [see (2.36)],Ωnp+2 (3.34) and 1 2 ∑ ®k∈Bnp+2 ln ḡ(k) (3.37) are used to write the part of the thermodynamic potential, which is correspondent to (3.20), as a sum of two terms Ω ′ = Ω (+) 0 +Ω ′ G . (3.38) The term Ω (+) 0 = −kBT NvE0(σ+) (3.39) is a part of the thermodynamic potential connected to the variable ρ0. For Ω′G , one has Ω ′ G = −kBT Nnp+2 fG . (3.40) The coefficient fG is as follows: fG = [ − 1 2 ln 3 + 2 ln s + 1 2 ln unp+1 − ln rR − ln U(xnp+1) − 3 4 y−2 np+1 − f ′′G ] / 2. (3.41) Here, unp+1 = u∗(1 + ΦqHc) , xnp+1 = x̄Hc(1 + ΦqHc) −1/2. (3.42) rR and ynp+1 are defined in (3.37) and (2.38), respectively, and for f ′′G , one gets f ′′G = ln ( 1 + a2) − 2 3 + 2 a2 − 2 a3 arctan a. (3.43) The part of the thermodynamic potential can be calculatedΩLGR (3.16) using the expressions of both Ω (+) TR (3.17) and Ω′ (3.38). The complete expression of the thermodynamic potential of a cell fluid model is obtained based on (3.2) in the way of gathering derived terms from all of the regimes of fluctuations for temperature T > Tc. The quantity ln gW [see (2.9)] contained in Ωµ [see (3.3)] is found as a result of transition to the spherical Brillouin zone and integration over k: ln gW = Nv fW , fW = − 1 2 ln(2π) − 1 2 ln [ βW(0) ] + f ′W , f ′W = − 1 2 ln [ 1 − (a′)2 ] + 1 3 + 1 (a′)2 − 1 2(a′)3 ln ��1 + a′ 1 − a′ �� , a′ = π c (2b)1/2. (3.44) The complete expression of the thermodynamic potential, which is equivalent to (3.2), is represented in the form of three terms Ω = Ωa +Ω (+) s +Ω (+) 0 . (3.45) The analytic part of the thermodynamic potential Ωa is as follows: Ωa = −kBT Nv ( γ01 + γ02τ + γ03τ 2) +Ω01 , (3.46) 43502-15 M.P. Kozlovskii, I.V. Pylyuk, O.A. Dobush where Ω01 = −kBT Nv ( Eµ + γa ) , (3.47) γa = f ′W − a0 + 1 4 ln ( 3 u0 ) + x2 4 + ln U(0, x). The quantities Eµ, u0, x and a′ are defined in (2.9), (2.16), (2.29) and (3.44). The term Ω(+)s is a sum of nonanalytic contribution. It has the following form Ω (+) s = −kBT Nvγ (+) s ( h̃2 + h2 c ) d d+2 . (3.48) Here, γ (+) s = fnp+1 − γ̄ (+) + fG/s3. (3.49) The shift of the variable ρ0 is determined by the quantity σ0, which is contained in the coefficients e(+)0 and e(+)2 of the term Ω (+) 0 = −kBT Nv [ e(+)0 h̃ ( h̃2 + h2 c ) d−2 2(d+2) − e(+)2 ( h̃2 + h2 c ) d d+2 ] . (3.50) The expressions for the coefficients e(+)0 and e(+)2 are presented in (3.36). 4. An equation of state of the model atT > Tc with effects of fluctuations taken into account We derived the thermodynamic potential Ω = −kBT lnΞ [see (3.45)] for the cell fluid model taking into account non-Gaussian fluctuations of the order parameter. Using the expression of the logarithm of the grand partition function lnΞ = Nv { Pa(T) + Eµ + [ γ (+) s − e(+)2 ] ( h̃2 + h2 c ) d d+2 + e(+)0 h̃ ( h̃2 + h2 c ) d−2 2(d+2) } , (4.1) Pa(T) = γa + γ01 + γ02τ + γ03τ 2 it is possible to get the expression of the pressure P as a function of the temperature T and the chemical potential µ using the well-known formula PV = kBT lnΞ. (4.2) Having the grand partition function, it is possible to calculate the average number of particles N̄ = ∂ lnΞ ∂βµ . (4.3) The latter relation is applicable to express the chemical potential via either the number of particles or the relative density n̄ = N̄ Nv = ( N̄ V ) v , (4.4) where v is the volume of a cubic cell, which is a parameter of the model in use. Combining the equalities (4.2) and (4.3), let us find the pressure P as a function of the temperature T and the relative density n̄, which is the equation of state of the model we study. Using (4.1), (4.3) and (4.4), one obtains n̄ = ∂Eµ ∂βµ + ∂ ∂βµ [ γ (+) s ( h̃2 + h2 c ) d d+2 ] + ∂ ∂βµ [ e(+)0 h̃ ( h̃2 + h2 c ) d−2 2(d+2) − e(+)2 ( h̃2 + h2 c ) d d+2 ] . (4.5) 43502-16 The EoS of a supercritical cell fluid model Based on (4.5) and taking into account (A.1) (see Appendix A) one has n̄ = −M − ã1 + 1 βW(0) a34 + σ (+) 00 ( h̃2 + h2 c ) d−2 2(d+2) . (4.6) Here, the coefficient σ (+) 00 = e(+)0 1[ βW(0) ]1/2 ( 1 + d − 2 d + 2 h̃2 h̃2 + h2 c ) + e(+)00 h̃( h̃2 + h2 c )1/2 + e(+)01 ( h̃2 + h2 c )1/2 , (4.7) beside e(+)0 from (3.36), contains the quantities e(+)00 = 2d d + 2 1[ βW(0) ]1/2 [ γ (+) s − e(+)2 ] (4.8) and e(+)01 = ∂γ (+) s ∂βµ − ∂e(+)2 ∂βµ . (4.9) The final expression for σ(+)00 is found having calculated the coefficient e(+)01 (see Appendix B). Taking into account the expressions for ã1 [see (2.9)], d(0) and ã2 [see (2.10)] the equation (4.6) gets the following form n̄ = ng − M + σ(+)00 ( h̃2 + h2 c ) d−2 2(d+2) . (4.10) Here, ng = −a1 − a2a34 + a4 3 a3 34 , (4.11) and the coefficient σ(+)00 is defined in (B.22). The quantity h̃ is a function of M , see (3.11). The nonlinear equation (4.10) describes a connection between the density n̄ and the chemical potential M . It can be rewritten in the form n̄ − ng + M = [ Mb(+)1 b(+)2 ]1/5 σ (+) 00 . (4.12) Herefrom we obtain Mb(+)1 = [ n̄ − ng + M σ (+) 00 ]5 b(+)2 (4.13) or b(+)3 M1/5 = n̄ − ng + M . (4.14) The coefficients b(+)1 , b(+)2 and b(+)3 are given by the formulae b(+)1 = [ βW(0) ]1/2 , b(+)2 = α (1 + α2)1/2 , b(+)3 = [ b(+)1 b(+)2 ]1/5 σ (+) 00 . (4.15) Following the equation M1/5 (b(+)3 −M4/5) = 0 at n̄ = ng, one has such values of the chemical potential M: M1 = 0 , M2,3 = ± [ b(+)3 ]5/4 . The points Mm1,m2 of the extremum of n̄ are found from the following equation 1 5 b(+)3 M−4/5 m1,m2 − 1 = 0. 43502-17 M.P. Kozlovskii, I.V. Pylyuk, O.A. Dobush Evidently, Mm1,m2 = ± [ b(+)3 5 ]5/4 . Easy to see that there exists an interval of values Mm2 < M < Mm1 , (4.16) where n̄ increases with the growth of M . For all Mm2 > M > Mm1, a decrease of n̄ is observed with an increase of M , which is not a reflection of a physical phenomenon. The interval (4.16) is physical. Figure 2 and figure 3 show, respectively, the plots of functions n̄(M) and M(n̄) which are obtained from the equation (4.14) for different temperatures τ. Note that the curves for temperatures τ = 0.001, τ = 0.0005, τ = 0.0001 coincide in these figures, as well as in figure 4. In case of M � 1, there is a possibility to express the chemical potential M as a function of the relative density n̄ using the equation (4.14). The fixed-point iteration method (method of successive approximations) is suitable for solving this equation. In the zero-order approximation, the term M on the right-hand side of the equation (4.14) is neglected and the consequent result is M (0) = [ n̄ − ng σ (+) 00 ]5 b(+)2[ βW(0) ]1/2 . (4.17) The outcome of the first-order approximation, assuming that M = M (0) on the right-hand side of the equation (4.14), is either M (1) = [ n̄ − ng + M (0) σ (+) 00 ]5 b(+)2[ βW(0) ]1/2 or M (1) = M (0) [ 1 + M (0) n̄ − ng ]5 . Henceforth, the zero-order approximation of (4.17) is used for M , namely, the case M = M (0) is considered. Note that the region M > 0 is in agreement with positive values of n̄ − ng, and n̄ < ng Figure 2. Plot of the relative density n̄ as a func- tion of the chemical potential M . Figure 3. Plot of the chemical potential M as a function of the relative density n̄. 43502-18 The EoS of a supercritical cell fluid model Figure 4. Plot of the pressure P as a function of the relative density n̄ for different temperatures τ. The critical point (n̄c = 0.997, Pc = 0.474) is denoted by the symbol ◦. meet M < 0. The chemical potential is equal to zero when n̄ = ng. Taking into account (4.1) and (4.2), at T > Tc the following equation of state is derived: Pv kBT = Pa(T) + Eµ + [ γ (+) s − e(+)2 ] ( h̃2 + h2 c ) d d+2 + e(+)0 h̃ ( h̃2 + h2 c ) d−2 2(d+2) . (4.18) Here, the quantity Eµ is expressed in (2.9) where the chemical potential from the formula (4.17) should be substituted for M . Using the formula (4.17) in the expression of h̃ [see (3.11)], one gets h̃ = [ n̄ − ng σ (+) 00 ]5 b(+)2 . (4.19) The definition of hc is in (3.11). The formula (4.10) is used for determining h̃2 + h̃2 c and then rewriting the equation (4.18) as follows: Pv kBT = Pa(T) + Eµ + [ n̄ − ng σ (+) 00 ]6 [ e(+)0 α (1 + α2)1/2 + γ (+) s − e(+)2 ] . (4.20) In case of T = Tc, it is possible to describe the behavior of the system by the expression (4.20). In this situation, α → ∞. Moreover, Hc → 0 and rnp+2, unp+2, xnp+2 fail to be the functions of the chemical potential. The coefficients σ0, e(+)0 , e(+)2 and b(+)2 become some constants. The equation of state at T = Tc(τ = 0) is of the following form: Pv kBTc = Pa(Tc) + Eµ \|Tc + 5 6 (n̄ − ng)6 [ βcW(0) ]1/2[ σ (+) 00 (Tc) ]5 , (4.21) where σ (+) 00 (Tc) = 6 5 1[ βcW(0) ]1/2 [ e(+)0 + γ (+) s − e(+)2 ] . (4.22) 43502-19 M.P. Kozlovskii, I.V. Pylyuk, O.A. Dobush Table 1. The values of the temperature (Tc), density (n̄c) and pressure (Pc) at the critical point from the present research (theory), simulations and experiment. The values are presented in the form of reduced dimensionless units, where temperature kBT = kBT ′/D, density n̄ = ρR3 0 , and pressure P = P′R3 0/D( the quantities T ′, ρ, P′ have dimensionality, for example, [T ′] = [K], [ρ] = [1/m3], [P′] = [Pa] ) . kBTc n̄c Pc Theory (cell fluid model) 4.028 0.997 0.474 Simulations (continuous system with the Morse potential) 5.874 1.430 2.159 Experiment 3.713 1.215 0.415 At the critical point, the density n̄c is expressed by the quantity ng, namely n̄c = ng . (4.23) The estimated values of parameters of the critical point for sodium are shown in table 1 . These values are obtained in different ways: from our present research for the cell fluid model, from Monte Carlo simulation data for the continuous system with the Morse potential in the grand canonical ensemble [23]. In addition table 1 contains the known values from experiments for sodium [29], which are not related to any interaction potential. Therefrom, one can see that the use of the continuous Morse potential and its lattice counterpart yields results of the same order. The expressions (4.20) and (4.21) match the case where the term M is neglected on the right-hand side of the equation (4.14). If this term is not neglected, the numerical results give the behavior of pressure P as a function of an increasing density n̄ for various values of τ which is shown in figure 4. The isothermal compressibility KT = (∂η/∂ p̃)T /η, where η = n̄/n̄c, p̃ = P/Pc, is presented in figure 5. It is known that discontinuous changes of the properties of the fluid along the first-order binodal that culminates in a critical point can be extended into the supercritical region as the Widom line [17, 18]. This line is defined by the maxima of some thermodynamic quantities such as the constant-pressure specific heat, the correlation length, etc. Taking into account the extremum values of KT (see figure 5) it is possible to plot the Widom-like line of a supercritical cell fluid. The temperature dependence of the pressure P at the extrema points of KT is shown in figure 6. Figure 5. Plot of the isothermal compressibility KT as a function of the density n̄ for different values of the relative temperature τ. Figure 6. Plot of the pressure as a function of tempe- rature at the extremum points of the compressibility. 43502-20 The EoS of a supercritical cell fluid model 5. Conclusions An explicit expression for the thermodynamic potential of the cell fluid model is obtained in the collective variables representation. The basic idea of the calculation of the thermodynamic potential near Tc at a microscopic level lies in the separate inclusion of contributions from short-wave and long-wave modes of the order parameter oscillations. The short-wave modes are characterized by the presence of the RG symmetry and are described by a non-Gaussian measure density. In this case, the RG method is used. The corresponding RG transformation can be related to the case of the one-component magnet in the external field [26]. The approach which we propose is based on the use of a non-Gaussian density of measure. The inclusion of short-wave oscillation modes leads to a renormalization of the dispersion of the distribution describing long-wave modes. The way of taking into account the contribution of long-wave modes of oscillations to the thermodynamic potential of the cell fluid model is qualitatively different from the method of calculating the short-wave part of the grand partition function. The calculation of this contribution grounds on the use of the Gaussian density of measure as the basis density. The dispersion of this Gaussian distribution becomes a non-analytic function of temperature and density due to consideration of short-range fluctuations. We have developed a direct method of calculating the thermodynamic potential including both types of oscillation modes in a supercritical region. The nonlinear equation which links the relative density n̄ and the chemical potential M is derived and investigated. The expressions for coefficients of this equation are represented as functions of the ratio of the renormalized chemical potential to the renormalized temperature. The quantity n̄ corresponding to M = 0 is obtained. The interval of values of the chemical potential, within which the density n̄ increases with the growth of M is indicated. Reduction of n̄ with an increase of M beyond this range does not reflect the physical nature of the phenomenon. The chemical potential is expressed in terms of the density. The equation of state for the case of temperatures above the critical value of Tc obtained in the present research provides the pressure as a function of temperature and density. The equation of state corresponding to the case of T = Tc is also derived. The main advantage of this equation of state is the presence of relations connecting its coefficients with the fixed-point coordinates and the microscopic parameters of the interaction potential. It shows the capability of the collective variables method to be efficient in describing both universal and non-universal characteristics of the system as functions of microscopic parameters. This possibility is rather unusual within an RG approach since, for example, it is well-known that the RG of perturbative field theory is incapable of explicitly accounting for a non-universal effect of a particular microscopic parameter of a specific system. Using the analytic expressions obtained in this study, we made numerical estimations. In particular, the results of the estimates are represented as plots of compressibility and isotherms of pressure as functions of density. It is also shown that the extrema of compressibility form a line on the (P,T) plane, which is similar to the Widom line. The technique elaborated here for the derivation of the equation of state at temperatures above the Tc is envisaged to be generalised to the case of T < Tc. The calculations can be extended to the higher non-Gaussian distribution (the ρ6 model) [24, 25]. Acknowledgements Authors thank their colleague Ihor Omelyan for his valuable help with numerical calculations. This work was partly supported by the European Commission under the project STREVCOMS PIRSES-2013-612669. 43502-21 M.P. Kozlovskii, I.V. Pylyuk, O.A. Dobush A. Deriving the expression of the relative density n̄ In order to derive the expression of the relative density n̄ (4.5), one should take the following partial derivatives ∂Eµ ∂βµ = −M − ã1 + a34 βW(0) , ∂ ∂βµ [ γ (+) s ( h̃2 + h2 c ) d d+2 ] = ( h̃2 + h2 c ) d−2 2(d+2) [( h̃2 + h2 c )1/2 ∂γ (+) s ∂βµ + 2d d + 2 γ (+) s × 1[ βW(0) ]1/2 h̃( h̃2 + h2 c )1/2 ] , ∂ ∂βµ [ e(+)0 h̃ ( h̃2 + h2 c ) d−2 2(d+2) − e(+)2 ( h̃2 + h2 c ) d d+2 ] = ( h̃2 + h2 c ) d−2 2(d+2) × [ e(+)0 1[ βW(0) ]1/2 ( 1 + d − 2 d + 2 h̃2 h̃2 + h2 c ) − 2d d + 2 e(+)2 1[ βW(0) ]1/2 h̃( h̃2 + h2 c )1/2 − ( h̃2 + h2 c ) 1 2 ∂e(+)2 ∂βµ ] . (A.1) Note that during the calculation of the latter formula (A.1), the derivatives of σ0 with respect to βµ give the expression which coincide with the condition (3.28). Therefore, the correspondent terms compensate each other. That is why, in the calculation scheme described above, the quantity σ0 is considered not to be a function of chemical potential. The derivative ∂e(+)2 /∂βµ is as follows: ∂e(+)2 ∂βµ = 1 2 σ2 0 s−3 ( ∂rnp+2 ∂βµ + 1 12 σ2 0 ∂unp+2 ∂βµ ) , (A.2) where ∂rnp+2 ∂βµ = 1[ βW(0) ]1/2 qE2 ∂Hc ∂ h̃ , ∂unp+2 ∂βµ = 1[ βW(0) ]1/2 u∗ΦqE2 ∂Hc ∂ h̃ . (A.3) The expression for Hc [see (3.10)] is applicable to find the derivative of Hc with respect to h̃ ∂Hc ∂ h̃ = − Hc p0 h̃ h̃2 + h2 c . (A.4) B. Calculation of σ(+) 00 Let us provide the expressions of derivatives contained in the relation (4.9). Taking into account the formulae (A.2)–(A.4), one can derive the equality ∂e(+)2 ∂βµ = − 1[ βW(0) ]1/2 qsσ2 0 ( 1 + ql 12 σ2 0 ) ( h̃2 + h2 c )−1/2 , (B.1) where the constants qs and ql are as follows: qs = E2 2p0 qs−3Hc α( 1 + α2)1/2 , ql = Φqu∗q−1, (B.2) 43502-22 The EoS of a supercritical cell fluid model and α = h̃/hc is the ratio of the renormalized chemical potential h̃ = M [ βW(0) ]1/2 (M = µ/W(0) − ã1) to the renormalized temperature hc = τ̃(d+2)ν/2 [τ̃ = τ(c11/q)]. The following is taken into account here: ∂Hc ∂βµ = ∂Hc ∂ h̃ ∂ h̃ ∂βµ = − 1[ βW(0) ]1/2 Hcd ( h̃2 + h2 c )−1/2 , where Hcd = Hc p0 α (1 + α2)1/2 . The explicit expression of the derivative ∂γ (+) s ∂βµ = ∂ fnp+1 ∂βµ − ∂γ̄(+) ∂βµ + s−3 ∂ fG ∂βµ (B.3) can be found by deriving the expressions of its each term. Let us continue with the derivative of fnp+1 (3.18) with respect to the chemical potential. Note that ∂ynp+m ∂βµ = ynp+mrp+m ∂xnp+m ∂βµ , rp+m = U ′(xnp+m) U(xnp+m) − 1 2 ϕ′(xnp+m) ϕ(xnp+m) , U ′(xnp+m) = 1 2 U2(xnp+m) + xnp+mU(xnp+m) − 1 , ϕ′(xnp+m) = 6U ′(xnp+m)U(xnp+m) + 2U(xnp+m) + 2xnp+mU ′(xnp+m) , (B.4) moreover, for xnp+m (3.19), we have ∂xnp+m ∂βµ = 1[ βW(0) ]1/2 gp+m ( h̃2 + h2 c )−1/2 . (B.5) Here, gp+m = −x̄HcdEm−1 2 ( 1 + ΦqHcEm−1 2 )−1/2 [ 1 − Φq 2 HcEm−1 2 ( 1 + ΦqHcEm−1 2 )−1 ] . (B.6) Taking into account the equalities represented above, ∂ fnp+1 ∂βµ = 1[ βW(0) ]1/2 fp ( h̃2 + h2 c )−1/2 , (B.7) where fp = 1 2 rpgp ( 1 − 9/y2 np ) − 1 2 gp+1U(xnp+1). (B.8) The derivative of γ̄(+) [see (3.7)] with respect to the chemical potential is of the form ∂γ̄(+) ∂βµ = − 1[ βW(0) ]1/2 γp ( h̃2 + h2 c )−1/2 . (B.9) Here, γp = Hcd (γ̄2 + 2γ̄3Hc) . (B.10) Calculating the derivative of fG (3.41) with respect to the chemical potential, one should take into account that the shift σ0 is a function of chemical potential. To derive ∂σ0/∂βµ, let us use the equality (3.28), which makes possible to obtain ∂σ0 ∂βµ = 1[ βW(0) ]1/2 gσ ( h̃2 + h2 c )−1/2 , (B.11) 43502-23 M.P. Kozlovskii, I.V. Pylyuk, O.A. Dobush where gσ = s5/2 rR 1 1 + α2 + σ0 rR HcdqE2 ( 1 + ql 6 σ2 0 ) . (B.12) The derivative of rR is ∂rR ∂βµ = 1[ βW(0) ]1/2 gR ( h̃2 + h2 c )−1/2 , (B.13) where gR = −qE2Hcd ( 1 + 1 2 qlσ2 0 ) + unp+2gσσ0 , (B.14) and the derivative of a from (3.37) is ∂a ∂βµ = 1[ βW(0) ]1/2 ga ( h̃2 + h2 c )−1/2 , ga = − agR 2rR . (B.15) The derivative of the quantity f ′′G (3.43) contained in fG [see (3.41)] is as follows: ∂ f ′′G ∂βµ = 1[ βW(0) ]1/2 gaag ( h̃2 + h2 c )−1/2 . (B.16) Here, ag = 2a 1 + a2 − 4 a3 + 6 a4 arctan a − 2 a3 1 1 + a2 . (B.17) Taking into account the expressions represented above, we come to the following expression ∂ fG ∂βµ = 1[ βW(0) ]1/2 fgv ( h̃2 + h2 c )−1/2 , (B.18) where fgv = − 1 4 u∗Φq unp+1 Hcd − 1 2 ( gR rR + gaag ) + gp+1 [ 3 4 rp+1 y2 np+1 − 1 2 U ′(xnp+1) U(xnp+1) ] . (B.19) Summing up the terms (B.7), (B.9) and (B.18), we obtain the derivative of γ(+)s with respect to the chemical potential based on (B.3) ∂γ (+) s ∂βµ = 1[ βW(0) ]1/2 fγ1 ( h̃2 + h2 c )−1/2 . (B.20) Here, fγ1 = fp + γp + fgv/s3. (B.21) The quantities fp, γp and fgv are functions of α only. The obtained expressions (4.8), (4.9), (B.1) and (B.20) are helpful in calculating the coefficient σ(+)00 in the equality (4.6). Therefore, σ (+) 00 = e(+)0 1[ βW(0) ]1/2 ( 1 + d − 2 d + 2 α2 1 + α2 ) + e(+)00 α( 1 + α2)1/2 + e(+)02 , (B.22) where e(+)02 = e(+)01 ( h̃2 + h2 c )1/2 = 1[ βW(0) ]1/2 [ fγ1 + qsσ2 0 ( 1 + qlσ2 0 /12 ) ] . (B.23) Obviously, σ(+)00 is not a function of either temperature or of chemical potential. It is a function of their ratio α = h̃ hc = [ βW(0) ]1/2 ( q c11 )p0 α0. 43502-24 The EoS of a supercritical cell fluid model The multiplier α0 = µ/W(0) − ã1 τp0 is defined by the ratio of initial µ and τ. References 1. Marr R., Gamse T., Chem. Eng. Process. 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Kozlovskii, I.V. Pylyuk, O.A. Dobush Рiвняння стану комiркової моделi плину в надкритичнiй областi М.П. Козловський, I.В. Пилюк, О.А. Добуш Iнститут фiзики конденсованих систем НАН України, вул. Свєнцiцького, 1, 79011 Львiв, Україна Використовуючи множину колективних змiнних та перетворення ренормалiзацiйної групи, розвинено аналiтичний метод розрахунку рiвняння стану комiркової моделi плину в областi вище критичної темпе- ратури (T > Tc).Математичний опис з урахуванням негаусових флуктуацiй параметра порядку виконано в околi критичної точки на основi моделi ρ4. Запропонований метод розрахунку великої статистичної суми дозволяє отримати, крiм унiверсальних величин, зокрема, критичних показникiв кореляцiйної дов- жини, рiвняння для критичної температури моделi плину. Побудовано кривi iзотермiчної стисливостi як функцiї густини. Також зображено лiнiю екстремумiв стисливостi у надкритичнiй областi. Ключовi слова: комiркова модель плину, критичнi показники, рiвняння стану, надкритична область 43502-26 Introduction Basic expressions A thermodynamic potential of the model An equation of state of the model at TTc with effects of fluctuations taken into account Conclusions Deriving the expression of the relative density Calculation of 00(+)

The equation of state of a cell fluid model in the supercritical region

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