Phase transition in a cell fluid model

We propose a method of describing a phase transition in a cell fluid model with pair interaction potential that includes repulsive and attractive parts. An exact representation of the grand partition function of this model is obtained in the collective variables set. The behavior of the system at...

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Published in:	Condensed Matter Physics
Date:	2017
Main Authors:	Kozlovskii, M.P., Dobush, O.A.
Format:	Article
Language:	English
Published:	Інститут фізики конденсованих систем НАН України 2017
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Cite this:	Phase transition in a cell fluid model / M.P. Kozlovskii, O.A. Dobush // Condensed Matter Physics. — 2017. — Т. 20, № 2. — С. 23501: 1–18. — Бібліогр.: 32 назв. — англ.

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spelling	Kozlovskii, M.P. Dobush, O.A. 2019-06-19T12:22:52Z 2019-06-19T12:22:52Z 2017 Phase transition in a cell fluid model / M.P. Kozlovskii, O.A. Dobush // Condensed Matter Physics. — 2017. — Т. 20, № 2. — С. 23501: 1–18. — Бібліогр.: 32 назв. — англ. 1607-324X PACS: 51.30.+i, 64.60.fd DOI:10.5488/CMP.20.23501 arXiv:1604.08406 https://nasplib.isofts.kiev.ua/handle/123456789/156986 We propose a method of describing a phase transition in a cell fluid model with pair interaction potential that includes repulsive and attractive parts. An exact representation of the grand partition function of this model is obtained in the collective variables set. The behavior of the system at temperatures below and above the critical one is explored in the approximation of a mean-field type. An explicit analytic form of the equation of state which is applicable in a wide range of temperatures is derived, taking into account an equation between chemical potential and density. The coexistence curve, the surface of the equation of state and the phase diagram of the cell Morse fluid are plotted. Запропоновано метод опису фазового переходу в комiрковiй моделi плину iз парним потенцiалом взаємодiї, що мiстить частини вiдштовхування i притягання. Отримано точне представлення великої статистичної суми цiєї моделi в формалiзмi колективних змiнних. У наближеннi типу молекулярного поля дослiджено поведiнку системи при температурах вищих i нижчих, нiж критична. Використовуючи рiвняння, що зв’язує хiмiчний потенцiал та густину, розраховано явну аналiтичну форму рiвняння стану, що справедливе для широкої областi температур. Представлено криву спiвiснування, поверхню рiвняння стану та дiаграму стану комiркового плину Морзе. This work was partly supported by the European Commission under the project STREVCOMS PIRSES-2013-612669, FP7 EU IRSES projects No. 269139 (DCPPhysBio). en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Phase transition in a cell fluid model Фазовий перехiд в комiрковiй моделi плину Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Phase transition in a cell fluid model
spellingShingle	Phase transition in a cell fluid model Kozlovskii, M.P. Dobush, O.A.
title_short	Phase transition in a cell fluid model
title_full	Phase transition in a cell fluid model
title_fullStr	Phase transition in a cell fluid model
title_full_unstemmed	Phase transition in a cell fluid model
title_sort	phase transition in a cell fluid model
author	Kozlovskii, M.P. Dobush, O.A.
author_facet	Kozlovskii, M.P. Dobush, O.A.
publishDate	2017
language	English
container_title	Condensed Matter Physics
publisher	Інститут фізики конденсованих систем НАН України
format	Article
title_alt	Фазовий перехiд в комiрковiй моделi плину
description	We propose a method of describing a phase transition in a cell fluid model with pair interaction potential that includes repulsive and attractive parts. An exact representation of the grand partition function of this model is obtained in the collective variables set. The behavior of the system at temperatures below and above the critical one is explored in the approximation of a mean-field type. An explicit analytic form of the equation of state which is applicable in a wide range of temperatures is derived, taking into account an equation between chemical potential and density. The coexistence curve, the surface of the equation of state and the phase diagram of the cell Morse fluid are plotted. Запропоновано метод опису фазового переходу в комiрковiй моделi плину iз парним потенцiалом взаємодiї, що мiстить частини вiдштовхування i притягання. Отримано точне представлення великої статистичної суми цiєї моделi в формалiзмi колективних змiнних. У наближеннi типу молекулярного поля дослiджено поведiнку системи при температурах вищих i нижчих, нiж критична. Використовуючи рiвняння, що зв’язує хiмiчний потенцiал та густину, розраховано явну аналiтичну форму рiвняння стану, що справедливе для широкої областi температур. Представлено криву спiвiснування, поверхню рiвняння стану та дiаграму стану комiркового плину Морзе.
issn	1607-324X
url	https://nasplib.isofts.kiev.ua/handle/123456789/156986
citation_txt	Phase transition in a cell fluid model / M.P. Kozlovskii, O.A. Dobush // Condensed Matter Physics. — 2017. — Т. 20, № 2. — С. 23501: 1–18. — Бібліогр.: 32 назв. — англ.
work_keys_str_mv	AT kozlovskiimp phasetransitioninacellfluidmodel AT dobushoa phasetransitioninacellfluidmodel AT kozlovskiimp fazoviiperehidvkomirkoviimodeliplinu AT dobushoa fazoviiperehidvkomirkoviimodeliplinu
first_indexed	2025-11-25T20:56:21Z
last_indexed	2025-11-25T20:56:21Z
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fulltext	Condensed Matter Physics, 2017, Vol. 20, No 2, 23501: 1–18 DOI: 10.5488/CMP.20.23501 http://www.icmp.lviv.ua/journal Phase transition in a cell fluid model M.P. Kozlovskii, O.A. Dobush Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii St., 79011 Lviv, Ukraine Received September 17, 2016, in final form February 5, 2017 We propose a method of describing a phase transition in a cell fluid model with pair interaction potential that includes repulsive and attractive parts. An exact representation of the grand partition function of this model is obtained in the collective variables set. The behavior of the system at temperatures below and above the critical one is explored in the approximation of a mean-field type. An explicit analytic form of the equation of state which is applicable in a wide range of temperatures is derived, taking into account an equation between chem- ical potential and density. The coexistence curve, the surface of the equation of state and the phase diagram of the cell Morse fluid are plotted. Key words: cell model, collective variables, equation of state, Morse fluid, phase transition PACS: 51.30.+i, 64.60.fd 1. Introduction Description of phase transitions in fluids remains a ponderable area of investigations in statistical physics both in macroscopic and microscopic scales. During many years of research in the field due to hard work of scientists it became known that the phase transition is possible either at the infinite volume or at the thermodynamic limit and is characterized by a density jump on the isotherm below the critical temperature [1]. Plenty of successful studies were dedicated to the explanation of the nature of the first-order phase transitions at a macroscopic level, but a resemblant theoretical description at the microscopic stage is still a question to be answered. Most approaches to a description of phase transitions and critical phenomena in fluids are based on the following: complete scaling ideas [2, 3]; methods of the theory of integral equations, in particular, self-consistent Ornstein-Zernike approximation (SCOZA) [4, 5]; perturbation series expansion such as hierarchical reference theory [6]; non-perturbative renormalization group approach [7]; method based on the study of the behavior of the virial equation of state with extrapolated coefficients [8]; numerical methods and computer simulations. There are only a few recent works devoted to finding possible ways of solving this problem in the framework of grand canonical ensemble. Among them, there is a collective variables method [9] used by Yukhnovskii to completely integrate the grand partition function of a system of interacting particles in the phase-space of collective variables and, therefore, investigate its behavior in the vicinity of a critical point, or the method developed by Tang [10] where he combined the grand canonical ensemble with density functional to explain the occurrence of a first-order phase transition in homogeneous fluids. The objective of our investigation is microscopic description of a first-order phase transition in a cell fluid model. We do this exclusively on the basis of grand canonical ensemble. The model that we used as an approximation of a continuous system is quite similar to the cell gas model [11]. Thermodynamic functions, correlation functions and free energy of the latter appeared to be close to the corresponding values of continuous systems if the cells size tends to zero [12]. The values of parameters of the interaction potential are required to obtain numerical results. For this purpose, we chose the Morse potential as a potential of interaction. The Morse fluid has already been well studied, e.g., within the integral equation approach [13], by molecular dynamics simulations 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. 23501-1 https://doi.org/10.5488/CMP.20.23501 http://www.icmp.lviv.ua/journal http://creativecommons.org/licenses/by/4.0/ M.P. Kozlovskii, O.A. Dobush of simple fluids [14], by Monte Carlo simulations using both NpT plus test particle method [15] and the grand-canonical transition matrix method [16]. The application of such a potential appeared to be sufficient to describe a liquid-vapor coexistence in liquid metals [16]. This paper is laid as follows: in section 2, the Jacobian of transition from individual to collective variables is calculated. A rigorous representation of the GPF of the cell fluid in the form of multiple integral over collective variables is obtained in section 3. In section 4, this expression is restricted to ρ4- model and is calculated in the mean-field approximation. Sections 4 and 5 are devoted to the exploration of the behavior of a system in a wide temperature range except the vicinity of the critical point, and an equation of state of the cell fluid applicable in this range is derived. Conclusions are presented in section 6. 2. A cell model for describing the behavior of a simple fluid To follow our goal, we calculate the grand partition function (GPF) of the cell fluid model. Similar to the case of a cell gas model, the idea of the cell fluid consists in a fixed partition of the volumeV of a systemwith N particles on Nv congruent cubic cells, each being of a volume v = V/Nv [17]. The GPF of this system can be written as follows: Ξ = ∞∑ N=0 zN N! ∫ V (dx)N exp − β 2 ∑ ®l1,®l2∈Λ Ũl12 ρ®l1 (η)ρ®l2 (η)  . (2.1) Here, z = e βµ is the activity, β is the inverse temperature, µ is the chemical potential. In this expression, integration is made over the coordinates of all the particles in the system ∫ 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, the difference between two cell vectors is noted as l12 = \|®l1 − ®l2 \|. Each ®li takes values from a set Λ, defined as follows: Λ = { ®l = (l1, l2, l3)\|li = cmi; mi = 1, 2, . . . , N1; i = 1, 2, 3; Nv = N3 1 } . Here, c is the side of a cell, N1 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) The characteristic functions (indicators) I∆ ®l (x) I∆®l (x) = { 1, if x ∈ ∆ ®l , 0, if x < ∆ ®l (2.3) identify the particles in each cubic cell ∆ ®l = (−c/2, c/2]3 ⊂ R3 and their contribution to the interaction of a model. This is the way to plunk all the particles and sort them to different cells. The following expression is obvious ∑ ®l∈Λ ρ®l (η) = N . Particles with different coordinates can get into the same cell, where they interact regardless of the distance between them by equally repelling each other. Interaction between the constituents of one cell and the particles hosted in another cell is expressed as the function Ũl12 dependent on the distance between cells. This function consists of repulsive and attractive parts, respectively Ũl12 = Ψl12 − Ul12 . Both Ψl12 and Ul12 are positive. As an example, we choose Ũl12 in the form of the Morse potential Ψl12 = De−2(l12−1)/αR, Ul12 = 2De−(l12−1)/αR, (2.4) 23501-2 Phase transition in a cell fluid model 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]. In terms of convenience, the R0-units will be used for length measuring. Rewriting the GPF (2.1) in terms of Fourier representation gives us a sum of diagonal terms in the exponent with interaction potential Ξ = ∞∑ N=0 zN N! ∫ V (dx)N exp − β 2 ∑ ®k∈Bc Ũ(k)ρ̂®k ρ̂−®k  , k = \| ®k \|, (2.5) which is more prospective for calculation than the expression (2.1). The variable ρ̂®k is the representation of the occupation number ρ ®l (η) in reciprocal space ρ̂®k = 1 √ Nv ∑ ®l∈Λ ρ ®l (η)e i®k®l . Vector ®k takes values from the set Bc corresponding to one cell Bc = { ®k = (k1, k2, k3) ��ki = −πc + 2π c ni N1 , ni = 1, 2, . . . , N1; i = 1, 2, 3; Nv = N3 1 } . The Fourier transform of the Morse potential (2.4) Ũ(k) = −U(k) + Ψ(k) is as follows: U(k) = U(0) ( 1 + α2 Rk2 )−2 , Ψ(k) = Ψ(0) ( 1 + α2 Rk2 4 )−2 , (2.6) U(0) = 16Dπ α3 R v eR0/α, Ψ(0) = Dπ α3 R v e2R0/α . (2.7) At this stage, the GPF (2.1) of N interacting particles will be written in the form of N integrals over their coordinates and Nv integrals over the collective variables (CV) ρ®k [18] Ξ = ∞∑ N=0 zN N! ∫ V (dx)N exp − β 2 ∑ ®k∈Bc Ũ(k)ρ̂®k ρ̂−®k  ∫ (dρ)Nv ∫ (dν)Nv exp 2πi ∑ ®k∈Bc ν®k(ρ®k − ρ̂®k)  . (2.8) Herewith (dρ)Nv = ∏ ®k∈Bc dρ®k , (dν)Nv = ∏ ®k∈Bc dν®k . Let us make an identical transformation of the potential of interaction Ũ(k) as follows: Ũ(k) = −W(k) + βc β χΨ(0). The effective interaction potential W(k) is expressed as follows: W(k) = U(k) − Ψ(k) + βc β χΨ(0). (2.9) Hence, χ is the real positive parameter (χ > 0), βc = 1/kBTc, kB is the Boltzmann constant, Tc is some fixed temperature which will be defined later. A part of the repulsive interaction βc χΨ(0) > 0 is now transferred to the Jacobian of transition from individual coordinates to collective variables. This allows us to write an accurate representation of the Jacobian of transition. As we will see subsequently, it will form a new measure of distribution of density fluctuations. 23501-3 M.P. Kozlovskii, O.A. Dobush As a result of the above-mentioned transformations, the GPF takes the form Ξ = ∞∑ N=0 1 N! ∫ V (dx)N ∫ (dρ)Nv ∫ (dν)Nv exp 2πi ∑ ®k∈Bc ν®k ( ρ®k − ρ̂®k ) × exp βµρ0 + β 2 ∑ ®k∈Bc W(k)ρ®k ρ−®k − βc 2 χΨ(0) ∑ ®k∈Bc ρ̂®k ρ̂−®k  , (2.10) here, exp(βµN) = exp(βµρ0) and ρ0 = ρ®k \|®k=0 . To integrate over the coordinates x1, . . . , xN it is enough to consider a part of (2.10) in terms of ρ ®l (η) exp − βc 2 χΨ(0) ∑ ®k∈Bc ρ̂®k ρ̂−®k  = Nv∏ l=1 exp [ − βc 2 χΨ(0)ρ2 ®l (η) ] and make the following transformation exp [ − βc 2 χΨ(0)ρ2 ®l (η) ] = gΨ ∞∫ −∞ dϕ ®l exp [ − 1 4p ϕ2 ®l + iϕ ®lρ ®l (η) ] , gΨ = (4πp)− 1 2 . The parameter p is real and positive. It is expressed by the following formula: p = βc χΨ(0)/2. (2.11) After integrating over the coordinates x1, . . . , xn (see appendix A) we obtain a functional representa- tion of the GPF of the cell fluid model in the way similar to the one we did before [19] 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) ] . (2.12) Now, the difference between the cell gas model [11] and the cell fluid model becomes clear. Molecules of a cell gas move in space in such a way that each cell can house no more than one particle. This is somewhat similar to the lattice-gas model, which is frequently used for describing simple fluids. It is known that in the lattice-gas, only one particle can reside on each site, otherwise the site is vacant. As regards the cell fluid model, we introduce the occupation numbers of cells ρ ®l (η) which can take values m = 0, 1, 2, . . . . This means that each cell of a cell fluid can host an arbitrary number of particles. Due to the term e−pm2 , the more m increases the less the m-th term contributes to the sum in (2.12). Therefore, the probability of finding many particles in one elementary cube is very small. 3. Functional representation of the grand partition function of the cell fluid model At this stage, we propose two ways to deal with the calculation of the GPF (2.12). The former is presented in our recent study [19]. The latter, which in our opinion provides more transparent results, will be considered here. Firstly, let us make the Stratonovich-Hubbard transformation exp  β 2 ∑ ®k∈Bc W(k)ρ®k ρ−®k  = gW ∫ (dt)Nv exp − 1 2β ∑ ®k∈Bc t®k t −®k W(k) + ∑ ®k∈Bc t®k ρ®k  , (3.1) gW = ∏ ®k∈Bc [2πβW(k)]−1/2 . 23501-4 Phase transition in a cell fluid model The variables t®k are complex values t®k = t(c) ®k − it(s) ®k , where t(c) ®k and t(s) ®k are real and imaginary parts, respectively. For an element of a phase volume, we have (dt)Nv = dt0 ∏ ®k∈Bc ′ dt(c) ®k dt(s) ®k . The prime above the product means that ®k > 0. Note that the mandatory condition for transformation (3.1) is W(k) > 0. Taking into account the form of the interaction potential (2.4), it is easy to see that the effective potential W(k) (2.9) is positive when ln 2 < R0/α < 4 ln 2 (3.2) and χ > 0. The left-hand side of inequality (3.2) is peculiar for the Morse potential, which was used [20–22] for a description of metals (particularly, Cu, Fe, Al, Au, Na and so on). Taking into account transformation (3.1), the GPF (2.12) can be written as follows: Ξ = gW ∫ (dt)Nv exp − 1 2β ∑ ®k∈Bc t®k t −®k W(k)  J(t̃ ®l). (3.3) The Jacobian of transition J(t̃®l) factors in l is as follows: J(t̃ ®l) = Nv∏ l=1 ∞∑ m=0 vm m! e−pm 2 ∞∫ −∞ dν ®l e−2πiν ®lN ∞∫ −∞ dρ ®l exp [ 2πi ( ν ®l + t̃ ®l 2π i ) ρ ®l ] , (3.4) t̃ ®l = t ®l + βµ, t ®l = 1 √ Nv ∑ ®k∈Bc t®ke−i®k®l . (3.5) Obviously, integrals over ρ ®l in (3.4) are delta-functions, which somehow remove the integration over ν ®l . Consequently, we come to the following expression of the GPF Ξ = gW ∫ (dt)Nv exp − 1 2β ∑ ®k∈Bc t®k t −®k W(k)  Nv∏ l=1 ( ∞∑ m=0 vm m! emt̃ ®l e−pm 2 ) . (3.6) The series under the product of l in (3.6) can be expressed as a cumulant series J̃l(t̃ ®l) = exp [ − ∞∑ n=0 an n! t̃ n ®l ] . (3.7) Coefficients an are explicit analytical functions of the parameters p and v (see appendix B) which is in contradiction to our previous work [19], where such coefficients were expressed in a complicated way via the integrals containing restricted cumulant series. Therefore they could be only numerically calculated. Consequently, we obtained an approximated expression of the GPF [19]. Evidently, p and v appeared to be instrumental parameters in the present approach. Moreover, there is a monosemantic dependence between them (for more detail see appendix C). It is worth using a space of t̃®k to make the following calculation of the GPF. In this case, the Jacobian of transition fails to be a function of chemical potential, which actually makes calculation much easier. So, let us transit in (3.6) to the variables t̃®k using (3.5), and rewrite it in the form Ξ = gW exp { − [ a0 + βµ2 2W(0) ] Nv } ∫ (dt̃)Nv exp {√ Nv [ µ W(0) − a1 ] t̃0 − 1 2 ∑ ®k∈Bc D(k)t̃®k t̃ −®k − ∞∑ n=3 an n! N 2−n 2 v ∑ ®k1,... , ®kn ®ki ∈Bc t̃®k1 . . . t̃®knδ®k1+...+®kn } . (3.8) 23501-5 M.P. Kozlovskii, O.A. Dobush W(0) is the effective potential of interaction at \| ®k \| = 0 and D(k) = a2 + 1 W(k)β . Note that the functional representation (3.8) is exact for the cell fluid model, in comparison with the visually similar expression obtained in our previous manuscript [19], which is approximate. To derive an equation of state and physical characteristics of the system, one should calculate the GPF. Computation of (3.8) would be an exact solution of the problem, but this task is rather complicated, whereas expression (3.8) contains an infinite number of terms in the exponent. Thereby, we use the ρ4-model approximation, which means that the cumulant series in (3.8) are restricted to n 6 4. We made sure [23–26] that this approximation provides a qualitatively satisfactory description of a second-order phase transition with non-classical values of critical exponents for the Ising model. So, we hope that the same approximation would work for the cell fluid model as well. 4. Calculating an explicit form of the GPF in the approximation of a mean-field type The expression obtained in the previous section is similar to the partition function of a three- dimensional Ising model in an external field [27]. In our case, µ plays the role of the field. To calculate the GPF (3.8), we should make a change of variables t̃®k = ρ®k + nc √ Nvδ®k , nc = − a3 a4 , which is aimed at destroying the terms proportional to the third power of the variable t̃®k . As a result, we obtain the following expression Ξ = gW eNv (Eµ−a0) ∫ (dρ)Nv exp [√ NvMρ0 − 1 2 ∑ ®k∈Bc d(k)ρ®k ρ−®k − a4 24 1 Nv ∑ ®k1,... , ®k4 ®ki ∈Bc ρ®k1 . . . ρ®k4 δ®k1+...+®k4 ] , (4.1) where Eµ = − βW(0) 2 (M + ã1) 2 + Mnc + d(0) 2 n2 c − a4 24 n4 c , M = µ W(0) − ã1, ã1 = a1 + d(0)nc + a4 6 n3 c . (4.2) The coefficient d(k) is of the form d(k) = 1 βW(k) − ã2 , ã2 = a4 2 n2 c − a2 . (4.3) Currently, we are interested in the behavior of the cell fluid in a wide range of temperatures. The results obtained for a magnet in an external field [28, 29] as well as for fluids [9, 30] point out that the variable ρ0 with ®k = 0 (®k is the wave vector of a fluctuation mode) gives a main contribution to a free energy in the range of temperatures far from the vicinity of the critical point. The terms that include ρ®k with ®k , 0 fail to play a part to the average density of the system n̄ = 1 Nv ∂ lnΞ ∂βµ , (4.4) since integration over these variables in (4.1) is not related to the chemical potential, namely a derivative of these terms with respect to the chemical potential is equal to zero. Obviously, such terms do not make the density dependent contribution to the pressure PV = kBT lnΞ. (4.5) 23501-6 Phase transition in a cell fluid model Thus, let us make such a type of mean-field approximation and consider the collective variables ρk with ®k = 0 both with neglecting all the ρk with ®k , 0 in expression (4.1). In this approximation, the GPF is of the form Ξ = g′W eNvEµ ∞∫ −∞ dρ0 exp [NvE(ρ0)] . (4.6) E(ρ0) is obtained using the change of variables ρ′0 = √ Nvρ0 E(ρ0) = Mρ0 − 1 2 d(0)ρ2 0 − a4 24 ρ4 0 . (4.7) The temperature of transition in the mean-field approximation [31, 32] can be determined from the following condition d(0) = 1 βcW(0,Tc) − ã2 = 0. (4.8) Hence, we obtain the expression for the critical temperature kBTc = ã2W(0,Tc), (4.9) where W(0,Tc) = W(0)\|T=Tc is the value of the effective interaction potential W(0) at the critical tem- perature. Since the coefficients an of the Jacobian of transition are the functions of p, this parameter influences on the value of Tc. This dependence is important for applying the approach proposed in the present paper to particular physical systems. Taking into account the form of the effective potential (2.9), it is easy to make sure that temperature dependence of d(0) can be expressed as follows: d(0) = ã2γτ, τ = (T − Tc) Tc , γ = A0 A0 + (τ + 1)χ = − Ũ(0) W(0) , A0 = 16e− R0 α − 1. (4.10) Recall that Ũ(0) = U(0)−Ψ(0) is the Fourier transformof the interaction potential between the constituents of the system. Since the value Nv in the exponent of (4.6) tends to infinity in the thermodynamic limit, we can use the Laplace method to make integration in this expression (see [19]) and obtain the GPF as follows: Ξ ' g′W exp [ NvEµ + NvE(ρ̄0) ] , (4.11) where the value ρ0 = ρ̄0 can be found from the equation ρ̄ 3 0i + pQ ρ̄0i + qQ = 0, pQ = 6d(0) a4 , qQ = − 6M a4 . (4.12) ρ̄0 should give a maximum of E(ρ0). Having an explicit expression of the GPF (4.11), one can write the average density of the system using (4.4) n̄ = 〈N〉 Nv = ∂Eµ ∂βµ + ∂E0(ρ̄0) ∂βµ ⇒ n̄ = ng − M + ã2κ ρ̄0 , (4.13) where ng = −a1 − a2nc + a4 3 n3 c , βW(0) = 1 ã2κ , κ = τγ + 1, (4.14) since ρ̄0 = ρ̄0(τ, M) equation (4.13) describes a link between the density n̄ and the chemical potential M . This equation is non-linear. It has a different number of real solutions depending on whether τ is positive or negative. At τ < 0, there exist three real solutions ρ̄0i , while for T > Tc, there is only one solution. For all τ, there are three possible values of M , which correspond to the same one n̄. It is important that we can find the chemical potential as a function of the average density, but it appeared to be not an 23501-7 M.P. Kozlovskii, O.A. Dobush easy thing to express M = f (n̄, τ) using (4.13). However, we can act in a different way to link n̄ with M by writing an equation that meets the condition of maximum of E(ρ0) (4.7) M = d(0)ρ̄0b + a4 6 ρ̄ 3 0b , (4.15) where according to (4.13) ρ̄0b = (n̄ − ng + M) ã2κ . (4.16) Using the denotation m = M + n̄ − ng , (4.17) we can reduce equation (4.15) to the following form m3 + mpb + qb = 0, (4.18) where pb = 6κ2ã3 2 a4 , qb = −(n̄ − ng)pbκ, and its discriminant Db = ( pb 3 )3 + ( qb 2 )2 . (4.19) Since pb < 0, the value of Db can be either positive or negative (see figure 1). This fact shows the influence on the number of real solutions of equation (4.18). If Db < 0, the equation (4.18) has three real solutions. If Db = 0, there are two values of density n̄. In case of Db > 0, the equation has a single solution describing a decrease of n̄ with an increase of the chemical potential, and this is an unphysical behavior. In the ranges of density 0 6 n̄ 6 n̄max, Db is negative, as it is plotted in figure 1. Solutions of equation (4.18) are of the form m1 = 2 √ 2κ2ã3 2 a4 sin αb 3 , m2 = −2 √ 2κ2ã3 2 a4 sin (αb 3 + π 3 ) , m3 = −2 √ 2κ2ã3 2 a4 sin (αb 3 − π 3 ) , (4.20) where αb = arcsin [√ 9a4 8ã3 2 (n̄ − ng) ] . 100 0 -100 -200 -300 1 2 3 Db n _ Figure 1. Plot of Db as a function of average density. n _ 4 2 0 -2 -4 1 2 3 mn 2 1 3 Figure 2. Solutions mn as functions of average den- sity n̄: m1 —curve 1, m2 —curve 2, m3 —curve 3. 23501-8 Phase transition in a cell fluid model Note that the solution m1 in (4.20) (see curve 1 in figure 2) is equitable for some (bounded) range of values of the chemical potential M = f (n̄, τ) M = m1 − (n̄ − ng). (4.21) At the parameters given in (C.1), the chemical potential take values −0.7 < M < 0.7 (figure 3). If M falls out of this interval, there exist two other solutions (m2 and m3) of equation (4.21), but they do not reflect a physical matter of the phenomenon. Going back to the calculation of the GPF (4.11), the caseT > Tc means a single real solution of (4.12) ρ̄0b = ( 3M a4 + √ Q ) 1 3 + ( 3M a4 − √ Q ) 1 3 , Q = [ 2d(0) a4 ]3 + ( − 3M a4 )2 . (4.22) Then, the equation of state or in other words the pressure as a function of temperature and density is of the form Pv kBTc = (τ + 1) [ f − (M + ã1) 2 2κã2 + d(0) 2(κã2)2 ( n̄ − ng + M + κã2nc )2 + a4 8(κã2)4 ( n̄ − ng + M + κã2nc )4 ] , (4.23) f = 1 Nv ln g′W − a0 + a4 24 n4 c . Now, considering T = Tc in (4.23), it is easy to write an explicit dependence of pressure on density at the critical temperature Pv kBTc = fc + M0 4ã2 ( M0 + 3n̄ �� T=Tc − ng ) , (4.24) fc = 1 Nv ln g′W − a0 + a4 24 n4 c − (a1 + a4n3 c/6)2 2ã2 . The following formula expresses the chemical potential M0 as a function of density M0 = 2 ( 2ã3 2 a4 )1/2 sin ( 1 3 αb �� T=Tc ) − ( n̄ �� T=Tc − ng ) . Index 0 denotes that M0 corresponds to T = Tc. The pressure P = P(n̄) expressed by (4.23) and P \|T=Tc = P \|T=Tc(n̄) expressed by (4.24) are plotted in figure 4. n _ 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 1 2 3 M Figure 3. Plot of the chemical potential as a function average density. 8 6 4 2 1 2 3 n _ 0 4 3 2 1 P Figure 4.Plot of the pressure as a function of average density at different temperatures (τ = 0 — curve 1, τ = 0.5 — curve 2, τ = 1 — curve 3, τ = 1.5 — curve 4). 23501-9 M.P. Kozlovskii, O.A. Dobush 5. Behavior of the system at temperaturesT < Tc The equation of state in case of T < Tc can be written in the form Pv kBT = 1 Nv ln g′W + Eµ + M ρ̄0i − 1 2 d(0)ρ̄ 2 0i − a4 24 ρ̄ 4 0i , (5.1) where Eµ is defined in (4.2), and values ρ̄0i , i = 1, 2, 3, are the three solutions of equation (4.12) [19] ρ̄01 = 2ρ0r cos αm 3 , ρ̄02 = −2ρ0r cos (αm + π 3 ) , ρ̄03 = −2ρ0r cos (αm − π 3 ) , (5.2) where ρ0r = √ − 2d(0) a4 , αm = arccos M Mq . (5.3) Mq is the value of the chemical potential M when the discriminant Q of equation (4.12) is equal to zero Mq = √ − 8[d(0)]3 9a4 . (5.4) Note that the region of values of M is restricted to the interval \|M \| 6 Mq . For all \|M \| 6 Mq , we have Q < 0, so there exist three real solutions of equation (4.12) in this interval of M . In case of \|M \| > Mq , we have a single root of equation (4.12), as well as in case of T > Tc (4.22). At M = −Mq we have Q = 0, so using (4.22) we obtain ρ̄ (−) 0q = 2 ( − 3Mq a4 ) 1 3 ≡ −2ρ0r , (5.5) and at M = Mq using (4.22) we can find ρ̄ (+) 0q = 2 (3Mq a4 ) 1 3 ≡ 2ρ0r. (5.6) Let us analyse an asymptotics of the solutions (5.2) at \|M \| = Mq . Here, at M = −Mq , we have cosα(−)q = −1, that is why α(−)m = π and ρ̄ (−) 01 = 2ρ0r cos π 3 = ρ0r , ρ̄ (−) 02 = −2ρ0r cos ( 2π 3 ) = ρ0r , (5.7) ρ̄ (−) 03 = −2ρ0r cos 0 = −2ρ0r. In case of M = Mq cosα(+)m = 1, or α(+)q = 0, then ρ̄ (+) 01 = 2ρ0r cos 0 = 2ρ0r , ρ̄ (+) 02 = −2ρ0r cos π 3 = −ρ0r , (5.8) ρ̄ (+) 03 = −2ρ0r cos ( − π 3 ) = −ρ0r. Thereby, for M = −Mq we have ρ̄ (−)0q = −2ρ0r defined in (5.5) coinciding with ρ̄ (−)03 expressed in (5.7). In case of M = Mq ρ̄ (+) 0q = 2ρ̄03 from (5.6), which is the same as ρ̄ (+)01 expressed in (5.8). Now we can draw the following conclusion. An increase of the chemical potential from −∞ to −Mq corresponds to a single solution of equation (4.12), which is given in (4.22) (region I in figure 5). For 23501-10 Phase transition in a cell fluid model n M I II III IV 0 n12 n20 n03 n34 nlim MmaxMq‐Mq 0 0Mmin Figure 5. Regions of values of the chemical potential and the corresponding values of the average density at temperature below the critical one. −Mq < M < 0, this solution transits to ρ̄ (−)03 expressed in (5.7) and remains to be applicable until M = −0 (region II) where cosα(−)0 = 0 or α(−)0 = π2 lim M→−0 ρ (−) 03 = ρ (−) 0 = −2ρ0r cos ( π 6 − π 3 ) = − √ 3ρ0r. (5.9) On the other hand, a decrease of M from +∞ to Mq corresponds to a single solution (4.22) (region IV), which at M = Mq transits to the solution ρ̄ (+) 01 expressed in (5.8) and remains to be applicable until M = +0 (region III): lim M→+0 ρ (+) 01 = ρ (+) 0 = 2ρ0r cos ( π 6 ) = √ 3ρ0r. (5.10) Thus, the equation of state at T < Tc can be written in the form of several terms using the Heaviside functions θ in accord with the values of the chemical potential M Pv kBT = − 1 Nv ln g′W + Eµ + E(ρ̄0) [ Θ(−M − Mq) + Θ(M − Mq) ] + E(ρ̄03)Θ(−M)Θ(M + Mq) + E(ρ̄01)Θ(M)Θ(Mq − M). (5.11) This information is sufficient to plot the function P = P(τ, M). The corresponding curves are shown in figure 6. There we can see lines broken at the point M = 0. In case of T > Tc such curves are smooth. However, the plot of the pressure as a function of density and temperature is more informative. To get this dependence, we should express the chemical potential as a function of density according to the interrelation (4.13) and use the obtained expression in (5.11). For this purpose, let us use (4.21) and compare the behavior of ρ̄0 as a function of chemical potential on the one hand and, on the other hand, as a function of density. Consider each of the regions depicted in figure 5. Region I, M 6 −Mq . At M = −Mq , we have expression (5.5) for ρ̄0, where ρ0r is a function of temperature (5.3). On the other hand, there is equality (4.13) n̄ = ng − M + ρ̄0κã2 , (5.12) which links the average density n̄with the chemical potential (in the present case M = −Mq) and the value ρ̄0 = −2ρ0r. Thereby, according to (4.13) we can find the density n̄12 (the point of transition between region I and region II), which corresponds to M = −Mq n̄12 = ng − 2κã2ρ0r + a4 3 ρ3 0r. (5.13) Region II, −Mq 6 M 6 −0. At M = −0, we have ρ̄03 = − √ 3ρ0r. A boundary value of density which is pertinent to n̄20 = lim M→−0 n̄ is of the form n̄20 = ng − √ 3κã2ρ0r. (5.14) 23501-11 M.P. Kozlovskii, O.A. Dobush 1 2 3 P M -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.2 0.3 0.4 0.5 0.6 0.7 Figure 6. Plot of the pressure P as a function of chemical potential and temperature at T < Tc (τ = −0.1 — curve 1, τ = −0.2 — curve 2, τ = −0.3 — curve 3). Region III, 0 6 M 6 Mq . This region starts from M = 0, where ρ̄01 = √ 3ρ0r and respectively n̄03 = ng + √ 3κã2ρ0r , (5.15) and ends up at M = Mq , which takes the consequences with ρ̄01 = 2ρ0r, therefore, n̄34 = ng + 2κã2ρ0r − a4 3 ρ3 0r. (5.16) Region IV, M > Mq . This region starts from n̄03 (5.15), which decreases up to some fixed value nlim. The value nlim is bounded. The plot of n̄20 and n̄03 as functions of temperature is shown in figure 7 (the binodal curve). Maximal value of M corresponds to nlim = 3.26. In case of n̄ > nlim, the solution m1 in (4.21) should be replaced by m3. However, this solution is non-physical, because it gives a decrease of the chemical potential with an increase of the density. Having a temperature dependence of the boundary densities n̄12, n̄20, n̄03 and n̄34, one can rewrite the equation of state (5.11) in terms of (τ, n̄). For this purpose, it is sufficient to use the formula (4.21) in equation (5.11). This formula gives a dependence of the chemical potential on temperature and the average density. The boundary densities described above should be substituted for corresponding values of the chemical potential in Heaviside functions Θ(Mi − Mq) in (5.11). As a result, the equation of state is expressed as follows: Pv kBT = − 1 Nv ln g′W + Eµ(n̄) + E1(ρ̄03)Θ(n̄12 − n̄) + E2(ρ̄03)Θ(n̄ − n̄12)Θ(−n̄ + n̄20) + E3(ρ̄01)Θ(n̄ − n̄03)Θ(n̄34 − n̄) + E4(ρ̄01)Θ(n̄ − n34). (5.17) Eµ is determined by the formula (4.2). Functions En(ρ̄0) are of the following form En(ρ̄0n) = M ρ̄0n − d(0) 2 ρ̄ 2 0n − a4 24 ρ̄ 4 0n , (5.18) where ρ̄0n is either ρ̄01 [see (5.2)] for E3(ρ̄01) and E4(ρ̄01), or ρ̄03 [see (5.2)] for E1(ρ̄03) and E2(ρ̄03). The pressure P as a function of density and temperature is plotted in figure 8. It is easy to see that in the temperature range τ < 0, there is a jump of density depicting a coexistence curve. Numerical results show that one can observe a liquid phase at T < Tc in the density region n̄ 6 nlim. On the phase diagram (figure 9), one can see the line of some limited pressure Plim = lim n̄→nlim P(n̄) connected to the maximal value of the fluid density nlim. This boundary value depends on the microscopic parameters of the system. In other words, it takes a specific value for each system. The line of maximal density may appear as a consequence of using the ρ4-model [see (4.1) which is the approximation 23501-12 Phase transition in a cell fluid model _ n/ n c T/Tc 2 1.5 1 0.5 0 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 Figure 7. Plot of the temperature of phase transi- tion as a function of density (binodal — curve 1, spinodal — curve 2). 0.8 0.6 0.4 0.2 0 1 2 3 n _ P 4 1 2 3 Figure 8. Plot of the pressure P as a function of density and temperature τ = 0 —curve 4, τ = −0.1 —curve 1, τ = 0.2—curve 2, τ = −0.3—curve 3. -0.8 -0.6 -0.4 -0.2 0.2 0.4 0.6 0.8 1 1.2 P τ High density region Liquid Gas Critical point Figure 9. Phase diagram. 1 2 3 -0.3 -0.2 -0.1 0 0.1 0 1 2 3 n _ τ P Figure 10. Surface of pressure as a function of aver- age density and temperature. of (3.8)]. This line sets the range of densities 0 6 n 6 nlim applicable to the description of a fluid behavior in this manuscript. Black line between the gray and the white zones of the surface of pressure as a function of temperature and average density (figure 10) corresponds to the range of temperatures close to the critical Tc, where the mean-field approximation is not applicable. 6. Conclusions To describe a first-order phase transition in the cell fluid model, we deal with the calculation of the grand partition function. A rigorous functional representation of the GPF (3.8) is obtained, which is reduced to the ρ4-model (4.1). This expression should be calculated in two stages. The former is described in this paper. It covers a wide range of temperatures except the critical point. The latter stage concerns a narrow vicinity of the critical point. It requires the consideration of contribution from all the variables ρ®k with ®k , 0 within the field-theoretical approach. 23501-13 M.P. Kozlovskii, O.A. Dobush In the present work we were restricted to the use of a simplest approximation, which is called the approximation of a mean-field type. Thereby, determination of the GPF consists in the computation of a single integral using the Laplace method. As a result, we obtained an explicit form of the GPF as a function of temperature and chemical potential in this approximation. The pressure as a function of temperature and average density [equations (4.4) and (4.5)] is found. Figure 5 is important in this respect. It shows a correspondence between the regions of chemical potential and density at temperatures below the critical one Tc. The critical temperature is defined by (4.9). Referring to figure 6 it is easy to see that transition of the effective chemical potential M across the value M = 0 changes the behavior of the pressure P at T < Tc. Here, we have lim M→−0 P = lim M→+0 P, although the derivatives of pressure on the left-hand side and on the right-hand side from M = 0 are different. The latter fact causes a jump of density starting from n20 to n03 (see figure 5). Each of these values are functions of temperature. In case of T > Tc, we do not observe such a jump (formally n20 = n03). A special attention is paid to the equation of state. Both the coexistence curve and the spinodal are plotted for the case of a particular parameter p (see appendix C), which takes a specific value for each system. Analyzing the surface of equation of state (see figure 10) we noticed that the curves of pressure sharply increase in the region of high densities (which exceed the liquid phase density). Apparently, the ρ4-model has some boundary density nlim (see figure 2) so that there is a range of physically existent densities 0 6 n̄ 6 nlim. To examine the behavior of the system in the high density region n̄ > nlim (see figure 9), it might be essential to consider (3.8) in the approximation of a ρ2m-model where m > 3. The forthcoming research is to calculate the GPF (4.1) using the renormalization group method [26] with correlation effects taken into account. Thereby, it will be possible to describe the behavior of a simple fluid in a wide temperature range including the critical point within a sole approach. Acknowledgements This work was partly supported by the European Commission under the project STREVCOMS PIRSES-2013-612669, FP7 EU IRSES projects No. 269139 (DCPPhysBio). A. Integrating over the coordinates x1, . . . , xN in the GPF Now expression of the GPF (2.10) becomes slightly different Ξ = gNv Ψ ∫ (dρ)Nv exp βµρ0 + β 2 ∑ ®k∈Bc W(0)ρ®k ρ−®k  ∫ (dϕ ®l) Nv exp ©«− 1 4p ∑ ®l∈Λ ϕ2 ®l ª®¬ × ∫ (dν ®l) Nv exp ©«2πi ∑ ®l∈Λ ν ®lρ ®l ª®¬ ∞∑ N=0 1 N! ∫ V (dx)N exp −2πi ∑ ®l∈Λ ( ν ®l − ϕ ®l 2π ) ρ ®l(η)  , ρ ®l and ν ®l are the representations of the collective variables ρ®k and ν®k in direct space ρ ®l = 1 √ Nv ∑ ®k∈Bc ρ®ke−i®k®l, ν ®l = 1 √ Nv ∑ ®k∈Bc ν®kei®k®l, (dν ®l) Nv = ∏ ®l∈Λ dν ®l . It can be seen from the latter expression that one can integrate over the coordinates of particles in the system. For this purpose, let us consider an integral over dx1 in detail I = ∫ V dx1 exp [ −2πi ( νl − ϕ ®l 2π ) ρ ®l(η) ] = ∫ V dx1 exp [ −2πi ( ν ®l − ϕ ®l 2π ) ∑ x∈η I∆ ®l (x1) ] . 23501-14 Phase transition in a cell fluid model A particle with the coordinate x1 can get into one cell, for example into ∆ ®l , and cannot get into any other cell, so I = ∫ v dx1 exp [ −2πi ( ν ®l − ϕ ®l 2π )] = v exp [ −2πi ( ν ®l − ϕ ®l 2π )] . The rest of integrals over each of xi give the same contribution. Now, the GPF can be written in the form Ξ = gΨ ∫ (dρ ®l) Nv exp βµ ∑ ®l∈Λ ρ ®l + β 2 ∑ ®l1,®l2∈Λ Wl12 ρ ®l1 ρ ®l2  × ∞∑ m=0 vm m! ∏ ®l∈Λ ∫ (dϕ ®l) exp ( − 1 4p ϕ2 ®l + imϕ ®l ) ∫ (dν ®l) Nv exp 2πi ∑ ®l∈Λ (ρ ®l − m)ν ®l  . Here, √ 4πp e−pm2 is the result of integrating over ϕ ®l . B. Cumulant representation of the Jacobian of transition Let us find an “entropy” part of (3.6) (the Jacobian of transition). To ease the process, let us use J(t̃ ®l) = Nv∏ l=1 Jl(t̃ ®l) Jl(t̃ ®l) = ∞∑ m=0 vm m! e−pm 2 emt̃ ®l . This expression can be represented as a cumulant series J̃l(t̃ ®l) = exp ( − ∞∑ n=0 an n! t̃ n ®l ) . Taking into account the equality J̃l(t̃ ®l) �� t̃ ®l=0 ≡ Jl(t̃ ®l) �� t̃ ®l=0 , we have e−a0 = ∞∑ m=0 vm m! e−pm 2 . Applying the following condition ∂n J̃l(t̃ ®l) ∂ t̃ n ®l �� t̃ ®l=0 = ∂nJl(t̃ ®l) ∂ t̃ n ®l �� t̃ ®l=0 , we can find the cumulants an with n > 1. The first four of them are of the form a1 = − T1(v, p) T0(v, p) , a2 = − T2(v, p) T0(v, p) + a2 1 , a3 = − T3(v, p) T0(v, p) − a3 1 + 3a1a2 , a4 = − T4(v, p) T0(v, p) + a4 1 − 6a2 1a2 + 4a1a3 + 3a2 2 , (B.1) where the functions Tn(v, p) Tn(v, p) = ∞∑ m=0 vm m! mne−pm 2 are rapidly convergent series for all p > 0. As it can be seen from (2.11), the parameter p is proportional to Ψ(0). The numeric values of these four coefficients in the case of parameters (C.1) a1 = −0.8668, a2 = −0.4006, a3 = −0.0529, a4 = 0.0299. 23501-15 M.P. Kozlovskii, O.A. Dobush C. Relation between the parameters p and v Note that the condition Db = 0, [recall that Db is expressed by (4.19)] yields two real solutions for the density n̄ and determines the range of density for the given values of the parameters p and v n̄ = ng ± ( 8ã3 2 9a4 ) 1 2 . Therefore, we obtain both the minimal and the maximal density nmin = ng − 2 3 ( 2ã3 2 a4 ) 1 2 , nmax = ng + 2 3 ( 2ã3 2 a4 ) 1 2 for the considered cell fluid model. Apparently, nmin = 0, and hence we have the equation ng − 2 3 ( 2ã3 2 a4 ) 1 2 = 0, allowing us to calculate a respective value of the parameter v (the volume of a cell) for each particular parameter p (see figure 11). Using figure 12, it is easy to make sure that not all the values of v in combination with the parameter p would give a correct picture. Curves 1, 2, 3 plotted in figure 12 match p = 0.8 and three different values of v. Curve 2, which is pertinent to p = 0.8 and v = 4.859808, starts at n̄ = 0. Increasing the value of v at fixed p, the least m1 is reached for n̄ < 0 (curve 3 for v = 6). Decreasing the value of v, the least m1 is for n̄ > 0 (curve 1 for v = 2). Thus, there is a clear criterion for the selection of values of the parameters p and v, which are the arguments of the special function Tn(p, v) (see appendix B). The former p is defined using the formula (2.11). This parameter varies upon the type of the investigated substance and depends on the relation R0/α [for R0 and α see (2.4)]. The latter parameter v provides a description of a particular density region of the system n̄ (0 6 n̄ 6 nlim, where nlim is the limited density of a liquid phase) and is unambiguously defined by the values of the parameter p. 25 20 15 10 5 0.5 0.6 0.7 0.8 0.9 1 p v Figure 11. Plot of v as a function of p. 4 2 0 -2 -4 1 2 3 4 m1 n _ 1 2 3 -1 -2 -3 0-0.2 0.2 0.4 1 2 3 n _ m1 Figure 12. Plot of the solution m1 (4.20) as a func- tion of average density at p = 0.8 and various values of v: v = 2 (curve 1), v = 4.859808 (curve 2), v = 6 (curve 3). 23501-16 Phase transition in a cell fluid model Moreover, p and v should meet the condition a4 > 0. This condition provides a convergency of such integrals as (4.6). This result is very important because all parameters of the model are uniquely linked with each other. So, making estimations for numeric values of the coefficients an expressed by (B.1) we set the value of parameter p, then find the corresponding value of v using the equation nmin = 0. Taking into account (4.9), the expression (2.11) gets the following form p = χΨ(0)/[2ã2W(0,Tc)]. Using the later, one can find the value of χ, which meets the condition W(k) > 0. 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Kozlovskii, O.A. Dobush Фазовий перехiд в комiрковiй моделi плину М.П. Козловський, О.А. Добуш Iнститут фiзики конденсованих систем НАН України, вул. Свєнцiцького, 1, 79011 Львiв, Україна Запропоновано метод опису фазового переходу в комiрковiй моделi плину iз парним потенцiалом вза- ємодiї, що мiстить частини вiдштовхування i притягання. Отримано точне представлення великої ста- тистичної суми цiєї моделi в формалiзмi колективних змiнних. У наближеннi типу молекулярного поля дослiджено поведiнку системи при температурах вищих i нижчих, нiж критична. Використовуючи рiвня- ння, що зв’язує хiмiчний потенцiал та густину, розраховано явну аналiтичну форму рiвняння стану, що справедливе для широкої областi температур. Представлено криву спiвiснування, поверхню рiвняння стану та дiаграму стану комiркового плину Морзе. Ключовi слова: колективнi змiннi, комiркова модель, плин Морзе, рiвняння стану, фазовий перехiд 23501-18 Introduction A cell model for describing the behavior of a simple fluid Functional representation of the grand partition function of the cell fluid model Calculating an explicit form of the GPF in the approximation of a mean-field type Behavior of the system at temperatures T<Tc Conclusions Integrating over the coordinates x1,…,xN in the GPF Cumulant representation of the Jacobian of transition Relation between the parameters p and v

Phase transition in a cell fluid model

Institution

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