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 ($T \geqslant T_\text{c}$) 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 $\rho^4$ 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.


Introduction
For over a century, scientists are trying to theoretically describe the nature of phase transitions and critical phenomena in liquid systems. For many years of strenuous work, there have been elaborated various worthy approaches to solve this problem at the macroscopic stage, but the development of resemblant investigations at the microscopic level remains topical. The latter direction is of particular importance because within the framework of a grand canonical ensemble the actual systems of atoms and molecules can be adequately represented, due to the presence of chemical potential. Only this thermodynamic parameter is responsible for the exchange of constituents between different parts of the system and with an environment, and also it quantitatively describes the tendency of the thermodynamic system to establish a composition equilibrium.
In articles [1,2], we calculated the grand partition function of a cell fluid model in the meanfield type approximation. In this way, it was possible to describe the first-order phase transition and, in general, the behavior of the system in a wide range of temperature above and below the critical point, except for its vicinity. In particular, using the Morse interaction potential, which well describes the interaction in liquid alkali metals, we obtained a state equation, coexistence curves, and the values of critical density and critical temperature for sodium and potassium. In spite of this, 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 are main. The solution of the problem will be the calculation of a grand partition function using the renormalization group (RG) transformation. Actually, we are aimed to apply the idea of the method elaborated for calculating the grand partition function of the 3D Ising-like model in an external field [4,5,6]and analyzing its critical behavior. This may allow us to thoroughly investigate critical properties of the correlation function and thermodynamic functions such as heat capacity, isothermal compressibility, isobaric expansion, to obtain the values of critical amplitudes and critical parameters, and plot the Widom line, which is important for studying a fluid in the supercritical region.
In this article, we propose a theoretical description of the behavior of a cell fluid model 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. Also, the recurrence interrelations (RI) between the coefficients of effective non-Gaussian (quadruple) measures of density, the solutions of these interrelations and the equation for a phase transition temperature are represented. The thermodynamic potential is considered in section 3. The total expression of the thermodynamic potential in case of temperatures T > T c 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 of T > T c and T = T c . Plots of the isothermal compressibility and the line based on the points of maximum isothermal compressibility are also represented in this section. Discussions and conclusions are presented in Section 5.

Basic expressions
In the previous article [2] we obtained the functional representation of the grand partition function of the cell fluid model in ρ 4 approximation in the following form (2.1) Here The following expression corresponds to the coefficient d(k) The effective potential of interaction 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 R 0 /α. If R 0 /α = 2.9544, α = 1.81686 (which is typical of Na (sodium)), we have the following The Fourier transform of the initial potential of interaction (Ψ − U ) at k = 0: Let us represent d(k) in the form of series in power k 2 , (2.8) where α R = α/R 0 . Note that at χ = 3/4 one has b = α 2 R R 2 0 . As can be seen from the structure of expressions for d(0) (2.3) and b (2.8) it is expedient to make a change of variables in the expression (2.1) As a result, we have moreover, (the sign of stress near ρ k is omitted) Let us make in (2.10) a staged integration keeping with the technique elaborated in [3]. One should start from the variables ρ k with large value of the wave vector and end with integration over the variable ρ 0 . The latter is the basis of a type of mean-field approximation. The contribution of ρ 0 is main for temperatures far from T c (T c is the critical temperature). It is essential to account for fluctuations in the vicinity of T c thus one has to consider for calculation terms ρ k with k = 0. Accordingly to [4] let us write the following The range of values k ∈ B 1 has the form Here c 1 = sc, moreover, the way of dividing the space of collective variables into intervals (s > 1) is determined by the parameter s = B/B 1 . The average value k 2 for k ∈ B\B 1 has the form (2.14) Then Rewrite (2.10) as and integrate over N v variables η k . Thus we have (2.17) Here ν k e i k l is a site representation of the variable ν k , j 1 = 2 (N 1 −1)/2 is the Jacobian of transition from variables ν k to ν l , Here m 0 = 2 is the approximation correspondent to the ρ 4 model. But there are no special difficulties when using higher approximations such as: ρ 6 , which corresponds to m 0 = 3 etc. Keeping with these higher approximations one has to use coefficients a 2m with m > 2 in (2.1). The coefficients P 2m are derived in the way it was done in [7] The term s −nd in (2.19) is connected with necessity of transition from the set of values k ∈ B to the set k ∈ B 1 (see [7]), d = 3 is the dimension of space. In terms of parabolic cylinder functions we have the following The special functions U (x) and ϕ(x) are expressed by the functions of parabolic cylinder U (a, x) The following expression is a result of integration over the variables ν l in (2.17) For g 1 (k) and a (1) 4 we formulas q is expressed in (2.15), (2.25) The following explicit form of the recurrence interrelations, which come of between the coefficients of the exponent in the grand partition function after first stage of integration. Taking into account the formulas for P 4 (see (2.21)), y and x (see (2.25)) one find where E(x) = s 2d ϕ(y)/ϕ(x). (2.29) Use the notion 4 . and write the expressions for a (1) 1 from (2.23), g 1 (0) from (2.26) and a (1) 4 from (2.28) in the following form (2.30) The special functions N(x) and E(x) are defined in above. According to [5,7] n stages of integration in Ξ we have , the recurrence interrelations can be represented as follows The initial conditions are r 0 = 1 −ã 2 βW (0), u 0 = a 4 (βW (0)) 2 , w 0 = M (βW (0)) 1/2 . Using the following conditions, one find the coordinates of the fixed point w * , r * , u * w n = w n+1 = w * , r n = r n+1 = r * ; u n = u n+1 = u * .
For w * there is w * = 0, since s > 1. The equation for u n+1 yields which juxtaposes own x * to each s. For the value s * = 3.5977 falls in with x * = 0 [5]. From the second equation we have (2.35) Therefore, there coordinates of the fixed point of the recurrence interrelations (2.35) are w * = 0, r * = −q, but u * is to be determined from (2.37). Note that the values of y n from (2.34) are large.
Using eigenvalues of the transformation matrix R which are equal to (in case of s = s * ) and also using eigenvectors of R, one has The expressions derived in [5] are valid for the coefficients c 1 and c 2 has the following form Since r * = −q, we obtain the equation (2.45) Here c 10 = 0, because of the equation (2.44), which, actually, is used to determine the critical temperature. For other coefficients we have the following For the coefficients c 2l (l = 0, 1, 2) we get (2.47)

A thermodynamic potential of the model
The later calculation of (2.31) is based on the method proposed in [5]. In case of T > T c , the thermodynamic potential is represented as where , In the critical region of fluctuations one has Ω (+) The coefficients γ 0l meet the expressions which coincide with similar relations for the Ising-like system in an external field [5]. The values f contains a nonanalytic function of temperature τ and chemical potential µ. Herē whereγ l are constants [5]γ which has the following values at s = s * For n p , H c and s −3(np+1) we have Hereτ = τ (c 11 /q) is the renormalized relative temperature, Let M 0 = 1 (for the 3-dimensional Ising-like system the similar parameter is of the order of unity [5]). Express the index in the following form Here ν is the index of the correlation length ξ = ξ ± |τ | −ν at M = 0. It is represented by the formula (3.14) The behavior of the correlation length at The values ν = 0.605 and p 0 = 1.512 are derived for the model ρ 4 at s = s * . Note that the values Ω µ and Ω r are contained in the expression for the thermodynamic potential Ω (3.2) (see (3.3)). A temperature dependence can be single out from these terms, but this would cause a renormalization of coefficients of the analytic part of the thermodynamic potential. The terms of the thermodynamic potential which are nonanalytic functions of temperature and chemical potential are of particular interest.
The part of the thermodynamic potential Ω LGR from (3.2) can be expressed as a sum of two terms [5] Ω LGR = Ω (+) The former term Ω (+) T R 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 (+) T R holds for the formula (see [4]) The expression for the coefficient f np+1 is It is easy to derive x np+1 from the relation in case of m = 1.
H c is defined in (3.10), and y np is to be derived from formula for y n (2.34) in case of n = n p . In order to calculate the term Ω (see (3.16)) we have to reckon the correspondent expression of the grand partition function Ξ (3.21) The expressions for the coefficients g np+2 (k) and a (np+2) 4 are as follows both r np+2 and u np+2 meet the expressions As a result, (3.25) Here (3.26) As in [4,8], the magnitude of the shift σ + is found from the condition The solutions of the gotten cubic equation for σ 0 , is analyzed by [4,5,8]. In general case (T = T c , M = 0), the solutio 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 σ 0 = A + B (see [9]), where Direct calculation shows that Q is positive in case of T > T c . The plot of real solution σ 0 as a function of the chemical potential M for various values of τ is shown on 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) terms with k = 0 and integrating over η k with k = 0 we obtain (3.31) Here The next stage of calculations is a return to the variable ρ 0 using the "reverse" to (3.24) change of variables in (3.32). The following expression is the result of integration over ρ 0 in the term (3.31) of the total thermodynamic potential using the Laplace method (3.34) where the following notation is used Note that both the change of variables (3.33) and the substitution ρ 0 = √ N v ρ in (3.32) cause an appearance of a sharp maximum of the subintegral expression at the pointρ. Both the extremum condition of the subintegral expression, from which it is possible to defineρ, and the representation ρ =ρ s −(np+2)/2 lead to the same equation (with the same coefficients) as (3.30), where the role of σ 0 is played byρ . The expressions ofρ and σ 0 coinside. The values ofρ and σ + (3.29) are also similar. So the subintegral expression E 0 (ρ) at the pointρ is presented as E 0 (σ + ) (see (3.35)) with coefficients (3.36), whereρ is changed to σ 0 . The sum over k ∈ B np+2 in (3.34) is calculated using a transition to the spherical Brillouin zone and integration over k: The expressions derived for Q(P np+1 ) (see (2.32)), Ω np+2 (3.34) and 1 2 k∈B np+2 lnḡ(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 The term Ω is a part of the thermodynamic potential connected to the variable ρ 0 . For Ω G one has The coefficient f G is as follows r R and y np+1 are defined in (3.37) and (2.34) respectively, and for f G one get The part of the thermodynamic potential can be calculated Ω LGR (3.16) using the expressions of both Ω (+) T R (3.17) and Ω (3.38). The complete expression of the thermodynamic potential of a cell fluid model is obtained on the base of (3.2) in the way of gathering derived terms from all of the regimes of fluctuations for temperature T > T c . The value ln g W (see (2.2)) contained in Ω µ (see (3.3)) is found as a result of transition to the spherical Brillouin zone and integration over k: The complete expression of the thermodynamic potential, which is equivalent to (3.2), is represented in the form of three terms Ω = Ω a + Ω (+) The analytic part of the thermodynamic potential Ω a is as follows where The values E µ , u 0 , x and a are defined in (2.2), (2.11), (2.25) and (3.44). The term Ω (+) s is a sum of nonanalytic contribution. It has the following form Here The shift of the variable ρ 0 is determined by the value σ 0 , which is contained in the coefficients e . (3.50) The expressions for the coefficients e

An equation of state of the model at T ≥ T c with effects of fluctuations taken into account
We derived the thermodynamic potential Ω = −kT ln Ξ (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 it is possible to get the expression of the pressure P as a function of the temperature T and the chemical potential µ applying the well-known formula P V = kT ln Ξ. (4.2) Having the grand partition function it is possible to calculate the average number of particles The latter relation is applicable to express the chemical potential via either the number of particles or the average densityn 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 average densityn, which is the an equation of state of the model we study. Using (4.1), (4.3) and (4.4), one obtain . (4.5) Here Note that during the calculation of the latter formula (4.6) the derivatives of σ 0 with respect to βµ gives 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 value σ 0 is considered not to be a function of chemical potential. The derivative ∂e (+) 2 ∂βµ is as follows where The expression for H c (see (3.10)) is applicable to find the derivative of H c with respect toh Based on (4.5) and taking into account (4.6) one has Here the coefficient (4.14) Here   Henceforth, the zero-order approximation of (4.21) is used for M , namely, the case M = M (0) is considered. Note that the region M > 0 is in agreement with positive values ofn − n g , andn < n g meet M < 0. The chemical potential is equal to zero whenn = n g . Taking into account (4.1) and (4.2), at T > T c the following equation of state is derived: Here the value a and the chemical potential M are defined in (3.44) and (4.21) respectively. E µ is expressed in (2.2) where the chemical potential from the formula (4.21) is to be substituted for M . Using the latter formula in the expression ofh (see (3.11)), one get h = n − n g σ (+) 00 The definition of h c is in (3.11). The formula (4.14) is applied for determiningh 2 +h 2 c and then rewriting the equation (4.22) as follows In case of T = T c the behavior of the system is possible to be described by the expression (4.24). In this situation α → ∞. Moreover, H c → 0 and r np+2 , u np+2 , x np+2 fail to be the functions of the chemical potential. The coefficients σ 0 , e where σ  The isothermal compressibility K T = (∂η/∂p) T /η, where η =n/n c ,p = P/P c , is represented on Figure 5.  It is known that discontinuous changes of fluid's properties along a first-order binodal that culminates in a critical point can be extended into the supercritical region as the Widom line [12]. The latter is characterized by the locus of points with maximum particular thermodynamic functions, for example, heat capacity, correlation length, etc. Taking into account extremum values of K T (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 extremum points of K T is shown on Figure 6.

Conclusions
The explicit expression for the thermodynamic potential of the cell fluid model is obtained in the CV representation. The basic idea of the calculation of the thermodynamic potential near T c on a microscopic level lies in the separate inclusion of contributions from short-wave and long-wave modes of 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 one-component magnet in the external field [6,13,14]. 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 in which the contribution from long-wave modes of oscillations to the thermodynamic potential of the cell fluid model is taken into account differs qualitatively from the method of calculating the short-wave part of the grand partition function. The calculation of this contribution is based 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 supercritical region.
The derived nonlinear equation which links the relative densityn and the chemical potential M is investigated. The expressions for coefficients of this equation are presented as functions of the ratio of the renormalized chemical potential to the renormalized temperature. The quantityn corresponding to M = 0 is found. The interval of values for the chemical potential, where the densityn increases with increasing M is indicated. Reduction ofn with increasing M beyond this range does not reflect the physical nature of the phenomenon. The chemical potential is expressed in terms of the density.
The obtained equation of state for temperatures above the critical value of T c gives the pressure as a function of temperature and density. The state equation corresponding to the case of T = T c is also derived.
The main advantage of the equation of state obtained in the present research is the presence of relations connecting its coefficients with the fixed-point coordinates and the microscopic parameters of the interaction potential. It shows the ability of the CV 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 cannot explicitly account for a non-universal effect of a particular microscopic parameter of a specific system.
On the basis of the obtained analytic expressions, we conducted the numerical calculations. In particular, the isotherms of pressure and compressibility are plotted as functions of density. It is shown that extremum compressibility form a line on the (P,n) plane, which is likely to the Widom line.
A technique elaborated here for deriving the equation of state at temperatures above T c is planned to be generalized to the case of T < T c . The calculations can be extended to higher non-Gaussian distribution (the ρ 6 model) [10,11]. The multiplier α 0 = µ/W (0) −ã 1 τ p 0 is defined by the ratio of initial µ and τ .