The phase transition of the first order in the critical region of the gas-liquid system

This is a summarising investigation of the events of the phase transition of the first order that occur in the critical region below the liquid-gas critical point. The grand partition function has been completely integrated in the phase-space of the collective variables. The basic density measure is the quartic one. It has the form of the exponent function with the first, second, third and fourth degree of the collective variables. The problem has been reduced to the Ising model in an external field, the role of which is played by the generalised chemical potential $\mu^*$. The line $\mu^*(\eta) =0$, where $\eta$ is the density, is the line of the phase transition. We consider the isothermal compression of the gas till the point where the phase transition on the line $\mu^*(\eta) =0$ is reached. When the path of the pressing reaches the line $\mu^* =0$ in the gas medium, a droplet of liquid springs up. The work for its formation is obtained, the surface-tension energy is calculated. On the line $\mu^* =0$ we have a two-phase system: the gas and the liquid (the droplet) one. The equality of the gas and of the liquid chemical potentials is proved. The process of pressing is going on. But the pressure inside the system has stopped, two fixed densities have arisen: one for the gas-phase $\eta_{g} = \eta_{c} ( 1 - {d}/{2})$ and the other for the liquid-phase $\eta_{l} = \eta_{c} (1 + {d}/{2} )$ (symmetrically to the rectlinear diameter), where $\eta_{c} = 0.13044$ is the critical density. Starting from that moment the external pressure works as a latent work of pressure. Its value is obtained. As a result, the gas-phase disappears and the whole system turns into liquid. The jump of the density is equal to $\eta_{c} d$, where $d = \sqrt{{D}/{2G}} \sim \tau^{\nu/2}$. $D$ and $G$ are coefficients of the Hamiltonian in the last cell connected with the renormalisation-group symmetry.


Introduction
The present research follows our previous consideration concerning the liquid-gas critical point. In this work we continue the investigations of the system of interacting particles in the vicinity of the critical point [1][2][3][4]. The great partition function was calculated. The potential of the interaction between particles consists of the sum of the van der Waals interaction and of the short-range interaction between the hardspheres. We used the collective variables (CV) method. The Jacobian of the transition from the Cartesiancoordinates space to the space of the collective variables was averaged with the statistical weight of a distribution function of the hard-spheres.
The problem was reduced to the Ising model in an external field, presented by the generalised chemical potential µ * . The critical indexes correspond to that of the Ising model. Their values, the sizes of the particles, the critical temperatures and densities are given in tables 1 and 2 [1].
The first results of the investigations in the area of temperatures below the critical point, τ 0, are given in [1][2][3][4]. It has been shown that the events connected with the phase transition of the first order from the gas phase to the liquid phase (and inversely) are described by the integral over the variable ρ 0 for k = 0. This integral embraces the results of the integration over all collective variables with the exception of the variable ρ 0 .
The collective variable ρ 0 is connected with the distribution of the number of particles. The integration over the variable ρ 0 was performed in the steepest descent method. It is shown that the discriminant Q of the equation concerning the maximum of integrand can take positive or negative values, or be equal to zero. When the discriminant Q is positive we have pure monophasic states of the gas or of the liquid. In the area Q < 0, the maximization problem possesses three real roots, However, only two of them, actually for n = 0 and n = 1, correspond to a maximum of the integrand.
The areas of the principal and of the secondary maxima were determined. It was shown that there is a break between the end of the principal maximum for the gas phase and the beginning of the principal maximum for the liquid phase. Those were the main results of the papers [1,2]. In this work we would like to pass the whole route of the phase transition of the first order. The work consists of the introduction and seven sections. In Introduction we define the problem and give the main results of the mentioned works. In section 2 the description of the monophasic states is given, which corresponds to the positive values of the discriminant Q. Section 3 is devoted to the coordinate form of the reference system. We work within the framework of the quartic density measure. The reference system is included in our problem through the cumulants M 1 , M 2 , M 3 and M 4 . In accordance with the accuracy of the consideration of the whole problem, it is needed to take a suitable expression for the pressure P 0 , the chemical potential µ 0 and the free energy F 0 of the reference system. Section 4 is an essential one in this work. The domain of the phase transition is considered. Here, the discriminant is negative Q < 0. In this region we have two maxima -the principal and the secondary one.
The principal maxima correspond to the pure gas or to the pure liquid phases. The secondary maxima are connected with the stable density fluctuations, which are either of the liquid type in the gas phase, or of the gas type in the liquid phase. Under consideration is the equation for the envelope curve of two maxima, the principal and the secondary ones: Having opened the second equation we get the probabilities of the system to be in the states corresponding to the principal or to the secondary maxima. We have got analytical expressions for the probabilities to be in the gas or in the liquid phases.
The central considerations concern the events along the line µ * = 0. Here, when the gas system is pressed, a droplet of liquid appears inside it, that is, a new phase arises -the phase of liquid. In the appearance of the droplet lies the essence of the phase transition of the first order (as a result of isothermal pressure on the gas system). The work for the formation of a droplet -the surface-tension energy -was calculated.
A fundamental quantity ∆ is introduced. It describes an inverse to the usual order parameter, characterizing the weight of the pure monophase state in the system. On the rectangle vertex of the Q = 0, the weight is equal to unity for the gas phase and zero for the liquid phase or, on the contrary, dependent on which of them the maternal is. The quantity ∆ is equal to zero on the line of the phase transition.
This means that on that line the gas phase (maternal) and the liquid phase (in a droplet-form) have equal statistical weights. In its initial determination ∆ was a function of the reference system cumulants M 2 , M 3 , M 4 and, therefore, was a function of the compressibility coefficients and its derivatives. The equality ∆ = 0 means that on the line µ * = 0, the compressibility coefficient equals zero. We have some metastable state -the state of phase transition of the first order.
In section 6, the equality of chemical potential of the gas phase and of the liquid phase on the line µ * = 0 is shown.
In section 7, the gas-condensation process is analyzed. An analytical expression for the surface-tension energy is obtained. The latent work A lat in the isothermal pressure process (an analogue to the latent heat in an isobaric process), which acts along the line µ * = 0 during the gas-liquid phase transition, was obtained.
And, in the end, in section 8, the equation of state was obtained. The equation for the isotherm P = P (η) τ=const has three parts: 1) a monophasic gas branch: P = P (η, τ) for densities η η g = η c (1 − d /2), d = D/2G (D and G are coefficients in Hamiltonian at the second and at the fourth degree of ρ 0 ); 2) a horizontal line µ * (η) = 0 for the two-phase system, here P = 0, and η g η η l , η l = η c (1 + d /2). The transition of gas into liquid is takes place at the expense of the latent work A lat. acting from without; 3) the liquid-branch for η > η l = η c (1 + d /2).
Thus, during the gas-condensation process we get a jump of the density of the amount of η l − η g = η c d ∼ τ ν/2 . As we see, the magnitudes of the densities of the beginning and of the end of the transition process from the gas state to the liquid state are located symmetrically to the rectilinear diameter.
When the system approaches to the critical point, the specific surface-tension energy disappears as τ 5/2ν , because D ∼ τ 2ν , G ∼ τ ν , and the jump of the density η is proportional to τ ν/2 .
In the preamble we stressed that within the framework of the Gibbsian statistics one can describe the phase transition of the first order [6].
In this research we made a number of assumptions, especially concerning the reference system, the dependence of the coefficients D and G on the density and others. In fact, by means of the method of collective variables it is possible to describe the critical point theory of the liquid-gas system with a more sufficient accuracy. Now we get to the theme.

How it should look like from the very start
Let a big volume V 0 in the form of a sphere with the radius R 0 , V 0 = 4π 3 R 3 0 and with the number of particles N , N = n 0 V 0 be picked out in a gas medium with the density n 0 , pressure P at the temperature T .
We consider the phase transition of the first order as a result of the quasistatic pressure on the gas sphere. The process takes place at temperatures close to the critical point T c . The dimensionless temperature τ is introduced τ = (T − T c )/T c , and |τ| 0, 01, τ < 0. The pressure inside the sphere of radius R 0 is equal to P +δP , where δP is the additional pressure, connected with the spherical form of the volume V .
The gas energy inside the sphere: where S 0 is the area of the sphere, S 0 = 4πR 2 0 , α is the surface-tension coefficient. The minimum of the Such a picture may be useful to remember when discussing the isothermal phase transition in the gasliquid system illustrated in this work. We consider the grand partition function where N is the number of particles, z N is the activity µ is the chemical potential, Z N is the configuration integral of N particles [1,2]. The collective-variables method is used {ρ k } [3][4][5] to calculate Ξ. The Jacobian of transition from the Cartesian coordinates to the collective variables was calculated with the statistical weight of the system of hard cores. The latter was taken as a reference system. As a result, for Ξ we get the expression [1,2]: where Ξ 0 is the partition function of the reference system (RS), 6) µ 0 is the chemical potential of RS, F 0 is free energy of RS. The quantity Ξ 1 is the partition function [1], which includes all interaction effects, with the exception of the effects connected with the formation of a new phase, i.e., the liquid phase within the maternal sphere of gas in the volume V 0 : Here, Ξ g is the partition function in which the effects appearing after integration over the collective variables ρ k for k > B are taken into account, where B is the point of the first zero of the Fourier transform of the attraction potential; the integration is performed with the Gaussian density measure; F CR + F IGR are the free energies arising after integration over ρ k with 0 < k B : F CR in the critical regime, where the renormalization group (RG) symmetry between the block-Hamiltonians exists and 2πn τ /a n τ k B , n τ is the number and a n τ is the periods of the block-lattice, on which, given τ, the renorm-group symmetry between the coefficients of the block-Hamiltonians terminates; the integration is performed in the quartic density measure [3][4][5]; F IGR appears as a result of the integration over ρ k performed here in an inverse-Gaussian regime [3] for 0 < k 2πn τ /a n τ . We suppose that all expressions in Ξ 1 are known, and that all of them are fixed quantities within the temperature region τ < 0.01. In this work, we consider only one

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part, Ξ ρ 0 , of the Ξ. It is connected with the density fluctuations of the number of particles inside of the sphere of the radius R 0 . The phenomena of the phase transition of the first order at τ < 0 are described by the expression Ξ ρ 0 . The chemical potential µ of the whole system is only within Ξ ρ 0 . As a result of calculations carried out in [1,2], the initial form of the partition function Ξ ρ 0 for integration over ρ 0 is as follows: where µ * is the generalized chemical potential of the system 1 , whereΦ(k) is the Fourier transform of the attraction potential; are coefficients of the limiting block-structure Hamiltonian H n τ at the variable ρ 0 , D 0 1.19; G 0 = 1.67; ν = ln s * / ln E 1 = 0.605 is the correlation length critical exponent, E 1 is the greater of two eigenvalues of the matrix for linearized recurrent relations of Wilson's type [3][4][5][13][14][15][16] for coefficients of blockstructure Hamiltonians, s * is the optimal parameter in the scale transformations s * = 3.58 . . ., (s * ) < E 1 < (s * ) 2 . We work in a critical region of temperatures and densities 2 , τ < τ * , η < η * . For the limit interval of the temperature τ * we have the expression: [4] C 1 τ * r * + βΦ(0)η 6 Here, r * is one of the fixed-point coordinates, a 2 is a coefficient in the Jacobian of transition. On both sides of (1.15) there are inverse radia of correlation in the quartic and in Gaussian density measure, correspondingly [4]. For the limiting point of the interval of densities we have: 6 , σ is a diameter of a particle (hard core), η * is the limiting value of density for the critical regime.
In general, coefficients D 0 and G 0 in (1.13) depend on density. In this work we take them as η = η c , where η c = 0.13044 is the critical density determined from (1.11) by the condition ∆ = 0. The Hamiltonian E (ρ 0 ) in (1.8) is obtained as a result of integration in (1.5) over all variables ρ k , with the exception of ρ 0 3 . 1 While investigating the characteristics of the integrand, the origin of the coordinates is transferred to the point ∆ = 0, µ * = 0, and thus the change of ρ 0 may accept both positive and negative values (see figure 1). 2 "critical" means that within this region there exists the RG symmetry. 3 The variable ρ 0 is introduced in Ξ in (1.5) as a result of integration over d Γ N and summation over N of the expression δ(ρ 0 − N / 〈N 〉), where N is the independent variable of the number of particles. In order to integrate over ρ 0 in (1.8) we surely have to take the grand partition function.

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The chemical potential µ appears only in Ξ ρ 0 . Thus, the variable ρ 0 is connected with the density and with the distribution of the number of particles.
By definition (1.17) Based on these two expressions we get the equation of state: (1.18) connecting such variables as pressure, temperature, density, p, τ, η. The curve of equations (1.17) is an envelope curve for pV /θ = ln Ξ, the latter depends on p, τ, η, µ. We work within the framework of the equilibrium thermodynamics. The number N is large. We use the presence of the thermodynamic limit: In this connection, when we integrate over ρ 0 in (1.8) we use the steepest descent method. The integration in (1.8) was carried out and discussed in [1,2]. We give here only the final conclusions.
In the integral exp[N E (ρ 0 )]dρ 0 in (1.8) we looked for the maximum of the subintegral function.
The derivative gives us the cubic equations: The equation (1.21) has three roots, depending on the sign of the discriminant Q: (1.23) Here, the first addend is positive, and the second is always negative. Therefore, the discriminant Q of equation ( (1.26) 4 Two other roots, ρ 2 and ρ 3 = − 1 2 3 µ * /G do not obey (1.22).

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Phase transition of the first order The thick lines correspond to principal maxima of the function E (µ * , ρ 0 ), thin lines correspond to secondary maxima, and the very thin line is a line of minimum. The origin of the system's coordinates is located in the point µ * = 0, ∆ = 0. The abscissa axis is along the line µ * = 0. The ordinate axis is along the rectilinear diameter ∆ = 0. where which corresponds to (1.24) at Q = 0.

Monophasic states of liquid and gas
The solution ρ 0 max = ρ 1 in (1.26), describes the values of densities of absolute maxima for the function exp[N E (ρ)] in integral (1.8) for Ξ ρ 0 . Considering N as a very big number, according to the steepestdescent method, we write:

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Eliminating µ * by means of (1. 17) ∂ ln Ξ ρ 0 Then, in correspondence with (1.20) -contribution from the reference system, is the contribution from the region k B and from the integrals over ρ k for 0 < k B . The latter includes integrals in the critical regime and in the inverse Gaussian regime, and the free energy of the RS up to the cumulant M 4 . Finally, The sum of (2.6), (2.7) and (2.12) in (2.5) presents the gas-state isotherm when ∆ −b, and the liquid-state isotherm when ∆ b. Our main task is to consider events in the region |∆| b. In this region, the discriminant (1.23) of equation (1.21) is negative. Note that we restrict ourselves to τ < 0, |τ| < τ * 0, 02 and By definition, the coordinates of the critical point are The expression for ∆ is given in (1.11). The quantity ∆ is also determined in (2.3). Here, for ρ 1 we have its value (1.26), the quantities D and G being connected with the attractive forces. Thus, owing to expression (2.2), both the connection of the effects of the long-range and of the short-range (of the RS) interactions takes place.
In the critical region, we can also use for ∆ the following expansion: (2.14) Here, the following expressions hold d∆/dη η=η c = 1/η c . According to the definition of the critical point ∆| η=η c = 0, and for (2.14) we get (see figure 2):

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Phase transition of the first order It follows that for gaseous state ρ 1 −b and for liquid state ρ 1 b in (2.15), one has some restrictions on values of ∆ and η. Besides, we should take into account the restrictions on density in (2.13), as well as those connected with cumulants M 4 [2,5].
For µ * we have expressions (2.4) and (1.9). The latter includes the chemical potential of the reference system, and (2.5) has a contribution into the pressure of the whole system.

On the reference system
Let us turn back to formula (1.5) for the grand partition function Ξ. The logarithm of Ξ gives the whole equation of state accounting for external shell, thus accordingt to (1.1) ln Ξ = P sph V sph − αS sph , where according to (1.2), P sph = p + δP 0 . In this section, we omit the terms connected with the existence of the external shell assuming ln Ξ = PV /θ. In accordance with (1.5) Our aim is to separate in the right-hand side the terms belonging only to the RS. At first it is ln Ξ 0 = P 0 V /θ, where P 0 is the pressure of the RS. Then, from (2.9), (2.10) and (1.9) for ln Ξ ρ 0 we may write: Thus, in the right-hand side we have two addends connected with RS, namely, N (ξ−βµ 0 ). They do not disappear, when excluding the attractive interaction, taking α = 0. Of course, in the right-hand side of (3.1) we should exclude some additional expressions, completing together the free energy F 0 of the RS. We are talking about the factor 5 lnQ −1 (−ξ, a 2 , a 4 ) that forms a part in the expression for Ξ 1 . Collecting together all terms remaining in the expression (3.1) at α = 0, we can write (3.1) in the form:

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where P 0 V /Θ = ln Ξ 0 , and As a result, we may write F 0 in the form Herein below in our computations, in the critical region we shall consider the quantities entering (3.4) as the known quantities. We have presented here the expressions for F 0 because we say that the pressure P 0 , and the chemical potential µ 0 ought to be determined from the expression for F 0 . The latter is obtained here using the quartic density measure while calculating the Jacobian. Thus, for the chemical potential of the RS we shall take the expression The extraction of the RS while calculating the equation of state according to (3.3), will not be considered here, and will be the subject of another research.

The field of the phase transition
We consider the partition function (1.5) in the case of negative values of the discriminant (1.23) of equation (1.20). The part of the expression (1.5) will be analysed in detail, which is connected with the chemical potential via Ξ ρ 0 according to (1.17). For Ξ ρ 0 , we have expressions (1.8)-(1.13). We shall remember here the main form In the region Q < 0, equation (1.20) for extremum of exp N E (ρ) has the roots of (1.27).
We shall investigate the equation (1.17) for the envelope curve in the case of Q < 0.
Let us consider more in detail the second of the two equations Taking into account (4.1), one has

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Phase transition of the first order We devide the integral In the first one, along with figure 1, the maximum of the subintegral function exp[N E (ρ)] takes place for ρ = ρ 02 = b cos ϕ+2π 3 , 0 ϕ π. This is the domain of the gas state of the matter. In the second term, the maximum for exp[N E (ρ)] takes place for ρ = ρ 01 = b cos ϕ 3 -this is the liquid domain. In the whole integral (4.3), the principal maximum for the gas-phase trips together with the secondary maximum for liquid, and vice-versa the principal maximum for the liquid phase trips together with the secondary maximum for the gas.
Let us go to a more detailed consideration of expressions (4.3). So, we carry out the integrals in (4.3), and denote by All of them are of the same type, and we shall calculate I g precisely. We expand the function E (ρ) at the maximum point ρ = ρ 02 = b cos  where For the second integral in I 11 , we introduce the probability integral and get 6

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Substituting into (4.8) and gathering together all the terms after shortening in the limit of big N , we find The integral I l when calculated undergoes the same transformations and we obtain (4.10) The integrals K g and K l are actually calculated while calculating I 11 . For K g and K l , we have: Here, the functions w g and w l play the role of the probabilities of the gas or of the liquid state, accordingly.
As we see, w g + w l = 1.

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Phase transition of the first order  Attention should be paid to the fact that the function ∆ as a sum ∆ = w g ρ 02 + w l ρ 01 = ∆ g + ∆ l turns into zero at ∆ g = −d /2 and ∆ l = d /2. In figure 4, the line (−d /2 ÷ d /2) is interpreted as the line of the gasliquid, first-order phase transition. The points ∆ g | ϕ=π/2 = −d /2 and ∆ l | ϕ=π/2 = d /2 do not lie on the curve The points −d /2 and d /2 are the points of fixed densities for gas in the gaseous phase and for liquid in the liquid stage in the two-phase system, which arise at the first-order phase transition. In figure 5, the closeness of the curves µ * (ρ max ) and E (ρ max ) is obvious, which makes it possible, in what follows, to write E (∆) ≈ µ * (∆).
The quantity E (∆), where ∆ is a solution to equation (4.2) specified in (4.12), is a curve that envelops the principal and the secondary maxima of the function E (ρ).
Therefore, in what follows, we shall write The value of ∆ is given in (4.12). For µ * we may take two equivalent expressions. In the first one, accord-

Properties of functions E (∆) and µ * (∆) in the region of Q < 0
In the region of Q < 0 for 0 ϕ π, there are two solutions to equations (1.20), (1.21): the solution of the gaseous type ρ 02 and that of the liquid type ρ 01 . Moreover, when ρ 02 (ϕ) for π/2 ϕ π describes the principal maximum, in the same interval of values ϕ the solution ρ 01 (ϕ) describes the secondary maximum. Inverse situation is seen in the region where 0 ϕ π/2: the principal maximum is due to ρ 01 , and the secondary one is due to ρ 02 . In general, this is governed by the probabilities w g and w l given in (4.12). In order to take into account both maxima, in the expression for µ * (ρ max ) we should write: µ * = µ * (∆(ϕ)) = −2D∆ + 4G∆ 3 ,  Figure 6. The comparison of the curves µ * = a cos ϕ and µ * = −2D∆(ϕ) + 4G∆(ϕ) 3 as functions of the angle ϕ, 0 ϕ π; N = 10 7 and N = 10 8 . As N grows, the attitudes of maximum and minimum remain unchanged, the widths go to zero. See also figures 3 and 4 for w g and w l as well as for ∆ g , ∆ l and ∆ = ∆ g + ∆ l . Figure 6 presents the curves for the generalized chemical potential µ * (ϕ) as functions of ∆(ϕ): one curve is for µ * = a cos ϕ and the other is for µ * = −2D∆(ϕ) + 4G∆(ϕ) 3 . We are going to work with the latter one. They almost fully coincide except for the narrow interval of values ϕ near ϕ = π/2. It follows from figures 1 and 6, in the region of Q < 0, that in the process of isothermal compression of the gas along ρ 02 (ϕ) for π/2 ϕ π, the system via ∆ l enters the region of liquid-like values of µ * (∆ g + ∆ l ). In other words, at compression of the gaseous phase in the region Q < 0, the density fluctuations of the liquid type emerge. Ultimately, at ϕ = π/2, a droplet of liquid appears in the gaseous phase.
What to do next depends on the type of the shell the droplet possesses when the two-phase system arises. The simplest case is when the shell of the droplet is presented as a geometric sphere confining the droplet. The shell does not contain particles. The question arises whether this is physically justified. In fact, the shell of the droplet has a volume, and is of the form of a spherical layer with a non-neglected thickness. All of them, i.e., two phases and the surface layer, should be present in the right-hand side of (5.1). The left-hand side of (5.1), which initially is of the form PV /Θ = ln Ξ ρ 0 , is written as a sum of terms dependent on the right-hand side of (5.1), in the expression ln Ξ ρ 0 .

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Our aim is to show that the liquid-gas phase transition occurs along the line ∆ g = −d /2÷∆ l = d /2, and that condition (5.5) is obeyed starting at ∆ g = −d /2 and finishing at ∆ l = d /2, and that this process can be described within the Gibbs statistics. Since the first-order phase transition at a constant temperature τ occurs due to the external latent work of the pressure, we should demonstrate that at ϕ = π/2, and w l = w g = 1/2, the following is true For this reason, we return to (5.2) and (5.3). Let us show that equations (5.5), (5.6) do take place along the line µ * (∆ g , ∆ l ) = 0 for ∆ in the interval ∆ g = −d /2 ÷ ∆ l = d /2. We write the expressions µ * (∆ g + ∆ l ) and E (∆ g + ∆ l ), selecting the mixed products ∆ m g ∆ n l and separating them for µ g and µ l and for E g and E l Obviously, the sum of these expressions is equal to the initial form of µ * (∆ g + ∆ l ) in (5.2). Next, The sum of these expressions equals E g (∆ g , ∆ l ) in (5.2). By substitution ∆ g = w g ρ 02 and ∆ l = w l ρ 01 we make sure that on both ends of the line of phase transition We conclude that the line −d /2 ÷ d /2 has the properties of the line of the first-order phase transition in the gas-liquid system, at arbitrarily large N .
The expressions for generalized chemical potentials µ * g and µ * l as well as for E g and E l consist of both "pure" phase terms such as −D∆ g + 4G∆ 3 g , and "mixed" products such as ∆ g ∆ 2 l , ∆ 2 g ∆ l , ∆ g ∆ l etc. On the line −d /2 ÷ d /2, the "mixed" products are of the same order of magnitude as the "pure" ones. Note that in (5.8), quantities µ * g , µ * l are proportional to τ 5ν/2 , and the expression D∆ 2 −G∆ 4 is proportional to τ 3ν . In figure 7, there are presented the curves for the mixed products ∆ g ∆ 2 l , ∆ 2 g ∆ l . Their non-zero values are located near the limiting value ϕ = π/2 and characterise the width of transition layer between gas and liquid. The size of the region of values ∆ m g ∆ n l depends on N . At N increasing, it narrows and the region of the surface shell joins the surface µ * = 0, for the angle ϕ = π/2. Hence, the magnitude of the maxima keep their values unchanged. As a result, the effect of the mixed products on the requirements (5.5), (5.6) remains unchanged.  Figure 7. The region of the surface layer described by the products ∆ g ∆ 2 l , ∆ 2 g ∆ l in the expressions for E (∆ g , ∆ l ) and µ * (∆ g , ∆ l ) in (5.7), (5.8). The width of this region of values ∆ m g ∆ n l depends on N . At N increasing, this region narrows and joins the surface ϕ = π/2, µ * = 0. In addition, the magnitudes of maxima remain constant, and the effect of the mixed products on the requirement (5.5), (5.6) does not change.

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ρ 02 = d , ∆ g = −d /2 at density η g = η c (1 − d /2), and the liquid-phase isotherm ends at the point ρ 01 = d , ∆ l = d /2 at density η l = η c (1 + d /2). On the line η c (1 − d /2) ÷ η c (1 + d /2), the phase transition occurs from the gas into liquid. This phase transition from gas into liquid along µ * g (−d /2; d /2) ÷ µ * l (−d /2; d /2) occurs due to the external latent work of the pressure. Therefore, according to (5.7), there are contributions into the energy density of the surface layer due to µ * g (∆ g , ∆ l ) and µ * l (∆ g , ∆ l ) Dd . (5.12) Since ∆ g = w g ρ 02 , ∆ l = w l ρ 01 , and the probabilities w g and w l depend on the number of particles N and the angle ϕ, as is seen in figure 8. At N increasing, the products of w g and w l degenerate into δ−functions placed along ϕ = π/2. This means that in the limit of large N , the two-phase liquid-gas system degenerates into the system gas -a droplet of liquid divided by the surface without particles. The energy of the phase layer transforms into the energy associated with the creation of a geometric surface of division. The two-phase system emerges: gas and a droplet of liquid of radius R dr . If for the gaseous phase P g is the pressure, V g is the volume, P g V g is the energy, then for the droplet of liquid the volume is 4π 3 R 3 dr , the pressure inside it P l = P g + 2α/R dr , where α is the surface tension coefficient, the energy P l V l = P g + 2α/R dr 4πR 3 dr /3 and plus the surface tension energy −αS dr = −α4πR 2 dr . Hence, the formation of the droplet with a geometric interface is concerned with an additional energy of the surface 2α R dr where the surface-energy density is −α/R dr . The magnitude of the surface energy is connected with quantity (5.11). Also, the change of the volume for the gaseous phase takes place, from V l to V g = V − (4πR 3 /3)(n l /n g ).
In the approximation accepted in this research, the chemical potential of the reference system is βµ 0 = ξ. (6.2) Then, for the chemical potential µ we get the following expression: On the line of phase transition the conditions (5.5), (5.6), µ * (∆ g , ∆ l ) = 0, hold together with N g + N l = N , dN g = −dN l , (6.4) where N g and N l are the number of particles in the gas phase and in the liquid phase, respectively, µ g , µ l are the chemical potentials of the liquid and of the gas, respectively. According to (5.6) βµ g = |α(0)| g (1 − ∆ g ), βµ l = |α(0)| l (1 − ∆ l ). (6.5)

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According to definition (1.12) We consider the beginning of condensation (boiling) The densities η g and η l are expressed via the value η c in the critical point and via ∆ based on the relationship (2.15): Then, for (5.4) we obtain Thus, µ g = µ l . (6.10) It was proved that on the line µ * = 0 there exists a two-phase system of a gas phase and of a liquid phase: the maternal sphere of the gas and the droplet of liquid inside it and the chemical potentials of both phases are the same. Let us imagine that conditions (6.7) and (6.8) necessary for equalities (6.9) and (6.10), may be also valid for other quantities ∆ g and ∆ l , not only for that of (6.7), which are located symmetrically to the point of the intersection of the rectlinear diameter with the line µ * = 0 that is ∆ g = −∆ l . Such points are located at the ends of the intervals which pass through the origin µ * = 0 and touch the lines ρ 02 and ρ 01 that are the curves of the maxima for gaseous and liquid phases, respectively, see figure 1. For each pair of points, denoted byρ 02 andρ 01 , the equalities E (ρ 02 ) = E (ρ 01 ) take place, sinceρ 02 = −ρ 01 , and E (ρ 0 ) is an even function. Thus, for each pair of end-point values there is w(ρ 02 ) = w(ρ 01 ). Hence, for each pair, the following is true: w(ρ 02 )ρ 02 = −w(ρ 01 )ρ 01 and∆ = w(ρ 02 )ρ 02 + w(ρ 01 )ρ 01 = 0, µ * = 0, E (0) = 0, and also µ g (∆ g ) = µ l (∆ l ). However, conditions (5.6) are not obeyed if ∆ g = −d /2, ∆ l = d /2 are substituted by any of∆ g = −∆ l .
Therefore, neither approaching nor moving away from the densities of gas and liquid compared to (6.7) occur. The conclusion is that densities for gaseous and liquid phases do not change at the firstorder phase transtition. The density of gas remains fixed The density of liquid remains fixed η l = η c ∆ l + η c = η c 1 + d /2 . This remains true as long as the phase transition takes place. As a result, under the action of latent work of the pressure, all the gas condenses into liquid. A jump-like change of the gas density takes place η l − η g = η c d .

The process of condensation of gas into liquid (the boiling of liquid)
This section begins with (4.22) Let us follow the process of isothermal quasistatic compression in the gaseous phase (see section 1). The process takes place along the line ∆ = w g ρ 02 + w l ρ 01 . Starting at ∆ = ∆ g = w gρ02 : where (see figures [1][2][3][4] there is a homogeneous gaseous system (see figures 1 and 5).
Next, after the quantities ρ 02 and ∆ increase, we get ρ 02 > −b, ∆ > −b, and the probabilities are as follows:  figure 5) in a gaseous system, there emerge stable density fluctuations of a liquid type because E (∆) > 0 therein. The argument ∆ is determined by equation (4.12). Here, both terms in ∆ become of non-zero value. However, the term w g ρ 02 in ∆ is a principal one. As earlier, we have only gas, and there is a single-phase system (see figure 4).
When the external pressure increases, the pressure in the gaseous system also increases. The density of a gaseous system grows, in accordance with (2.15). The gaseous system evolves along the same isotherm µ * g (ρ 02 ). Now we get to the limiting densities η g = η c (1 + w g ρ 02 ), This state describes the end of the curve for the principal maxima for a gaseous phase and the beginning of the principal maxima for a liquid phase. Here, we arrive at (5.10) and (7.1). When work is done on the gaseous system, within the gas system there arise processes that evolve along the curve ∆ l = w l ρ 01 , although under the conditions of a secondary maximum, as is seen in figures 1 and 5. The system does an internal work to create stable density fluctuations of liquid type, which in the case of ρ 02 = −d , ∆ g = −d /2, ρ 01 = d , ∆ l = d /2 causes the creation of a droplet.
In order to determine the properties of a droplet, it should have exact boundaries. Equations (5.7) and (5.8) for µ * , µ * g , µ * l , E , E g , E l are "bulk" formulas. In these formulas, the interface is a phase layer with a number of particles. Our further calculations are performed for the case of very large N . The model of a two-phase region is considered as gas and a droplet of liquid arising in gas, the interface is a geometric surface without particles. The mixed products ∆ m g ∆ n l , entering µ * (∆ g , ∆ l ) and E (∆ g , ∆ l ), This is true for the case when the thermodynamic potential Φ ρ 0 from the beginning to the end of the process of condensation (i.e., boiling) remains constant Φ ρ 0 = N g µ g + N l µ l = N µ g = N µ l , (7.5) since N g +N l = N remains constant and is equal to the initial number of particles, and dN g = −dN l , which, in fact, holds. The amount of gas decreases, while the amount of liquid increases till all the gas transforms into liquid.
Liquid phase originates in the form of a droplet confined by the interface S of some surface energy.
The sharp bend of the isotherm in figure 8 is related to the instant of the creation of a droplet, and the sum of the moduli of the mixed products (5.11) is equal to the energy of the droplet.
For the basic relationship (7.1), the left-hand side takes the form P g V g + P l V l − αS l , (7.6) where P g , P l are the pressures in the gas and in the liquid phases, V g and V l are the volumes of the gas and of the liquid phases, S l is the surface of the droplet of liquid, α is the surface-tension coefficient. In the right-hand side of (7.1) we obtain: According to the conclusions drawn in (6.5) and (6.6), the quantities µ * g , µ * l take on fixed values. In accordance with (6.5)-(6.7): (7.8) and the process of the phase transition is reduced to redistribution of N g and N l . The width of the interface between gas and liqud droplet is considered to be zero.
Liquid phase is a droplet of radius R that originates in the gaseous phase. The droplet has strict boundaries. Combining (7.5)-(7.8), for (7.1) on the line µ * = 0 we obtain: The emergence of a liquid droplet, restricted by a shell, in a gaseous phase is the essence of the first order phase transition. The droplet is of a spherical form. The surface tension energy is minimal. For (7.9), we get: where V is the initial volume, R dr is the radius of the first droplet, η l , η g are the densities of gas and liquid, respectively. For the energy of the droplet surface, we have: 3 . (7.11) In the right-hand side of (7.9), it is written (7.12) Collecting (7.8)-(7.10) in equation (7.7), we obtain: The change of gas volume by 4π 3 R 3 dr η l /η g − 1 , that takes place in the product P g V − 4π 3 R 3 0 η l /η g − 1 in (7.13) corresponds to those changes in the quantity E g (∆ g ) which were observed at the transition from (7.14)

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Phase transition of the first order Hence, for the specific 7 surface tension energy of the droplet of radius R, we have − α R = n l (E l − E g )Θ. (7.15) Let us replace E l and E g with (7.8), and n l with n l = N l /(4πR 3 dr /3) = η l 6/πσ 3 . In the right-hand side of (7.15), we obtain: which conforms to (5.10). Therefore, based on (7.15), there is a contribution to specific pressure P total from the surface tension energy of the liquid droplet of radius R: whereΘ = Θ 6/πσ 3 . The quantity P total is proportional to τ 5/2ν . It governs the rate of decreasing the surface tension energy at approximatiion to the critical point τ = 0.

Latent work of condensation (latent heat of boiling)
Condensation begins at the moment of the creation of the droplet with the surface energy (7.17). As was mentioned above, condensation of a gas starts from the state and terminates in a liquid system in the state N E (∆) = N µ * l (∆ l ), ρ 01 = d , ∆ l = 1 2 d and η l = η c + 1 2 d η c . At the moment when condensation starts, the pressure is equal to P g = n g ΘE g (∆ g ) = n g Θµ * (∆ g ).
Dd . Then, Dd . (7.20) The initial volume of gas is a sphere of radius R 0 , of volume V 0 = 4πR 3 0 /3. The final volume of gas after transferring into a liquid during condensation at a fixed pressure is the volume of liquid V l in the form of a sphere of radius R l : All N particles of a gaseous sphere V 0 passed into a sphere of liquid V l . Thus, the total change of the volume is The work performed is the latent work (i.e., heat) of the process of condensation: 7 By a volume unity.

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Having substituted (7.20), the specific latent work of condensation is 8 ΘDd n 2 g . (7.22) Condensation at isothermal compressibility is a dynamical process. Equation (7.9) concerns the states of a system in which two phases have already been created: the initial gaseous phase and a new liquid phase in the form of a droplet. The conditions ∆ = 0, µ * = 0, E (∆) = 0 require an external pressure to be applied in order to do the latent work of condensation. As a result, the gaseous phase disappears (or remains as a nascent bubble of gas), and we write: [For a nascent bubble of gas, if any, P g V nascent g −αS nascent g is energy, and we can write: For the two-phase system along the line µ * = 0, E (∆) = 0 for external pressure we write down: Using L, we get ΘDd .
The condensation of gas at isothermal compression is a process analogous to the boiling at an isothermal decrease of pressure of a liquid. This is a quasiequilibrial directed process. Equation (7.9) concerns the states of a system in which two phases have already been formed: the initial gaseous phase and a new liquid one, in the form of a droplet. (The droplet has got a shell that separates liquid from gas, and its creation is the essence of the first-order phase transition.) Conditions ∆ = 0, µ * = 0, E (∆) = 0 require an external pressure to be applied in order to do the latent work of condensation (7.22). As a result, in (7.9) the gaseous phase disappears (or remains as a nascent bubble of gas).

(8.2)
On the line of the first-order phase transition, equations (7.9) hold. To obtain these equations, the mixed products ∆ m g ∆ n l were nullified in the exact relationships (5.7)-(5.8), in particular those in (5.11). Instead of them, the surface tension energy of a droplet was introduced as well as an additional internal pressure of a droplet. This allowed us to describe the magnitude of energy associated with the creation of the liquid droplet. The droplet has got a strictly defined interface. We have described the quantity of the latent work of the pressure. 8 Dimension of L coincides with the dimension of pressure.

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Phase transition of the first order The jump of the density at phase transition is η l − η g = η c d . The slope of the gaseous branch ∆ g while approaching the point ∆ g = −d /2 is equal to 0. The slope of the liquid branch while turning away from the point ∆ l = d /2 is equal to 0.
The task of our further research will be to clarify the problem of the thermodynamic limit and to compare the theory with experimental results. We will need a total expression for ln Ξ, not only ln Ξ ρ 0 . These urgent problems are easier to consider farther from the critical point, since there are no requirements following from the renormalization-group symmetry near T = T c .
Thus, the general expression for the equation of state takes the form and (5.8) between E g and E l in such a way that in the fixed limiting points of a phase transition ∆ g = −d /2 and ∆ l = d /2, the fundamental equalities (5.6) hold.
The third stage is considered to be the result of the action of the external latent work of pressure on the two-phase system, presented in (7.23)-(7.24). Pressure and temperature do not change, accoridng to (6.5)-(6.6). Gas transfers into liquid according to (7.5). The external latent work of pressure has been done. The density jump ∆η = η l − η g = η c d takes place. Everything ends with the fourth stage, which is a liquid with a nascent bubble of gas. Finally, the fifth stage is a liquid of density growing from the value η l = η c d /2 + η c to η l = η c b + η c . Thus, the events in the region Q < 0 finish. The region of liquid densities begins, where the discriminant Q > 0. This is a homogeneous liquid phase.
We started with an initial sphere of gas in a boundless system. By isothermal compression, we reached a point of the first-order phase transition. Gas transferred into liquid, condensed into a droplet. The quantities that describe the process were determined. We worked within the framework of the grand canonical distribution for the Gibbs statistics. The successful solution of the problem is ensured by the choice of a related phase space of collective variables, among which there is a variable ρ 0 . This variable is responsible for a description of the phase transition and the density jump. Calculations are carried out using the quartic basic measure density. The zero-approximation performed in this work provides good results on both the critical indexes and the description of the mechanism of density jump at the condensation of gas into liquid.
We have gone through all the events of the first-order phase transition below the critical point. The curve for the equation of state has a form presented in figure 9. A peculiar thing is the density jump between the points η g = η c (1 − d /2) and η l = η c (1 + d /2), where d = D/2G, where D and G are the coefficients at the second and fourth power of ρ 0 , respectively, in the initial Hamiltonian E (ρ 0 ). Figure 1 resembles the Van der Waals isotherm. The theory put forward by a great physicist [17], was the basis for numerous researches devoted to the gas-liquid phase transition, including the present research.
At the same time, this work is not void of numerous evident drawbacks. Among them we note: • we worked in a very narrow regions of temperatures and densities, near the critical point; • when calculating the Jacobian of transition to the phase space of collective variables, the statistical weight was introduced, the distribution over short-range interaction between hard spheres. The cumulants of the Jacobian, M 1 , M 2 , M 3 and M 4 , were taken at k = 0. This was connected with the fact that they have wide "shelves" at k = 0. The cumulants were determined via compressibility of the reference system. We neglected the terms proportional to k 2 , which were proportional to the binary correlation function for the system of hard spheres; • we neglected the effect of integration over ρ k for k 0 on the form of the isotherm of phase transition.
• we restricted the expression for the chemical potential of the reference system to βµ 0 = ξ.
We postulated that the dependence E (ρ max ) = E (∆), being accurate in the region of Q > 0, remains also accurate for Q < 0, if the sum ∆ = ∆ g + ∆ l is assumed as ∆.
All the above demands a more detailed consideration. The mutually-coordinated short-range and long-range interaction being taken into account will be one of the main tasks in further research.