phase transition of real gases. journal 1

Theoretical and Mathematical Physics, 167(2): 654–667 (2011)

PHASE TRANSITIONS IN REAL GASES AND IDEAL BOSE GASES

V. P. Maslov∗

Based on number theory, we present a new concept of gas without the particle interaction taken into

account in which there are first-order phase transitions for T < Tcr on isotherms. We present formulas for

new ideal gases, solving the Gibbs paradox, and also formulas for the transition to real gases based on the

concept of the Zeno line.

Keywords: first-order phase transition, phase transition of the second kind, Einstein paradox, gasmixture, cluster, ideal gas, Bose gas

1. Introduction

This paper is a continuation of [1].According to standard courses of thermodynamics, the ideal gas is a gas satisfying the relation

PV = RT, (1)

where P is the pressure, V is the volume, T is the temperature, and R is the gas constant. In other words,

PV

RT= 1. (2)

The quantity PV/RT is called the compressibility factor and is denoted by the number Z. This relation isalso well satisfied for real gases and mixtures of gases under room conditions. And why is the ratio PV/RT

namely unity? This value was adopted for convenience, appropriately choosing the gas constant.The Boltzmann distribution leads to this expression for any gases. But the so-called Gibbs paradox

arises here. If the answer obtained from the Boltzmann distribution is divided by 1/N !, where N is thenumber of particles, then the Gibbs paradox in its original statement can be resolved for any gas consistingof molecules of the same type. We must note here that the particles of this gas are indistinguishable.Relation (1) still holds for a mixture of gases, but dividing by 1/N ! is already clearly impossible becausethe gases are distinguishable.

And what happens if the gases are different isotopes of the same matter? If two mixed gases areabsolutely identical, then we must divide by 1/(N1 + N2)!. And what must we do if they differ onlyslightly? We obtain the same jump as in the original Gibbs paradox with a partition between the two gases.Incidentally, a solution of the original paradox was proposed in the form of a semitransparent partition(see, e.g., [2]–[5]). Finally, what happens if we mix Bose gases with different masses? This paradox, similarto the Gibbs paradox, is called the Einstein paradox.

We used purely mathematical theorems [6]–[18] to show that the value of Z for noninteracting particlesis not equal to unity for each separate gas consisting of molecules of the same type and the principal role

Lomonosov Moscow State University, Moscow, Russia, e-mail: [email protected].

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 167, No. 2, pp. 295–310, May, 2011. Original

article submitted February 8, 2011.

654 0040-5779/11/1672-0654

c©

is played by the value Zcr, i.e., by the compressibility factor expressed in terms of Tcr, Pcr, and ρcr, whereρcr is the critical density. This has a number theory meaning, namely,

Zcr =ζ(γ + 2)ζ(γ + 1)

, (3)

where ζ(x) is the Riemann zeta function, γ = d/2, and d is the fractional dimension of the Bose gas, whichhas no meaning as a quantum imperfect Bose gas but can be regarded as the problem of Diophantinerelations (7) and (8) in [1] (see below).

An analogy with number theory permits solving the van der Waals problem of calculating the “physical”entropy exactly. In 1908, van der Waals used namely the entropy calculation problem to explain the Gibbsparadox. He wrote [19]: “The entropy itself is not a directly measurable quantity, and we hence cannot

expect that the Gibbs paradox can be explained by measurements.”

In [1], we introduced the potential Ωγ in number theory. According to the physical representations, theentropy is determined as the derivative of this potential with respect to the temperature. Direct calculationsshow that it coincides with the Hartley entropy for the number of solutions of relations (7) and (8) in [1](see, e.g., [20]).

The problem of a gas mixture is still relevant. Relation (1) holds for new ideal gases as μ → −∞(where μ is the chemical potential). This relation also holds for their mixtures. In contrast to the new idealgases, two mixture types are known for real gases: homogeneous and heterogeneous. The homogeneousmixture is a mixture in which the gas percentage composition does not vary. Air is an example of such amixture. A mixture of air and water vapor remains homogeneous in a very narrow temperature range, untilsufficiently many water clusters form in the vapor. Then the mixture becomes heterogeneous. The initialpercentage composition of gases inside the droplets of fog varies: the mixture becomes heterogeneous. Weconsider only mixtures whose homogeneity regions contain the critical points of both mixture components.

Moreover, we assume that the “ideal curve” (the Bachinskii law) holds in this region, i.e., the curveZ = 1 on the (ρ, T ) diagram is a segment of the straight Zeno line. We can formally continue this curveto the points T = TB (the Boyle temperature) and ρ = ρB (the Boyle density). These points may never beattained in actuality. We define them for convenience in calculations.

The experimental data [21] indicate that a mean Zeno line arises for a mixture of two gases. Themethod proposed here does not permit calculating this line, although the scattering problem solution in [1]relates Zcr and the Zeno line. It is possible to choose a one-parameter family of Lennard-Jones–typepotentials for different values of the parameters uniquely relating different values of Zcr and the Zeno line.We believe that the solution of such a problem is simplified because obtaining Zcr and the Zeno line usingthe Lennard-Jones–type potential requires only the problem skeleton (see [1]), i.e., the set of point pairscorresponding to the rest points and not a functional of the interaction potential.

Therefore, if we obtain the ideal (“pure”) gas corresponding to the sum of the pure gases, then we canuse the formulas given below for the transition from an ideal gas to an real gas.

Remark 1. We here replace the standard thermodynamic normalization N/V → const as N → ∞with a van der Waals–type statement: Tred = T/Tcr, where Tcr is the critical temperature at the spinodalpoint of the liquid phase. This normalization significantly changes the notion of thermodynamic limit instatistical physics because in an ideal Bose gas, we have relations (see (7) and (8) in [1]) that are Diophantinerelations in number theory, where N → ∞ and E → ∞. This implies β = 1/T → 0. The normalizationTred = T/Tcr replaces the “balancing” of N in the right-hand side of relation (8) in [1] by using the volumeV and retains the problem in the framework of pure number theory. The difference from the van der Waalsformulation is that the equality Tred = 1 holds at the spinodal point of the liquid phase. In the ideal gas,the temperature of that point is uniquely determined by the value Zcr.

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2. Transition from an ideal gas to a real gas in gas, fluid, andliquid phases

The notion of an real gas indeed differs from the new notion of gas, introduced in [1], only in that theline Z = 1 corresponds to μ = −∞ in ideal gases and to μ > −∞ in real gases. Moreover, the behaviors ofthe temperature and pressure densities for Z = 1 are related quite definitely. On the plane (ρ, T ), this is aslanting line called the Zeno line (see [1]). In other words, it suffices to know the constants TB and ρB .

If the value γ = γ0 corresponds to Zcr, then we multiply by a function of V in our distribution (37)in [1], i.e., in the Bose–Einstein distribution, we change

V → ϕγ(V ),ϕγ(V )

V→ const as V → ∞. (4)

This function is constant for γ ≥ γ0, where γ0 = 0.2 is the critical dimension corresponding to the valueZcr = 0.29 [1]. The quantity Z obtained for μ = 0 therefore has the form

˜Z =V ϕ′

γ0(V )

ϕγ0(V )· ζ(γ0 + 2)ζ(γ0 + 1)

= 0.29. (5)

In the gas phase, the constant in (4) is equal to unity for γ ≥ γi. To obtain the value

PcrVcr

RTcr= Zcr =

ζ(γ0 + 2)ζ(γ0 + 1)

for the new ideal gas, we must take γ0 = 0.222.We repeat the calculations performed in [1] in more detail. For any Z < 1, we have

Z =V ϕ′

γ0(V )

ϕγ0(V )· Γ(γ0 + 1)Γ(γ0 + 2)

(∫ ∞

0

εγ0+1 dε

eξ−κ − 1

)(∫ ∞

0

εγ0 dε

eξ−κ − 1

)−1

=V ϕ′

γ0(V )

ϕγ0(V )Ψ(κ),

κ =μ

T, ϕ′

γ0(V ) =

∂ϕ

∂V, V =

ZT

P,

(6)

where Γ( · ) is the gamma function. For κ = 0, we obtain (5). Further, we find μ(T, V ) as a function of V

from the condition Z = 1:V ϕ′

γ0(V )

ϕγ0(V )Ψ(κ) = 1, κ = κ(V ). (7)

On the other hand, the condition on the Zeno line determined by its slope

P = ρBT

(

1 − T

TB

)

(8)

or

P = TBρ

(

1 − ρ

ρB

)

(9)

and also

T = TB

(

1 − ρ

ρB

)

(10)

is satisfied for Z = 1. The dependences P (T ), T (ρ), and P (ρ) are therefore known: P (T ) is the Bachinskiiparabola, T (ρ) is a straight line, and P (ρ) is a parabola.

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We obtain P from the Bose–Einstein distribution, where V is replaced with ϕγ0(V ):

P =ϕ′

γ0(V )T γ0+2

Γ(γ0 + 2)

∫ ∞

0

εγ0+1 dε

e−κeε − 1,

ϕ′γ0

(V ) Liγ0+2(y) =ρ

T γ0+1B

(1 − ρ/ρB)γ0+1, ρ =

1V

,

V ϕ′γ0

(V )ϕγ0(V )

Liγ0+2(y)Liγ0+1(y)

= 1,

(11)

where Li is the polylogarithm. The initial condition for γ < γcr and T < Tcr is determined by the conditionϕγ0(V )/V → 1 as V → ∞. The value1 ϕγ0(V ) does not vary for γ > γ0.

For T > Tcr as well as for γ < γ0, the initial condition is

ϕ(V )V

→ const. (12)

This constant is chosen such that the ideal gas isotherm for T < Tcr and for Z = 1 falls on the correspondingpoint T on the Zeno line. And for T > Tcr and Z = 1, the isochore issuing from the critical isotherm towardthe plane (Z, P ) falls on the corresponding point ρ on the Zeno line.

Of course, the notion of the new ideal gas [1] must generalize the notion of the usual ideal gas and passinto it for certain values of the parameters (temperature, pressure, density). The new ideal gas related tonumber theory, just as the Bose and Fermi gases, must become the old ideal gas as the chemical potentialtends to −∞, and we obtain Z = 1. Because the condition on the Zeno line is satisfied (see [1]) andis observed experimentally, it turns out that only one additional number, namely, the ratio of the Boyletemperature to the critical temperature, is needed to obtain the thermodynamics of real gases from the newideal gases by rather simple transformations (solving a first-order equation and “fitting” its initial conditionby the shooting method).

Remark 2. For Tcr = 1 and Pcr = 1, we obtain ρcr = 1/Zcr. For the Zeno line, we have

ρcr = ρB

(

1 − 1TB

)

and therefore

1 = ZcrρB

(

1 − 1TB

)

.

Hence, on the Zeno line, we obtain

P = ZcrρB

(

1 − T

TB

)

T = ZcrTB

(

1 − ρ

ρB

)

ρ

(if Tcr = 1 and Pcr = 1).For T < 1 in the the Bose–Einstein distribution of dimension γ0, the volume V is replaced with

ϕγ0(ρcrV ). For T > 1, the volume V is replaced with ϕγ(ρcrV ), where γ > γ0.The initial condition is replaced with the boundary condition. For an ideal gas T > Tcr (fluids), the

isochores–isodims are constructed (constant density and constant dimension γ, corresponding to the givenvalue Z). For T < Tcr, the isotherms–isochores are constructed for the liquid phase. Substituting theboundary condition for the initial condition also occurs in the case of the liquid phase.

1There was an unfortunate misprint in [1]: multiplication by the volume V was lost in formulas (26)–(29). The conditionϕγ(V )/V → 1 as V → ∞ cannot be satisfied without this factor.

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3. Normalization to the mercury temperature: Regularizationand correction to the Bose–Einstein distribution

We first perform the necessary renormalization of the Bose–Einstein distribution. The integral of thedistribution must be equal to unity in this renormalization. Such an integral can generally diverge. Forexample, this is the case for the Pareto-type distributions if the distribution decreases sufficiently slowly.

We assume that the distribution decreases as 1/ log(|x| + 1). Even its square cannot be integrated.From the standpoint of Hilbert spaces, this is a generalized function. As an example, we consider thefunctions {eiπn} on the interval (−π, π). If the coefficients of the expansion in these functions decrease as1/ log(|x| + 1), then this is a generalized function in the Hilbert function space L2 on the interval (−π, π).We further assume that its maximum is attained for n = 6 · 1023. This means that from the standpoint ofthe Hilbert space L2, they oscillate rapidly and hence represent a generalized function equal to zero on thespace of infinitely differentiable finite functions up to 10−23. Nevertheless, we normalize to this maximumand study the asymptotic behavior with respect to this maximum as if we were transported into anotherworld. For the origin (i.e., for the unity), we take the temperature in the hottest tropics, i.e., the tropicaltemperature.

Van der Waals, already knowing the standard normalization of the “terminological limit” N/V → const,noted a new normalization that gave the famous law of corresponding states, which allows classifying gases.The principal normalization is Pred = P/Pcr. We note that if precisely this relation is treated as the“thermodynamic limit” in the original statistical physics, then in the original number theory problem ofDiophantine relations (7) and (8) considered in [1], it is already unnecessary to introduce the additionalparameter V (or simply to set V = 1) because N is finite in this remarkable normalization (see (13) below)and can denote the density ρ, and 1/N is a “specific volume.”

In the Bose gas, Tcr is the degeneracy temperature determined as a quantity tending to infinity. Thisin fact corresponds to the normalization of the distribution to its maximum, very far from zero kelvins.The number of particles at which the condensate is formed is equal to

Ncr = Γ(γ0 + 1)ζ(γ0 + 1) (13)

in such a normalization and hence is not too large.For γ ≤ 0, it is necessary to introduce a correction to the Bose–Einstein distribution, namely, to use

formula (37) in [1].2 We find the asymptotic form of the relation

N =∫ ∞

0

tγdt

[

1(et/z) − 1

− N

(etN/zN) − 1

]

= Γ(γ + 1)[

Liγ+1(e−ξ) − Liγ+1(e−Nξ)Nγ

]

(14)

at the point Ncr given by (13). Relation (14) holds near point (13) and refines the Bose–Einstein formula.We note that the kelvin calibration is indefinite because absolute zero cannot be reached. In general, it

is more natural to consider log T for the kelvin calibration. Zero then becomes minus infinity, which cannotbe reached, and one kelvin becomes zero. The unattainability of minus infinity is more obvious than theunattainability of zero.

It then makes no difference from what point to measure. Therefore, log T is determined up to aconstant, and T is hence determined up to multiplication by a constant. It is therefore impossible todetermine Tcr for a single gas in some kind of units (for example, in kelvins). It is therefore more naturalfor the ideal gas to consider the normalization to Tcr just as the Celsius calibration, for example, is relatedto the real gas H2O [23], [24].

2This term is obtained from the parastatistics because the number of particles at each energy level cannot exceed N , thenumber of particles in the gas. On the other hand, N is sufficiently large for the Euler–Maclaurin estimates [22] to be used inthe parastatistics.

658

Therefore, to resolve the Einstein paradox, we must include the “fractal” fractional dimensions arisingfor a mixture of two Bose–Einstein gases for which there are two different degeneracy temperatures T1

and T2 (let T1 > T2) depending on the percent composition of the gas volumes. We then accept thenormalization Tred = 1 at the point Tcr for the gas with the minimal Zcr for the mixture components. Byvirtue of the equations given below, we find the temperature T sum

cr and the corresponding dimension γsum0 .

And because we take into account that particles are indistinguishable when counted, this refinement justleads to the solution of the Gibbs paradox.

If we consider the entire gas scale, then we discover that the values of Zcr are bounded above. Mercuryhas the greatest value of Zcr (0.39). The value of Zcr is close to it in the van der Waals model. The valueof Zcr is bounded below by the value for the gas He3, for which semiclassical effects are apparent (see [1]).Therefore, the normalization for the gas scale must be performed by setting T/Tcr as for mercury.

For a particular gas, the normalization Tcr = 1 and Pcr = 1 is possible. But if we compare differentgases and study their homogeneous mixtures, then it is most natural to consider the normalization on Tcr

for mercury because the value of Zcr for it is the maximum among all gases. In this case, for a classical gasof dimension γ, the critical temperature has a value less than unity:

T (γ)cr =

(

ζ(2 + γ0)ζ(2 + γ)

)1/(γ+2)

, (15)

where γ0 = 0.33 for mercury. Here, we find the corresponding spinodal point of the liquid phase andrenormalize with respect to it.

4. New classical ideal gas as an alternative to the Bose condensate

We considered all solutions of Diophantine equations (7) and (8) in [1] equiprobable. As we previouslyshowed with the example of distributing E bills over N spectators, if N > Ncr, then N − Ncr spectatorsend up with no bills. Of course, we then assumed that N/Ncr ≥ 1 + δ, where δ > 0 is any fixed numberand E → ∞ and E � N as N → ∞. As we have written many times, this fact corresponds to the Bosecondensate in physics and can, for example, determine the unemployment rate in economics.

An alternative to this phenomenon in number theory is association into pairs for N = 2Ncr and intoclusters for greater values of N . As previously noted, this association means a decrease in the dimension,which corresponds to a decrease in the number of degrees of freedom of the group components.

The question of how to group optimally, i.e., how to decrease the dimension most profitably, arises. Adecrease in the dimension (related to the “number of degrees of freedom”) can occur in economics eitherbecause of grouping or because of restricting “market freedom” by laws. How to choose the best way?

In the Korov’ev example,3 often discussed in our papers, it was stated that it is sufficient for thespectators to unite into groups of ten, and each group then gets at least one banknote with a very largeprobability. By dividing the banknotes among the members of the group, everyone obtains a share anddoes not die of hunger. This procedure results in a significant decrease in entropy because the moneyis distributed in each group according to some rule, for example, equally among all the members of thegroup. But this million can also be distributed equally among the spectators. The entropy then vanishes,and communism will arrive, as proposed by another person, Sharikov, in Bulgakov’s novel. In both cases,the main goal is achieved: all the spectators obtain money and survive, which in thermodynamic termsimplies that the number of particles is preserved. But the problem is to preserve the number of spectators(particles) under the condition that the number of spectators (particles) in each group is minimum. Thismeans that we must obtain the maximum dimension given that the number of particles is preserved. We

3In a variety show, Korov’ev used the trick of throwing 1 000 000 banknotes to 10 000 spectators.

659

call this transition to the maximum dimension transition to the liquid state. While we take the dimensionγ = γ0 for the gas branch of the isotherm and do not change this value for γ > γ0, we see that for the liquidbranch, the value of γ and of Zγ

cr decrease as the temperature decreases.Obviously, as the temperature decreases, the number of particles in the groups, i.e., the number of

molecules in the clusters, increases. To obtain this increase, we must raise the original temperature (forγ = γ0) for the entropy such that the original number of solutions of the Diophantine equations increases.The number of molecules in the group (cluster) is then also greater. This means that in the original formulafor the entropy, we must increase Tcr. Hence, as the real temperature of the liquid phase decreases, we mustincrease the relative temperature Tcr in the original formula for the entropy. This implies that the relativetemperature for Zγ0 increases and the original entropy becomes

S =(

T

Tliq

)γ+1(

Liγ+2(e−ξ)(γ + 2) + ξ Liγ+1(e−ξ))

, (16)

where ξ = −μ/T (see [18]).

Remark 3. The polylogarithm Liγ+2(e−ξ) has the form

Liγ+2(e−ξ) =∑ e−kξ

kγ+2=

∑ e−kξ−γ log k

k2.

Its derivative with respect to γ is equal to

∂

∂γLiγ+2(e−ξ) = −

∑ log ke−kξ−γ log k

k2.

For μ = 0, we obtain

S|μ=0 =(

T

Tliq

)γ+1

ζ(γ + 2)(γ + 2). (17)

We hence obtain the simple condition for maximizing γ:

dS

dγ

∣

∣

∣

∣

ξ=0

= 0, log(

T

Tliq

)

ζ(γ + 2)(γ + 2) + ζ′γ(γ + 2)(γ + 2) + ζ(γ + 2) = 0, (18)

− d log ζ(γ + 2)dγ

= log(

1Tliq

)

+1

γ + 2. (19)

For mercury, just as for the highest value of Zcr for all real gases, we set Tcr = 1. The solution of (18)then yields γ = 0.14, and therefore Zγ1(γ0) = 0.24.

Because N = const, we consequently have

Pγ1 = const · Zγ1

on the isotherm. The points Zγ1 and Pγ1 are the spinodal points of the liquid phase, i.e., the points wherethe metastable state of the liquid phase terminates.

Remark 4. The gradient descent method for the entropy describes the unstable part, and its trajec-tory is not observed in experiments. But to avoid discontinuities in the graphs for the gas and liquid phases,it is more customary to solve the gradient descent equation from γ0 to γ1(γ0) from the point γ = γ0, ξ = 0up to the point γ1 = 0.14.

660

For mercury, we must set Tred = T/Tcr, i.e., Tred = 1 for T = Tcr. If T < Tcr, then T liqred = T/Tcr.

Because the temperature is less than Tcr at each point of the liquid for γ < γ1 but each point isassociated with a particular value of Zγ

cr, it is convenient to normalize each value of Zγcr by the corresponding

value T γliq, which means

Tred =1

Tcr,

i.e., the liquid temperature T γ0cr and also therefore the entropy S increase by the value of the new Tcr equal

to 1/Tred,Sγ0 = T γ+1

cr

(

Liγ0+2(e−ξ)(γ0 + 2) + ξ Liγ0+1(e−ξ))

. (20)

We therefore set T = 1/Tliq in formulas (18) and (19).We say that a gas is quasi-ideal if ϕ′

γ(V ) = const (see [18]). In this case, the Zeno line appears inthe definition of the spinodal points of the liquid phase, but the liquid remains incompressible. Only theconsideration of the function ϕγ(V ) bends the straight lines and takes the compressibility of the liquidinto account. But for Tliq � 1, i.e., for sufficiently small (but still positive) values of γ, the value of thefunction ϕγ(V ) already has an insignificant effect. Therefore, a quasi-ideal gas and especially a mixture ofquasi-ideal gases provides a sufficiently good approximation to the spinodals of the liquid phase.

The geometric locus of the points of the quasi-ideal spinodals4 is given by

P = TρB

(

1 − T

TB

)

Zγcr, (21)

where ρB and TB are the Boyle density and Boyle temperature. Here,

Zγ(Tliq)cr =

ζ(2 + γ(Tliq))ζ(1 + γ(Tliq))

, T = Tliq. (22)

We recall that γ(Tliq) can be calculated from algebraic relation (19).Each point on the quasi-ideal spinodal is joined to the point on the Zeno line corresponding to the

given value of γ(Tliq) to the left of the critical line for γ(Tliq) = 1. For γ(T ) < 0, we use our previouslyintroduced correction to the Bose–Einstein distribution.

We find the constants b and κ from the relations

∫ ∞

0

ξ

(

1eb(ξ+κ) − 1

− N

ebN(ξ+κ−1)

)

ξγ dξ = E , (23)

∫ ∞

0

(

1eb(ξ+κ) − 1

− N

ebN(ξ+κ−1)

)

ξγ dξ = N, (24)

where κ = −μ and b = 1/T .We set n = E . For κ = 0, we have

n =∫

ξ dξα

ebξ − 1=

1b1+α

∫ ∞

0

η dηα

eη − 1, (25)

where α = γ + 1. Hence,

b =1

n1/(1+α)

(∫ ∞

0

ξ dξα

eξ − 1

)1/(1+α)

. (26)

4These are the endpoints of the metastable state of the liquid phase.

661

Setting Ncr = k0, we obtain

k0 =∫ ∞

0

(

1ebξ − 1

− k0

ek0bη − 1

)

dξα

=1bα

∫ ∞

0

(

1eξ − 1

− 1ξ

)

dξα +1bα

∫ ∞

0

(

1ξ− 1

ξ(1 + (k0/2)ξ)

)

dξα −

− k1−α0

bα

∫ ∞

0

(

kα0

ek0ξ − 1− kα

0

k0ξ(1 + (k0/2)ξ)

)

dξα. (27)

We set

c =∫ ∞

0

(

1ξ− 1

eξ − 1

)

ξγ dξ.

Replacing k0ξ = η, we obtain

k1−α0

bα

∫ ∞

0

(

kα0

eη − 1− kα

0

η(1 + η/2)

)

dξα

=k1−α0

bα

∫ ∞

0

(

1eη − 1

− 1η(1 + η/2)

)

dηα

=k1−α0

bα

{∫ ∞

0

(

1eη − 1

− 1η

)

dηα +∫ ∞

0

dηα

2(1 + η/2)

}

= −ck1−α0

bα+ c1

k1−α0

bα. (28)

Because1

η(1 + η/2)=

1η− 1

2(1 + η/2),

setting

c1 =∫ ∞

0

dηα

2(1 + η/2),

we can write

∫ ∞

0

(

1ξ− 1

ξ(1 + k0ξ/2)

)

dξα =k0

2

∫ ∞

0

dξα

1 + k0ξ/2=

=(

k0

2

)1−α ∫ ∞

0

dηα

1 + η= c1

(

k0

2

)1−α

. (29)

Consequently,

k0 = − 1bα

c1 +1bα

c

(

k0

2

)1−α

− k1−α0

bα

∫ ∞

0

{

1eη − 1

− 1η(1 − η/2)

}

dηα −

− 12

∫

dηα

1 + η/2· k1−α

0

bα= − 1

bαc +

k1−α0

bαc. (30)

Because k0 is the number of particles, b = 1/T , and α = 1 + γ, it follows that k0bα for γ > 0 is the

Riemann function ζ(1 + γ). Therefore, for γ < 1, kγ+10 increases, and we can neglect the first term in the

right-hand side of (30). Setting Tcr = 1 in this asymptotic form, we see that it is natural to regard the

662

function

M(γ + 1) =(

c(γ)Γ(γ + 1)

)1/(1+γ)

(31)

as the continuation of the Riemann zeta function to the additional domain of the arguments 0 > γ ≥ −1.

Remark 5. For μ < 0, the asymptotic form of (23) can be expressed as

∫

tγ dt

k2e−μk1et/k2 − 1− Γ(γ + 1)Liγ+1(eμ), (32)

and for k1 → ∞, k2 → ∞, and μ → 0, it becomes c(γ).

The compressibility factor Zγ = −ζ(γ + 2)/M(γ + 1) changes by a jump5 from γ = 0 to γ < 0.We see that the metastable region of negative pressures terminates. This implies the paradoxical effect ofincreasing density as the temperature decreases, which was observed experimentally (see [26] and also thevideo on the site http://www.iem.ac.ru//staff/kiril, where this phenomenon is observed).

In the considered approach, we find that the trajectory satisfies the relation

μ dN = 0. (33)

Hence, the new ideal gas corresponds to discontinuous isotherms composed of the gas branch and theincompressible liquid branch.

From the standpoint of economics, this means that the decrease in “freedom” is a result of firstassociation and then strengthening of laws such that, for example, there would be no unemployment andthe number N would be preserved. From the physical standpoint, this means that the increase in N forT > 1 leads to a cascade increase of associations into clusters, and the incompressibility phenomenon thenarises, i.e., an increase in the pressure no longer leads to a decrease in the volume.

The trajectory, for T > 1 consisting of two nonintersecting straight lines on the plane (Z, P ), followingfrom number theory, is the skeleton of thermodynamics and corresponds to the new ideal gas. The smoothing(“enveloping by soft tissue”) of this skeleton reflects the effects of the interaction.

Remark 6. The first-order phase transition on the plane (P, Z) at T < Tcr occurs for P = Pcr

and density and chemical potential variations, and the first-order phase transition at T < Tcr occurs onthe isotherm for different pressures and different chemical potentials. This transition corresponds to thetransition on the van der Waals diagram from the gas branch to the fluid branch on the entire interval ofmetastable states: from the spinodal point on the gas branch to the spinodal point on the liquid branch.

To obtain a phase transition of standard form for T < Tcr, we must find a point of equal chemicalpotentials on the two isotherms (the Maxwell rule). This is a separate condition of projection on the plane(T, P ) of two Lagrangian manifolds with an edge (the edge corresponds to the spinodal) under which theGibbs potential takes the maximum value. In other words, the chemical potential along the isotherm isminimized.

The so-called geometric quantization of these two manifolds can be performed by the tunnel canonicaloperator, where 1/N is a small parameter. This is the only small parameter in number theory. Accordingto [1], it is natural to take the gas diffusion as a small parameter for real gases (see Sec. 6).

5The jump of the spinodal as it passes through the point P = 0 from positive pressures (compression) to negative pressures(stretching) is natural for a liquid because like rubber with small pores for which Hooke’s law for strain and stress undergoesa jump, the derivative of pressure with respect to density also changes by a jump. Of course, this jump is smoothed for a realgas [25].

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5. Einstein paradox and ideal gas of a mixture

The notion of a mixture of ideal gases according to the old concept of an ideal gas corresponds to theGibbs paradox. We acted radically: we rejected the old notion of an ideal gas and accepted a new notionof it and the corresponding new notion of a mixture of ideal gases.

Describing a real gas, we can construct a potential Ω that differs somewhat from the Bose gas offractional dimensions in the gas phase and differs significantly for fluids and the liquid phase [1].

We present the formulation of the Bose-statistics paradox, which is often compared to the Gibbsparadox [27]. Einstein wrote ([28]): “In conclusion, I want to draw attention to a paradox that I could not

explain. Using the method presented here, we can also easily consider a mixture of two different gases. In

this case, each type of molecule has its own specific ‘cell,’ whence the additivity of the entropy of the mixture

components follows. Therefore, each component behaves in the sense of the molecular energy, pressure, and

statistical distribution as if the mixture volume contained only it alone. But for a given temperature, the

mixture of n1 molecules of one type and n2 molecules of another type in which the difference between the

molecules of the first and the second types is arbitrarily small (especially with respect to the molecular

masses m1 and m2) has a different pressure and a different state distribution than the simple gas with

the n1+n2 molecules of practically the same molecular mass occupying the same volume. But this seems

impossible.”

If we consider the quantum Bose gas for which the initial function ψ is symmetric with respect to n1

particles of the first type and n2 particles of the second type, then the solution is obvious. Our Hamiltonianis not symmetric with respect to permutations of p

(1)i , q

(1)i and p

(2)i , q

(2)i , where the superscript distinguishes

particles of the first and second types. Even if we symmetrize the initial function ψ, this symmetryimmediately disappears for t > 0. And the symmetrization itself is a jump of the symmetry operator.

As we saw in [1], the Bose-gas statistics corresponding to the number theory problem is a quiteanother matter. If we deal with calculations of the Bose-condensate point, then the calculation of the sumdimension also corresponds to the Bose condensate at the same Zsum

cr , as follows from [1]. We hence obtainthe Einstein rule of addition of Bose gases of any fractional dimension with the corresponding values of theBose condensates (i.e., critical points). Each of the masses m1 and m2 is associated with its own values T

(1)cr

and T(2)cr or N

(1)cr and N

(2)cr according to the van der Waals law of corresponding states and the normalization

Pcr = 1: gases with the same value of Zcr are equivalent.We speak about a mixture of gases with different values of the critical quantities if we define the density

of this mixture. This again reduces to a number theory problem and to Diophantine relations. Our problemis to introduce a new notion, a new ideal gas of a mixture of gases. It follows from the mass conservationlaw for N

(γ1)cr , N

(γ2)cr , and N

(γ)cr that

(αm1 + βm2)γ+1(T (γ)cr )γ+1ζ(γ + 1) = αmγ1+1

1 (T (γ1)cr )γ1+1ζ(γ1 + 1) +

+ βmγ2+12 (T (γ2)

cr )γ2+1ζ(γ2 + 1), (34)

where α + β = 1. It follows from the additivity of entropy that

((αm1 + βm2)γ+1(γ + 2)ζ(γ + 2)(T (γ)cr )γ+1 = mγ+1

1 α(γ1 + 2)ζ(γ1 + 2)(T (γ1)cr )γ1+1 +

+ mγ+22 β(γ2 + 2)ζ(γ2 + 2)(T (γ2)

cr )γ2+1 (35)

(both identities are in a nonequilibrium state). An analogue of the Zeno line and the function ϕγ0(V ) willbe considered in detail in the next paper.

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6. Critical exponents, the scaling hypothesis, the tunnel canonicaloperator, and the Maxwell rule

Not only the considerations presented above but also the arguments involving the Lennard-Jonesinteraction potential are based on the role of attraction between particles or, equivalently, on the formationof combinations (dimers and clusters) in the definition of the new ideal gas. The role of repulsion was onlyauxiliary when we spoke of the value of small viscosity for a dimer captured in a trap (potential well).Namely, the dimer must eventually descend to the bottom of the well because of small viscosity.

In the notion of ideal gas thus introduced, the states that are metastable on the liquid and gas branchesof the ideal gas do not differ from the stable state corresponding to the Maxwell rule for which the chemicalpotentials on these branches must be equal. This remarkable fact is explained by repulsion and thus byviscosity. In our opinion, Widom’s scaling hypothesis is related to this point.

Because the data of the classical theory of critical exponents did not coincide with the experimentaldata and Widom (1965) developed a hypothesis to some extent contradicting the basic postulates of ther-modynamics, it would seem natural that the solution of the problem concerning the Gibbs paradox wouldalso abolish this contradiction. Indeed, the nonanalyticity arises in the new theory concerning the transitionfrom the new ideal gas to the imperfect one. Namely, the derivative of ϕγ(V ) with respect to γ vanishes forγ ≤ γ0 and T < Tcr, where γ0 corresponds to the critical point, while this derivative is nonzero for γ > γ0.

But the independence of the critical exponents evaluated according to the scaling hypothesis for sixcritical values of gas leads us to suggest that the principal point of contradiction involves an additional factor.For a homogeneous mixture, no experimental confirmations of the critical parameters evaluated using thescaling hypothesis are known to us, but we have no doubt that these parameters will be confirmed.

After constructing a rigorous mathematical conception of thermodynamics, it is necessary to give ananswer concerning this disagreement. In this section, we answer this question, although the appropriatederivations and numerical coincidences are still not available.

The relation between viscosity and diffusion is well known in molecular physics. The diffusion parameterand the parameter ih, where h denotes the Planck constant, are closely related (see, e.g., the formulas forinstantons in [29] and [30]).

We have also written several times that the so-called thermodynamic potentials in the phase spaceT, P, V, −S form a two-dimensional surface, which we called a Lagrangian manifold in the phase space(q1, q2, p1, p2) in [31], and the thermodynamic potentials are a representation of the action S =

∫

p dq indifferent local charts of this manifold.

As noted above, the thermodynamics of an imperfect gas that we have constructed is not analytic atthe critical point. But the leading term of the tunnel canonical operator [32], where the role of a smallparameter is played by diffusion, is characterized in the Maxwell phase transition by the jump of the gasbranch to the liquid branch for the same value of the chemical potential. Diffusion “erodes” this densityjump in the same way that viscosity in thermodynamics smooths the shock wave, although this is not soclear in experiments. But at the critical point, as at the instant of generating a shock wave (for example,for the Burgers equation), the focus formed at the point is more significant the stronger the influence ofthe canonical operator and diffusion. Therefore, the contribution near the focal point (in our case, near thecritical point) gives an essential correction to the critical index obtained from the asymptotic behavior ofthe canonical operator. This contribution is independent of the value of the critical point.

The universal property, i.e., the independence from diffusion and dependence only on the defect of thefocal point [33], is related to the fact that the distance from the critical point is also a small parameter thatis much less than the diffusion parameter. The scaling rule holds for the simplest focal points expressedusing the Airy function or the Weber function. The defect of a focal point [33] is an invariant related to theprojection of this point on the coordinate plane (T, P ). Similarly, the projection to other coordinate planes

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gives other values of the critical parameters. In mathematics, this procedure (in the case of an oscillatingnontunnel canonical operator) is known as “geometric quantization” (see [34]). Accordingly, the creationoperators for dimers and other clusters also generalize the notion of secondary quantization, extend the areaof application of thermodynamics to the semiclassical case (see [35]), and, in particular, allow obtaining thesuperfluidity of classical gases (e.g., H2O) in nanotubes [36]–[38], which was confirmed experimentally.

Acknowledgments. The author deeply thanks V. S. Vorobiev for the useful discussions.

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