ISSN 0001-4346, Mathematical Notes, 2011, Vol. 89, No. 2, pp. 266–276. © Pleiades Publishing, Ltd., 2011. Original Russian Text © V. P. Maslov, 2011, published in Matematicheskie Zametki, 2011, Vol. 89, No. 2, pp. 272–284.
Mathematical Solution of the Gibbs Paradox V. P. Maslov * Moscow State University Received December 20, 2010
Abstract—In this paper, we construct a new distribution corresponding to a real noble gas as well as the equation of state for it.
DOI: 10.1134/S0001434611010329 Keywords: Zeno line, partitio numerorum, phase transition, cluster, dimer, critical tempera- ture, Boyle temperature, jamming effect, Bose–Einstein distribution, Gibbs paradox.
Since all the papers of the author dealing with this problem1 were published in English, here we present a short survey of these papers in both Russian and English. The Gibbs paradox has been a stumbling block for many physicists, including Einstein, Gibbs, Planck, Fermi, and many others (15 Nobel laureates in physics studied this problem) as well as for two great mathematicians Von Neumann and Poincare.´ Poincare´ did not obtain the mathematical solution of the Gibbs paradox, but attempted to solve this problem from the philosophical point of view.2 A careful analysis of Poincare’s´ philosophy shows that it is especially constructed so as to justify the contradiction between two physical theories. He writes, in particular, “If a physicist finds a contradiction between two theories that are equally dear to him, he will sometimes say: let’s not worry about this; the intermediate links of the chain may be hidden from us, but we will strongly hold on to its extremities. ... It may turn out that both theories express real relations, while the contradiction lies only in the symbols with which we have attired them.” [2, p. 104]. Just as the rules of arithmetic are used in mathematics, so are those of phenomenological thermody- namics in hydrodynamics, chemistry, and biology. For mathematicians, the Gibbs paradox is equivalent to the statement that 5=2. Therefore, this contradiction in the calculation of entropy obtained by Gibbs is so significant that, from Lenin’s materialistic point of view, Poincare´ “had fallen into heresy.” If symbols for attiring reality are not searched for, then, in order to provide a final solution of this problem, the Maxwell–Boltzmann distribution must be replaced by a new one-particle distribution which coincides with the Maxwell–Boltzmann distribution for certain parameter values and leads not only to the results of natural experiments carried out at the time of Poincare,´ but also correspond to contemporary observations, especially, those performed in the state of weightlessness. In fact, the new distribution presented by the author in this paper implies not only all phenomenologi- cal laws that were discovered by great physicists, but also touches on modern attempts to explain certain aspects of phase transitions, such as fractal dimension, percolation, dimers, and clusters. Up to now, these new theoretical considerations have not been fully tied up to the old conceptions of the classics. The new distribution completely removes this divergence: it contains fractional (fractal) dimensions and dimers, and, most importantly, it contains metastable states, which were, in principle, rejected by the old thermodynamic conception, although they appear in the empirical Van-der-Waals formula, although were not close to the natural experimental data. Later numerous attempts were made to improve similar empirical formulas (to make them fitspecific experiments).
*E-mail: [email protected] 1These papers were published in “Mathematical Notes” (the English version of “Matematicheskie Zametki”) beginning with Vol. 85 (5) and ending with Vol. 88 (6). 2TheexistenceoftheGibbsparadoxinthemodelofthePoincare´ ideal gas was established by V.V. Kozlov [1].
266 MATHEMATICAL SOLUTION OF THE GIBBS PARADOX 267 As Dirac put it, one must understand the solution of the main (central) problem of a particular science, and the solution of the other problems will appear automatically. This reflects a general property of physics. The distribution given here depends on the prescribed interaction potential for the particles. The numerical results that we refer to are related to the Lennard–Jones potential, which is the most often used potential. But even if we take another interaction potential, then we still have the same scheme of defining the critical constants (Tcr, Zcr, ρcr and TB, ρB), and the fundamentals of the theory of the distribution of integers remain unchanged. All the data given by the author coincide with those for noble gases inasmuch as the noble gases coincide with one another by the law of corresponding states. The construction of the distribution in question consists of two parts. The first part is number theory and number-theoretic distributions for fractional (“fractal”) dimensions,3 varying between 3 and 2. This part is the same for all interactions. It is divided into two alternatives (instant stopping, which is an analog of the Bose condensate, and slow descent, which is the alternative to the Bose condensate). The second part is the classical statistics for the scattering of pairs of particles corresponding to the Lennard–Jones potential. This part can be divided into two parts: 2a) without regard for the “dressed” interaction, i.e., only the interaction of two particles is taken into account, which corresponds to a negligibly small interaction with the other particles (Tcr and TB are determined); 2b) the interaction of all particles is taken into account, which allows us to find the constants Zcr, TB, ρB, and the so-called Zeno line. The first part implies that the focus at these points is so large that these quantities are also preserved in the statistics of the number theory of fractional dimension. And the Zeno line Z =1already implies a transition to the classical Boltzmann statistics.
1. NUMBER THEORY 1.1. A Famous Problem of Number Theory The problem of the partition of integers into summands goes back to ancient times. Euler’s survey “Introduction to the analysis of infinite quantities” (“Introductio in analysin infinitorum”) published in 1748 contained a whole chapter dealing with “partitio numerorum” (the partition of numbers into summands). The author has already discussed, several times, the scene from Bulgakov’s novel “Master and Margarita" in which Korov’ev throws money at the spectators of a variety show. The number of spectators in this example corresponds to the number of particles N, while the number of banknotes E , up to a certain unit of energy, corresponds to the energy. Korov’ev threw E banknotes to N spectators, and he must determine the probability of the event that a given number of spectators gets at least one banknote. The probability question is quite difficult in principle. The great mathematician Henri Poincare,´ who headed the Chair of Probability and Mathematical Physics at the University of Paris since the age of 32, devoted Chap. XI of his book [2] to the calculus of probabilities. In it, he gives the standard definition of probability as the ratio of the number of favorable outcomes to the total number of outcomes, and then presents a counterexample to such a definition of probability. He writes that, to this definition, one should add the additional phrase “provided that all outcomes are equiprobable" [2, p. 116 (Russian transl.)] and notes that we have completed a vicious circle—defined probability in terms of probability. Thus, the problem is to define what outcomes should be regarded as equiprobable. “We must look for mathematical ideas," Poincare´ writes, “where they remain pure, i.e., in arithmetic" [2, p. 13 (Russian transl.)]. And that’s what we do. A famous problem in arithmetic (number theory) “partitio numerorum” is to find the number of partitions of a natural number E into summands, for example, 5=1+1+1+1+1=2+1+1+1=2+2+1=2+3=4+1=3+1+1=5, (1)
3Here we do not prove that fractional dimensions considered by the author coincide with Hausdorff dimensions, but use the definitions given by Vershik [3] and Manin [4].
MATHEMATICAL NOTES Vol. 89 No. 2 2011 268 MASLOV where there are 7 possibilities, provided that the order of summation is not taken into account. If the partition is chosen at random (see [2, Chap. IV “Randomness"] then all the variants are equally possible, and it is natural to regard them as equiprobable, i.e., the probability of any variant is 1/7. Thus, we are returning to the “purest" mathematical ideas. The initial well-known identity for the above problem in number theory has the form of the following Diophantine equation for the unknown integers Ni: ∞ ∞ Ni = N, iNi = E . (2) i=0 i=0 If there are no more than N spectators (the theater has N seats, but, possibly, not all are occupied), then when the ratio of E to N varies, the probability may undergo a jump, i.e., a phase transition may occur. Indeed, in the case N =1and in the case N = E (whatever the value of N) there is only one variant (each spectator gets one banknote), its probability for large E is very small. Therefore, there exists a value Ncr (not necessarily unique) for which the number of cases (variants) of the partition of E into Ncr summands is maximum. This number Ncr was calculated by Erdos˝ in [5]. It is √ 1 E Ncr = ln E (1 + o(1)). (3) 2 π2/6
We consider the original mathematical problem in a different setting. Given E and N ≤ N,where N is a prescribed number, we must find N
In other words, the entropy undergoes a jump when Ncr is passed. Thus, If N
MATHEMATICAL NOTES Vol. 89 No. 2 2011 MATHEMATICAL SOLUTION OF THE GIBBS PARADOX 269 We have thus obtained an analog of the Bose condensate, because the energy E in relation (2) does not change, while the number of particles increases due to an increase of the number N0,whichdoesnot affect E , because it is multiplied by i =0in formula (2).
1.2. Transition to Fractional Dimension The following remark allows us to transfer the results to fractional dimensions.4 This is a most important generalization for the whole subsequent conception. Consider the line and the plane. On the line, we plot the points i =0, 1, 2,..., and, on the coordinate axes x, y of the plane, we plot the points x = i =0, 1, 2,... and y = j =0, 1, 2,... . With this set of points (i, j) we associate points on the line, namely, the entire sequence of natural numbers l =1, 2 ... . To each point l we assign a pair of points i and j such that i + j = l.Thenumbernl of such points is l +1. This is the two-dimensional case. Consider the three-dimensional case. Let us plot z = k =0, 1, 2,... on the z axis, i.e., we set i + j + k = l. In this case, the number nl of points (i, j, k) is (l +1)(l +2) n = . l 2 − For the d dimensional case, it is easy to verify that the sequence of weights (of multiplicities) of the d number of variants i = k=1 mk,wherethemk are arbitrary natural numbers, is of the form (i + d − 2)! q (d)= . (6) i (i − 1)!(d − 1)! (Here d is the topological dimension). Therefore, ∞ ∞ qi(d)Ni = E , Ni = N. (7) i=0 i=0 In√ the theory we are constructing, the difference between the entropies of these two problems is at most N ln N, which means that the difference of the “specific” entropies is small as N →∞. Instead of the number of variants (solutions) of relations (2) and (7), it is natural to consider their logarithm to base 2 (Hartley’s entropy). Then it turns out that the logarithms of the number of solutions to the problems (2) and (7) and to the problem ∞ ∞ Ni = N, Niqi(d)=E , i=0 i=0 ∞ ∞ (8) Ni = N, Niqi(d) ≤ E i=0 i=0 respectively coincide in absolute value with the given accuracy. This consideration makes it possible to generalize the given number theory to noninteger dimensions. We write out expressions of the form ∞ Γ(d + i − 1) N ≤ E , (9) Γ(i +1)Γ(d) i 0 ∞ Ni = N (10) 0 and consider the number of solutions satisfying inequality (9) and relation (10) for noninteger d (the “fractal dimension”); here Γ(d) is the gamma function. Only for N =1and N = E do indistinguishable variants of the type (4+1, 1+4) coincide with dis- tinguishable variants, such as those in Boltzmann statistics, Shannon’s information theory, Kolmogorov complexity theory, and the theory of Gibbs ensembles.
4Fractional dimensions in a Bose gas were first studied by A. M. Vershik.
MATHEMATICAL NOTES Vol. 89 No. 2 2011 270 MASLOV 1.3. Alternative to the Bose Condensate. Clusterization and Decrease in the Number of Degrees of Freedom In Example 1 corresponding to the Bose condensate, 9000 spectators died. However, if they had formed groups (clusters) of 10 persons each, then they would have survive, because each group with large probability would obtain banknotes and could divide them among themselves. This, however, would lead to a decrease in the “number of their degrees of freedom”, i.e., would decrease the “fractal dimension.” But nobody would die. This is an alternative to the Bose condensate. But how must the spectators unify into groups in the most profitable way? The answer is given in this section. Let us determine the parameters β and µ<0 from the system of equations (β =1/T ) ∞ ∞ q jq j = N, j = M. (11) eβj−µ − 1 eβj−µ − 1 j=0 j=0 Theorem. For an arbitrary integer l ≥ 0, the following inequality holds: ∞ ∞ qj −s PM Nj − > ∆ ≤ CsN ,s=1, 2,..., (12) eβj−µ − 1 j=l j=l 2/d 2 where ∆=N ln N,andPM is the ratio of the number of variants satisfying condition (12) to the total number of solutions of the Diophantine relations (9) and (10).5 Passing from the sums in the second term under the sign of the modulus to integrals and using the Euler–Maclaurin formulas, we obtain an expression in terms of polylogarithms. If we assume that the parameters µ, κ = µβ, β =1/T ,andγ are related by the condition asserting the number of variants L of the solution of the Diophantine equations for E and N is maximum, then this means that the “specificentropy” S, i.e., the number-theory entropy [6], [7], takes the maximal values κ κ Sγ = N(Liγ+2(e ) − κ Liγ+1(e )) → max, (13) where Li is the polylogarithm and κ<0, subject to the condition6 [9] µdN =0. (14) Using the method of maximal gradient descent, we obtain the equations ∂S ∂Li (eκ) ∂Li (eκ) γ˙ = = γ+2 − κ γ+1 , ∂γ ∂γ ∂γ (15) ∂S κ˙ = = −κLi (eκ). ∂κ γ In the limit, this leads to the relation Liγ+1(ξ)=const, т.е. dN =0. Here we encounter a difficulty near κ the point κ =0, because the derivative of Liγ+1(e ) with respect to κ lowers the dimension by 1, while, at the point κ =0, there exists a singularity whose regularization was carried out by the author in [8]. For T
5For sharper estimates, see [6]. 6A condition of chemical equilibrium.
MATHEMATICAL NOTES Vol. 89 No. 2 2011 MATHEMATICAL SOLUTION OF THE GIBBS PARADOX 271 2. STATISTICS AND TRAPS FOR COLLIDING MOLECULES. FORMATION OF DIMERS 2.1. Critical Temperature and Boyle Temperature Thus, an alternative to the Bose condensate is the union of particles. It begins with the formation of a dimer, which is the association into a pair of particles. In a gas, pairs of particles fly towards one another from an infinitely large distance with initial energy equal to the difference of the squared initial velocities divided by 2m. The interaction potential for them is denoted by Φ(r1 − r2). If there is no small viscosity, then they will move according to the laws of the standard scattering problem. But if there is an attractive part of the interaction and small viscosity, then the latter can promote the formation of a dimer. In order to coordinate the formation of a dimer with the scattering problem, it is necessary to coordinate the problem of rotation of particles relative to each other along the ellipse with the initial data for the scattered particles. First, we consider the usual assumption adopted in molecular physics that the gas is isotropic and find how many particles of the gas move toward one another. The usual argument in molecular physics is based on the assumption of symmetry of the average motion of molecules along all six directions. Therefore, 1/12 of all particles move toward one another. Since there are three axes, 1/4 of all molecules collide. As is well known, in the radially symmetric case, the total energy is mv2 M 2 + +Φ(r)=E. (17) 2 2mr2 For the original scattered particles, the energy E and the impact parameter B are given. Both the momentum M and the energy E are preserved. As is also well known, in the scattering problem, we have M 2 = B2E. (18) For the case of attraction, from (17) and (18), we obtain the energy (mv2)/2+Φ(r) E = (19) 1 − B2/2mr2 in the domain where r ≤ B. In what follows, we shall obtain analytic formulas for the Zeno line and the binodal; these formulas describe the dependence on the potential. In the scattering problem, for the interaction potential we take the Lennard-Jones potential 12 6 a a Φ(r ,r )=4ε − ,r= r − r , (20) r − r12 r − r6 where ε is the energy of the depth of the well, a is the effective radius, and r − r is the distance between two particles with radius vectors r and r. In the two-particle problem, the problem reduces to the one-dimensional radially-symmetric one in the absence of the external potential. In problem (19), other barriers and wells appear for different values of B. At the rest points Emin and Emax, the velocity is zero; therefore, they can be determined only from the potential term. We now deal not with one particle, but with a pair of particles with mass center captured by the trap. Therefore, the difference Emax − Emin is the energy required to release this pair (dimer) from the trap. The percentage of dimers in a gas can be determined experimentally. It can be observed that dimers are created and split by monomers. Next, their average number can be computed. The higher the temperature, the greater is the average energy of the monomers, and the less is the number of dimers. The main thing is that, under such an approach, only two quantities Emax and Emin remain in the framework of the scattering problem. When Emax = Emin, then the well vanishes. For the attractive part of the Lennard–Jones potential, this energy is 0.8ε. In view of the isotropy indicated above, we find that the average energy of the particles is (16/5)ε. The average energy is the temperature T =(16/5)(ε/k). Above this temperature, there is no well. In thermodynamics, according to the physical meaning, this is the so-called Boyle temperature TB. Using the data for ε from the table (given below) and the formula for
MATHEMATICAL NOTES Vol. 89 No. 2 2011 272 MASLOV the Boyle temperature (T = TB), we find that, for argon (Ar), TB = 382 and, for krypton (Kr) TB = 547; the experimental tables from [10] furnish the following data: for argon, TB = 392,forkrypton,TB = 538. The divergence of theoretical and experimental values is about 2–3%. The critical temperature Emax must correspond to the deepest well; this corresponds to the maximal value of the difference Emax − Emin for all impact parameters B. This difference is equal to the decrease in the energy of a dimer after its capture by the “trap” and thus determines the energy that a monomer must have to release the dimer from the well (which results in the disintegration of the dimer). The height of the barrier “protects” the new pair whose reduced mass found itself in the trap of “dimers” and clusters from the “impact” of monomers. As the temperature T Table Substance ε,K Tcr/4 Ecr · ε/k Ne 36.3 11 10.5 Ar 119.3 37 35 Kr 171 52 50 N2 95, 9 31 28 CH4 148.2 47 43 C2H6 243.0 76 70 C4H10 313.0 106 98 AsH3 281 93 82 GeH4 237 77 59 H2S 301 93 87 H2Se 320 102 93 NH3 300 101 87 PH3 251.5 81 73 Remark. The determination of the critical temperature and the Boyle temperature from the energy ε of the depth of the well of the interaction potential is highly inaccurate and different reference books present different data. Therefore, in what follows, we shall consider the dimensionless quantity T B =2.79, Tcr which is in better agreement with the experimental data (for argon, this value is 2.73). In our simplified problem, the only dimensionless quantity is Emin/Emax, i.e., we now have a problem in which there are just two rest points. Since Emin/Emax is a dimensionless quantity and Emax and the average energy is related to the temperature (multiplied by the density ρ), it follows that the dimensionless quantity in thermodynamics is the so-called compressibility factor denoted by the letter Z, i.e., Z = PV/T if T is measured in energy units (T = Rθ,whereθ is measured in absolute degrees) [7]. MATHEMATICAL NOTES Vol. 89 No. 2 2011 MATHEMATICAL SOLUTION OF THE GIBBS PARADOX 273 2.2. Determination of Zcr and the Zeno Line The dressed, or “thermic,” potential Ψ(r) is attractive. In addition, since the volume V isalarge parameter, it follows that if we expand ar2 Ψ(r)=Ψ , V where a is the effective radius, in terms of 1/V ,then ar2 C ar2 1 Ψ = C + 2 + O . (21) V 1 V V 2 Expanding (r − r )2 (r + r )2 r2 = r2 + r2 = 1 2 + 1 2 , (22) 1 2 2 2 we can separate the variables in the two-particle problem and obtain the scattering problem for a pair of particles and the problem of their joint motion for r1 + r2. In that case, in the scattering problem, the Lennard–Jones interaction potential is complemented by an attractive quadratic potential (inverted parabola), i.e., (mv2)/2+Φ(r) − ρr2 E = , (23) 1 − B2/2mr2 where ρ = C2a/2V (in what follows, the constant multiplying 1/V will not be taken into account, because we shall pass to dimensionless quantities). This addition corresponds to the addition of the third virial coefficient. The minimum value of the compressibility factor Z for a given ρ is Emin Zmin(ρ)= . (24) Emax B→∞ Obviously, the inclusion of the thermic potential must be consistent with what has been said above about the maximal impact parameter before its inclusion. This inclusion involves some “nonsmooth- ness” in the transition from formula (19) to formula (23). The curve Emin = Emax is called the Zeno line. It turns out to be a segment of the inclined line joining the point T = TB with the point ρ = ρB on the graph T,ρ. Let us find the critical points of the curve 1 − Zmin.Thefirst critical point is the point at which the derivative along the segment joining the endpoints of this curve vanishes. The value of Z at this point is denoted by Zcr. The value of ρ at this point is denoted by ρcr. The endpoint of this curve on the axis ρ is denoted by ρB. In [11], this point was called a hypothetical point ρB (of Boyle). Calculating the value of Zcr, we obtain Zcr =0.296, which, up to the third digit after the decimal point, coincides with the values of Zcr for noble gases. The ratio ρcr/ρB also coincides with values of this quantity for noble gases. The value of Zcr can be determined very accurately in experiments and it is 0.29 for noble gases, nitrogen, oxygen, and propane. The value of ρcr/ρB (the ratio of the critical ρ to ρB, i.e., to the whole length of the interval along ρ, where the Zeno line “cuts off” the abscissa in formulas (24)) coincides with the corresponding values for water, argon, xenon, krypton, ethylene, and some other gases. MATHEMATICAL NOTES Vol. 89 No. 2 2011 274 MASLOV 3. DISTRIBUTION FOR AN NONIDEAL GAS First, let us describe the distribution for Z ≤ Zcr. The analog of the potential Ωγ for the number-theoretic distribution for γ =(d/2 − 1)/2,whered is the “fractal” noninteger dimension, is of the form ˆ π1+γ T 2+γ ∞ 1 µ 1 Ωid = ξ1+γ dξ, κ = ,T= . γ (ξ−κ) (25) Γ(2 + γ) 0 e − 1 T β If the value γ = γ0 corresponds to Zcr, then multiplication by a function of V occurs in our distribution, which corresponds to the following changes in the Bose–Einstein distribution: ϕ (V ) V → ϕ (V ), γ → const as V →∞. (26) γ V The constant was defined in [9] and is coordinated with the solution of Eqs. (13), (14). This function is constant for γ ≥ γ0,whereγ0 =0.2 is the critical dimension corresponding to Zcr =0.29. Therefore, Zmin obtained for µ =0is of the form Vϕ (V ) γ0 ζ(γ0 +2) Zmin = · =0.29, (27) ϕγ0 (V ) ζ(γ0 +1) where ζ is the Riemann function. For any Z<1,wehave ˆ ∞ ˆ ∞ − Vϕ (V ) Γ(γ +1) εγ0+1 dε εγ0 dε 1 Z = γ0 · 0 · ξ−κ − ξ−κ − ϕγ0 (V ) Γ(γ0 +2) 0 e 1 0 e 1 Vϕγ0 (V ) µ ∂ϕ = Ψ(κ),κ= ,ϕγ0 (V )= , (28) ϕγ0 (V ) T ∂V where Γ( · ) is the gamma function. For κ =0, we obtain (27). Further, we obtain µ(T,V ) as a function of V from the condition Z =1: Vϕ (V ) γ0 Ψ(κ)=1,κ= κ(V ). (29) ϕγ0 (V ) On the other hand, for Z =1, the following condition holds on the Zeno line: T P = ρBT 1 − , (30) TB which follows from its slope, or ρ P = TBρ 1 − , (31) ρB as well as ρ T = TB 1 − . (32) ρB Therefore, the functions P (T ), T (ρ),andP (ρ) are known: P (T ) is the Bachinskii parabola, T (ρ) is a straight line, and P (ρ) is a parabola. Let us find P from the Bose–Einstein distribution, where V is replaced by ϕγ0 (V ): ˆ ∞ ϕ (V )T γ0+2 εγ0+1 dε P = γ0 =1, (33) −κ ε − Γ(γ0 +2) 0 e e 1 ρ 1 ϕ (V )Li (y)= ,ρ= , γ0 γ0+2 γ +1 0 − γ0+1 V TB (1 ρ/ρB) Vϕ (V ) Li (y) γ0 · γ0+2 =1. ϕγ0 (V ) Liγ0+1(y) MATHEMATICAL NOTES Vol. 89 No. 2 2011 MATHEMATICAL SOLUTION OF THE GIBBS PARADOX 275 Example 2. The curve Tr =1for an ideal and a nonideal Bose gas in Z, P coordinates. The formula for an ideal gas is Z Li (a)=ζ(γ +2), P γ0+1 0 where a = a(P ) is determined from the equation Li (a(P )) γ0+2 = P. ζ(γ0 +2) The formula for a nonideal gas is Z f Li (a)=ζ(γ +2), P γ0+1 0 where a = a(Z, P) is determined from the equation Liγ0+2(a) P = . ϕγ0 (Vcr)ζ(γ0 +2) ϕγ0 (Z/P) The calculations by the last formula coincide with the experimental graph for Tr =1up to the point Z =0.29. The compressibility factor Z in this distribution is given by the following relation: P V Z = r , (34) Tr where Pr = P/Pcr and Tr = T/Tcr. Thus, the fractal dimension d is uniquely determined for Z = Zcr = Vcr. For γ<γ0, the equation for ϕγ (V ) is of the form ϕ (V )Liγ+2(y) ρ γ = , (35) P γ+1 − γ+1 cr TB (1 ρ/ρB) 1 Vϕ (V )Liγ+2(y) ρ = ,γ≤ 0.2, γ =1. V ϕγ (V )Liγ+1(y) The variable y can be eliminated from these equations. The initial condition for V = Vcr is equal to the value of ϕγ0 (Vcr) (for γ0 =0.2). For µ ≤ 0, the equation for γ(µ) [9] follows from (14) with the initial conditions on the Zeno line corresponding to (35). The curve T = const can be obtained in the Z, P coordinates if (γ+2)/(γ+1) κ ϕγ (V )T Liγ+2(e ) ZTr P = ,V= , (36) ϕγ0 (Vcr)ζ(γ +2) Pr where ζ(γ +2)=Liγ+2(1), Vcr =0.29,andγ = γ(µ) for γ<0.2 in view of the condition µ → min,and κ Vϕγ (V )Liγ+2(e ) ZTr Z = κ ,V= . (37) ϕγ (V )Liγ+1(e ) Pr 7 Here Tcr is chosen so that, for γ<γ0 and for T = Tcr, there is no spinodal point, while, for T 7A spinodal point is a point limiting the domain of instability. MATHEMATICAL NOTES Vol. 89 No. 2 2011 276 MASLOV Combining the isotherms T ACKNOWLEDGMENTS The author wishes to express deep gratitude to the corresponding member I. V. Melikhov, Professor V. S. Vorob’ev and Professor G. A. Martynov for valuable discussions. This work was supported by the Russian Foundation for Basic Research (grant no. 09-01-12063- ofi_m). REFERENCES 1 . V. V. K o z l o v, “Kinetics of a collisionless gas: temperature equalization, increase of rough entropy, and the Gibbs paradox,” Nelineinaya Dinamika 5 (3), 377–383 (2009). 2. Henri Poincare,´ About Science (Nauka, Moscow, 1983) [in Russian] (Translation from the French of Poincare’s´ books “Science et Hypothese,"` “Valeur de la Science," “Science et Methode,"´ and “Dernieres` Pensees").´ 3. A. M. Vershik, “Statistical mechanics of combinatorial partitions, and their limit shapes,” Funktsional. Anal. i Prilozhen. 30 (2), 19–39 (1996) [Functional Anal. Appl. 30 (2), 90–105 (1996)]. 4. 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