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Mathematical Solution of the Gibbs Paradox V 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<N for which the number of solutions of the Diophantine inequality ∞ ∞ Ni ≤ N, iNi = E i=0 i=0 is maximum (and hence more probable than any other number).
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