ISSN 0001-4346, Mathematical Notes, 2010, Vol. 88, No. 3, pp. 402–413. © Pleiades Publishing, Ltd., 2010.

Number Theory, Dimension Theory, and the Crisis of Overproduction* V. P. Maslov** Moscow State University Received April 21, 2010

Abstract—We consider the relationship between probability theory and the economics of assets. We show that the notion of Bose condensate can be applied in economics. DOI: 10.1134/S0001434610090130 Key words: Bose–Einstein distribution, Bose condensate, clusterization, degrees of freedom, Gibbs paradox.

Sciences are divided into two types—the humanities and the natural sciences... I must admit that I don’t believe in this division of the sciences. Ernst Mach, Popular Science Essays

1. THE BOSE CONDENSATE AS A PHENOMENON IN THE SOCIAL SCIENCES We have 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 [1]. The number of spectators in this example corresponds to the number of particles N , while the number of bills E, up to a certain unit of energy, corresponds to the energy. Korov’ev threw E bills to N spectators, and we must determine the probability of the event that a given number of spectators gets at least one bill. 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 Chapter 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" (p.116 of the Russian edition) 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. 1 “We must look for mathematical ideas," Poincare´ writes, “where they remain pure, i.e., in arithmetic." (loc.cit., p. 13). And that’s what we do. A famous problem in arithmetic (number theory) 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)

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], Chapter 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. ∗

The text was submitted by the author for the English version of the journal. E-mail: [email protected] 1 In that book, Poincare´ gives a detailed analysis of the Bertrand paradox. As I pointed out in my paper [3], my arguments related to fluids from that paper also lead to a similar paradox and, therefore, must be regarded as purely heuristic. In the paper [4], where the discrete case is considered, the Bertrand paradox is avoided.

**

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NUMBER THEORY, DIMENSION THEORY, AND THE CRISIS OF OVERPRODUCTION 403

Thus, we are returning to the most “pure" mathematical ideas. 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 = 1 and in the case N = E (whatever the value of N ) there is only one variant (each spectator gets one bill), 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. If N < Ncr , then the maximum probability that a spectator gets at least one bill will occur when all the N seats of the theater are occupied. For N > Ncr , the maximum number of variants will always be ˝ in the paper [5]. It equals Ncr . The number Ncr was computed by Erdos √ 1 E ln E(1 + o(1)). (2) Ncr =  2 2 π /6 Therefore it is probable that when the number of spectators is more than Ncr , some will get zero bills. Note that zero does not appear as a summand in partitions such as (1). Example. Assume that Korov’ev has E = 1 000 000 bills and there are N = 10000 spectators. According to number theory and to the relations indicated above, only 1 thousand spectators will get one bill or more. The other 9 thousand will get nothing, and we will then assume that they die of hunger. Then this situation is completely similar to the situation of Bose condensate in the statistical physics of bosons. In the physics literature, as a rule, it is explained that there is no Bose condensate in the twodimensional case. The physicist H. Temperley [6] noticed the relation between the Bose distribution and the number theory problem discussed above. However, his paper and other works, especially those that did not deal with the Bose condensate directly, were drowned in the loud choir of physics papers (and textbooks) that supported the belief that there is no Bose condensate in the two-dimensional case. The outstanding physicist Ya. I. Frenkel wrote: “We easily get used to the monotonous and unchanging, we stop noticing it. What we are used to seems natural to us, things we are not used to seem unnatural and non-understandable. Essentially, we are unable to understand, we can only get used to." ([7], p. 63). The continuation of the sequence (1) by zeros is in contradiction, on the other hand, with the universally accepted mathematical definition of p(N ), the number of partitions of N into positive summands, since we must exclude zeros from the partition in the initial problem. The leading expert I. R. Shafarevich was so amazed by the appearance of zeros in my approach that he said in an interview published in the book Alumni of Mekhmat Recall (Mekhmat MSU, 2009) that Maslov (along with N. N. Bogolyubov) is the most “abstruse" of all mathematicians. Often new unusual conceptions due to a scientist from some specific field are understood by experts in other fields much better than by their colleagues from the same field. Thus, the outstanding physicist V. L. Ginsburg, known for his errorless intuition, unexpectedly appreciated my application of the notion of Bose condensate to the economics of debts. When, in our conversation, I mentioned classical particles, he said: “Ask Pita" (meaning Pitaevskii). I did, but have received no answer (Pitaevskii was one of the experts who asserts 2 , that there is no Bose condensate in the two-dimensional case). An analog of the Bose condensate is taken into consideration in the study of debts. A debt crisis occurs when the debtor reaches bankruptcy. During the age of slavery, the bankrupt debtor would become a slave, and during feodalism, a serf; in Russia, such a debtor could become a krepostnoy (a specific form of serfdom). Thus, bankrupty was the cause of the phase transfer from one type of society to another.3 2 3

See the last edition of the book [8]. Enslavement as the result of debts was not unusual, especially in the Russian krepostnoy society, where a free peasant unable to return his debts within a year would become a krepostnoy (enslaved peasant or servant). The passage from a society of free peasants dominated by the boyars to the society dominated by the serf-owning landed nobility is explained by the historian Klyuchevsky as the enslavement of peasants not by degree from above, but as the result of economic changes related to the crisis at the beginning of the 17th century (the so-called smuta—troubled times). If one is to single out the economic historical theory nearest to our mathematical conception, it should be the theory of commercial capitalism of Academician M. N. Pokrovsky (which was denounced by the ideologues of the Communist Party in the 1920ies).

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The last world economic crisis, as the author showed in [9], was related to the huge internal debts of the USA. The general global asymptotic distribution, which yields, in particular, formula (2), was first obtained by the author. It shows how natural numbers are distributed in the partition of the number E into N summands. The formula for this distribution appears in the papers [3], [1]. This formula holds only for very large values of E and N , and only in the weak probabilistic sense. The initial well-known identity for the above problem has the following form: ∞ 

∞ 

iNi = E.

(3)

These relations correspond to physical relations of the form   εi Ni = E, Ni = N,

(4)

Ni = N,

i=0

i=0

where Ni is the number of particles on the ith energy level, while the εi are discrete collections of energy. If the particular nonrelativistic case is considered, in which the Hamiltonian equals P 2 /2m, where p is the momentum, then the two-dimensional case corresponds to problem (3). The number-theoretic problem corresponding to the three-dimensional case is as follows:  (i + 2)!  Ni = E. (5) Nj = N, i!6 Here the Bose condensate exists for all values of the topological dimensions greater than zero. However, the higher is the dimension, the worse is the estimate. Estimates of the probability of the Bose condensate appear in the paper [1]. ¨ In his famous popular philosophical book [10], Schrodinger notes that √ the accuracy of statistical laws as N → ∞, where N is the number of particles, is no better than O( N ). In this connection, we state ¨ the following slightly modified conjecture concerning Schrodinger’s law formulated by him as the “law of nature.” If the number of particles, molecules and genes,√in a chromosome is N , then one can obtain statistical relations whose accuracy is no better than O( N ln N ). In the mathematical sense, this means that it is impossible to refine the estimate given in [11, formula (7)]. ¨ According to Schrodinger, it is also impossible to refine estimates in distribution theorems of number theory. However, there is a positive moment in such a rough estimate. The following remark shows how to transfer the results to fractional dimensions. This is the most important generalization for the subsequent theory. Instead of the number of variants (solutions) of relations (3) and (5), it is natural to consider their logarithm to base 2 (Hartley’s entropy). Then it turns out that the logarithm of the number of solutions to the problem (3) and (5) and to the problem ∞ 



Ni = N,

i=0

∞ 

iNi ≤ E,

i=0

 (i + 2)!

Ni ≤ E (6) i!6 coincides in absolute value with the given accuracy. In other√ words, in the theory we are constructing, the difference of 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 → ∞. This consideration makes it possible to generalize the given number theory to noninteger dimensions. Consider expressions of the form  Γ(d + i) Ni ≤ E, (7) Γ(i + 1)Γ(d + 1)  (8) Nj = N, Nj = N,

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and the number of solutions satisfying inequality (7) and equality (8) for noninteger D. Only for N = 1 and N = E do indistinguishable variants of the type (4 + 1, 1 + 4) coincide with distinguishable variants, such as those in Boltzmann statistics, Shannon’s information theory, Kolmogorov complexity theory, and Gibbs ensemble. √ But since, as previously mentioned, the accuracy of our formulas is no better than N , it follows √ √ that, at best, E and E − N coincide for N ≈ N and N . It would be very useful to write computer programs to estimate these intervals in percentage terms. This could be used in various sciences, where the numbers N and E are not so large as in the classical statistical physics, such as in mesoscopic physics, nanophysics, genetics, and biology. The distribution associated with this number-theoretic problem for the dimension do > 2 corresponds to the Bose–Einstein distribution for N ≤ Ncr . Namely, let us find the following constants “conjugate” to N and E: ∞  (j + 1)(j + 2) , (9) N= β(j−μ) − 1) 2(e j=1 where β is the solution of the equation E=

∞  j(j + 1)(j + 2) j=1

2(eβ(j−μ) − 1)

.

(10)

Then the distribution takes the form of the Bose–Einstein distribution [12], in which V = 1. In the general case, for d > 2, E=

∞  j=1

jqj β(j−μ) e

−1

,

(11)

.

(12)

where qj =

Γ(i + d) Γ(d)Γ(i + 1)

and N=

∞  j=1

qj β(j−μ) e

−1

The distribution has the form Nj ∼

qj β(j−μ) e

−1

.

(13)

√ Here the estimate for d0 ≥ 2 (see [13]) corresponds to N ln N . For d ≤ 1, an additional term, which is small for d0 > 1, should be added. In (11)–(13), this term is of the form N qi . (14) − βN (j−μ) e −1 Using this additional term, we can obtain Ncr for d0 > 0, i.e., for any dimension greater than zero. The parameter μ in the distributions (12)–(14) is called differentia in number theory, because, as μ → ∞ the passage to Boltzmann statistics and Shannon’s entropy occurs. The parameter T = 1/β is called movability. For dimensions greater than 4, the next term of the asymptotics can be computed, and this leads to a correction to the Stefan–Boltzmann law [10]. In the one-dimensional case (i.e., on the real line) Ncr can be computed according to the formula  2  W2 4 1+ 1− , (15) Ncr = 4 W MATHEMATICAL NOTES Vol. 88

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where  W = (2n)1/3 0

∞

√ −1/3 ∞  1 ξ dξ 1 dξ > 0. − eξ − 1 ξ 2 eξ 2 − 1

(16)

0

This is important for nanotubes in physics and for the so-called Waring problem in number theory . 2. CLUSTERIZATION AND DECREASE OF THE DEGREES OF FREEDOM AS AN ALTERNATIVE TO THE BOSE CONDENSATE AND THE SOLUTION OF THE GIBBS PARADOX The truth did not bring as much good to the world, as the semblance of truth brought evil to it. Larochefoucault.

Classical particles, when we study their density, i.e., the number of particles in the unit volume, lose their individuality, just as the bills in the scene from Bulgakov’s novel analysed above, and therefore obey the same statistics from number theory. Their interchange of positions does not affect the density. In this sense, the Gibbs paradox, related to entropy, disappears. The great Poincare´ himself tried to solve this problem: in the model of ideal gas that he proposed, an increase of entropy occurs if one removes the barrier separating the gas into two subvolumes [11]. Since he did not find a mathematical solution, he argued in the framework of philosophical logic. Throughout his whole book [2], Poincare´ attempted to bring together two incompatible physical theories, in particular, to explain the Gibbs paradox. ´ philosophy, shows that it is especially constructed so as to justify this A careful analysis of Poincare’s contradiction. 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 4 ." This argument, which reminds one of the reasoning of some mixed-up Bible commentator, would be ridiculous if physical theories were regarded from the point of view of nonspecialists (les profanes in French). From that point of view, at least one of the theories must be wrong in the situation considered by Poincare.´ But this is not necessarily so if one starts looking for something other than what one is supposed look for in the theories. It may turn out that “both theories express real relations, while the contradiction lies only in the symbols with which we have attired them." (loc.cite, p. 104) In politics and in ideology, such contradictions may be hammered into people’s heads, 5 as Frenkel asserts (see his remark from [7] cited above), but in mathematical physics one of two theories that contradict each other must be declared wrong, and no explanation by virtual symbols, whether it comes from a profane like the present author or a genius like Poincare,´ can change this state of affairs. And we must honestly admit that the model of gas in a parallelopipedon bounded by reflecting walls proposed ´ point of view on the subject did not allow him to get rid of the notion by Poincare´ is wrong. Poincare’s of ether, which contradicts Einstein’s relativity theory [14]. In his article [15], submitted for publication a month later than the above-mentioned paper by Einstein, he writes: “If the dispersion of gravitational forces occurs with the speed of light, this cannot be the result of some random circumstances, but must be caused by one of the functions of ether; then the problem of obtaining a deeper understanding of this function arises and of relating it to the other functions of ether." (Poincare,´ loc.cit., p.55). The numbers E and N are large; however, if E is relatively small, then, in the scheme that we √ √ considered above, either part of the spectators Ndead = N − const E ln E when N E, or they get together into clusters, for example, into groups of 10 people in the Example, and then each group divides among its members the bills it has acquired so that all members of the group survive. 4

´ philosophy doesn’t hold water. Poincare´ himself saw the Lenin [12] didn’t know any physics, but felt that Poincare’s obvious contradiction in Gibbs’ approach to thermodynamics and had the brilliant idea that it is related to a contradiction in probability theory. And he pointed out that the way out is via arithmetic. At the end of his life, Kolmogorov understood that the axiomatization of probability theory does not save the day, and used the idea of encoding to define the notion of ´ book not being translated into Russian. Incidentally, complexity. Unfortunately, Lenin’s criticism resulted in Poincare’s Lenin’s even more gross criticism of P. P. Maslov had serious repercussions on the author’s life. 5 Thus, in the heads of contemporary Russians, A. Sakharov and A. Solzhenitsyn are undoubted authorities, although their conceptions are opposite and exclude each other; see [13]. MATHEMATICAL NOTES Vol. 88 No. 3 2010

NUMBER THEORY, DIMENSION THEORY, AND THE CRISIS OF OVERPRODUCTION 407

Here the groups (clusters) must be divided into two classes—those whose members were sitting near each other and decided to form a group, establishing “holonomic relations," and those who noticed each other from far away (e.g. spectators of different gender) and liked each other (say, because of a contrasting color of hair), formed a couple (a dimer), and then decided to spend the bills obtained by one of them together. Such Cooper pairs appear in the theory of superconductivity—when the particles have opposite spin and momentum. The following real-life question arises: How stable are such groups? At this point, we must digress and describe the well-known model of a human being devised by a mathematician (Norbert Wiener) and a physicist (Yakov Frenkel). Wiener wrote that the statue of a human being is unstable—it falls if it is not placed on a pedestal. A live human being stands stably without any pedestal, because, as Frenkel puts it, he controls himself like a juggler controlling a vertical pole. The position is unstable, but sufficiently metastable. 6 Only, in old age, do humans start using a cane when they walk. The stable state is death. In just the same way, in our economic example, in order to survive, people must unite and juggle with the unstable system, bringing it to a metastable state. Applying the Bose–Einstein distribution to classical particles, and, in the case of dimension less than or equal to 1, this distribution as modified by the author [1], [3], we replace the Bose condensate by the grouping together of particles into clusters. Then “no one dies," i.e., the number of spectators (particles) does not change. This fact considerably simplifies the distribution. While the Bose distribution was regarded as purely quantum, it was customary to introduce the socalled exchange interaction between formally noninteracting Hamiltonians. But since we regard the Bose distribution as a distribution of classical particles (this is one of the main points that physicists can’t get used to), it is obvious that, for two Hamiltonians whose Poisson bracket is zero, the Bose distributions are not multiplied. This means that the additivity of energy and its various generalizations: white noise, Wiener process, and their combinations excessively narrow probability theory. It is precisely on the basis of that misunderstanding (it should be replaced, as explained in [16]), that the Gibbs–Maxwell–Boltzmann distribution appeared. Gibbs, in order to avoid the paradox, introduced the notion now called the Gibbs ensemble. Note that as soon as the value Ncr , for which the Bose condensate appears according to the old Bose–Einstein theory, is reached, the new theory of the author asserts that a pair (a dimer) appears. Besides, the more particles (or spectators) unite, the greater is the decrease of the number of degrees of freedom. 3. NUMBER-THEORETIC DISTRIBUTION WITHOUT THE BOSE CONDENSATE For d1 > 2, we define the dimension greater than d0 from the condition Ncr (d) = N.

(17)

ς(d) E d/(d+1) = N, (dς(d + 1))d/(d+1)

(18)

More precisely,

where d = D/2, D is the dimension in the Bose–Einstein physical equations, i.e., the three-dimensional case D = 3, and ς(x) is the Euler zeta function. Here Nmax = Ncr (d1 ), i.e., for N > Ncr , instead of the Bose condensate, we have a decrease in the dimension d < d0 according to the law (17) in the Bose– Einstein distribution. 6

Cyclones and anticyclones can be decomposed into rotational and potential components. In complicated atmospheric flows, the rotational component maintains the metastable equilibrium just like the juggler controlling a vertical pole. When an anticyclone generates a heat wave and, as a result, forest fires (especially underground peat bog fires), the surface of the earth heats up and a secondary convective vertical flow appears. If the rotational component is strong enough, then it can keep the anticyclone in place, just as a human being can lean on a telephone pole to stay effortlessly in vertical position without moving. The law of economy of energy works in both cases. Of all possible states, Nature chooses the most “energetically advantageous" one, which corresponds to the maximum number of variants, i.e., to maximum entropy.

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However, for N > Nmax , in general, the Bose condensate must appear. As we shall see below, there is no such phenomenon in the theory of classical particles. Returning to Korov’ev’s trick, we see that after an increase in the number of particles of the form N > 2Ncr , pairing is not sufficient for the Bose condensate to appear—trimers and clusters of many particles. This means that Boltzmann statistics and Shannon’s entropy play a significant role here, because dimers, monomers, and trimers are distinguishable from one another. The combinatorial aspect of this process is sufficiently simple; hence the exact relationship of these two statistics can be calculated on a computer. The presence of Nmax as μ → ∞ means the “complete” victory of Shannon’s entropy. We obtain a noble gas out of glassy dust when most of it particles are distinguishable from one another.7 . Namely, in noble gases, there are no preferences for dimers, trimers, etc. Therefore, the thermodynamics of noble gases models well the mixture of Bose–Einstein statistics and Boltzmann statistics. Here there is a remarkably simple law of statistics mixture. However, “preferences” may appear in the case of more complex molecules. In that case, such a mixture depends significantly on another parameter, namely, Nmax as μ → ∞. This law is called the Zeno line. In a particular case, it was first discovered by the Russian scientist Bachinskii. This makes it possible to get rid of E in formula (18) and find the function d(N, d0 , d1 ). 4. ON BASIC QUANTITIES IN DIMENSION THEORY. TEMPERATURE AS AN STABLE EQUILIBRIUM POINT IN THE ISOTROPIC THEORY OF PARTICLE COLLISIONS IN A MEDIUM WITH WEAK VISCOSITY The question of basic quantities in physics, believed Rayleigh, concerns logic and philosophy, and he pragmatically considered temperature as an independent quantity. D. Ryabushinskii (a member of a family of staroobryadsy-dvoedany8 ) objected to Rayleigh, saying [17] that the basic quantity is the mean kinetic energy of molecules determined, in turn, by the dimensions of length, mass, and time. Expressing the stable value of energy in the scattering problem in terms of the temperature [4], we side with Ryabushinskii on this question. This is one of the key points of the conception developed by the author. It follows from Sec. 2 that the phase transition to the Bose condensate in our approach is replaced by the formation of the smallest cluster—the pair of particles or dimer. Suppose that, in a gas, starting at a sufficiently large distance from each other, a pair of particles with initial energy equal to the squared difference of their initial velocities divided by 2m fly towards each other. Let us denote their interaction potential by Φ(r1 − r2 ). In the case where there is no viscosity, they collide and fly away from each other according to the standard laws of scattering theory. But, in our case, weak viscosity is present: it is caused by the repulsion of the colliding particles, just as this happens for the Boltzmann equation, generalized by Bogolyubov. If there is an attraction component in the interaction, then it is this component that causes the formation of a dimer. In order to include the formation of dimers in the framework of scattering theory, one must relate the initial data of the given particles to the rotation of pairs of particles around each other as their center of gravity moves along an ellipse. First of all, we will consider, as is customary in molecular physics, the isotropy of the gas: how many particles are flying towards each other. The usual argument used in molecular physics consists in assuming the symmetry of the averaged motion in all six coordinate directions in 3-space. Therefore 1/12th part of the particles move towards each other. There are three such axes, so that 1/4th of the particles collide. 7 8

The hydrodynamics of such a dust was studied in [29]. The dvoedany were citizens who had to pay a double tax because of their religious beliefs; they were excommunicated for being the strongest opponents to the official church. (Only the staroobryadsy were dvoedany.) Despite all the difficulties that the establishment church created, the Ryabushinskii family, very pragmatical in its business dealings, prospered. MATHEMATICAL NOTES Vol. 88 No. 3 2010

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For the initial moving particles, we specify the energy E and the aiming parameter B. The momentum M , as well as the energy E, is preserved. It is known that M 2 = B 2 E.

(19)

Expressing the energy E in the case of attraction, we obtain E=

(mv 2 )/2 + Φ(r) 1 − B 2 /r 2

(20)

in the domain where r ≤ B. The dressed or “thermal" potential Ψ(r) is attractive. Besides, since the volume V is a large parameter, if we decompose the relation  2 ar , Ψ(r) = Ψ V where a is the effective radius, in powers of 1/V , we obtain   2 1 C2 ar 2 ar = C1 + +O . Ψ V V V2

(21)

Writing (r1 − r2 )2 (r1 + r2 )2 + , (22) 2 2 we can, as was done in [19], separate the variables in the two-particle problem, reducing it to the scattering problem for two particles and the problem of their joint motion for r1 + r2 . Then, to the Lennard-Jones interaction potential in the scattering problem, we must add an attracting quadratic potential (an upside-down parabola). The Kolmogorov spectrum was obtained by that great mathematician from considerations of isotropy and dimension. We use dimensional considerations in thermodynamics as well in our study of the scattering problem and relate the stable values of energy with temperature. In the scattering problem thus obtained, there are, as a rule, two equilibrium points: the stable one Emin and the unstable one Emax . PV Their ratio is a dimensionless quantity. In thermodynamics, the dimensionless quantity is Z = N T, where P is the pressure, N is the number of particles, T is the temperature, and since the stable equilibrium point by its meaning corresponds to the temperature, the relation r 2 = r12 + r22 =

Z=

Emax − Emin PV = NT Emax

allows us to construct, on the graph T , ρ = N V , the curves Z = const. The curve Z = 1 is known as the Zeno line, while the locus of the extremity of Z = const as B → ∞ is bimodal. The dimension theory that we use here is undoubtedly related to similitude, and therefore to group theory. However, as Kolmogorov (who applied a similar approach to the chaos of turbulent pulsations) pointed out, a rigorous (in the sense of mathematical logic) approach to the problem is impossible. Nevertheless, this approach is more rigorous than various heuristic methods of closing up the Reynolds equations, in the sense that it is the consequence of simple and uniform postulates [18]. In the same way, the virial decompositions in Bogolyubov chains or in the theory of Gibbs ensembles, the various ways of closing up the Orenstein–Cernike equations in accordance with some experiment or other, the refutation by the authors of [20] of the conjecture of Marc Katz, show that there is no road leading to a rigorous mathematical foundation of these methods. In mathematical experiments of molecular dynamics based on the Monte-Carlo method, powerful computers are used; expensive American software is applied in two week-long computational sessions. The difficulties related to fluctuations at critical points do not allow one to obtain sufficiently precise answers, whereas calculations on the basis of explicit formulas, obtained from considerations of MATHEMATICAL NOTES Vol. 88

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dimension, can be performed for a given interaction potential by an ordinary laptop in less than an hour. This allows one to set the inverse problem: to specify the potential corresponding to the given gas or mixture of gases, in particular, the problem of finding the class of potentials corresponding to experimental values of the slope of the Zeno line (see the second plot in the paper [20]). 5. NEW DISTRIBUTION FOR CLASSICAL PARTICLES For dimensions 3 or more, the distribution from number theory is related to the distribution known in quantum physics as the Bose–Einstein distribution. The difference is only in that, for these dimensions, the number theory distribution is multiplied by another important quantity—the volume V . The derivative with respect to V corresponds, in thermodynamics, to pressure, which obviously no longer has a direct dependence on V , and therefore coincides exactly with the distribution in number theory. At this point, the Bose–Einstein distribution has another “puncture." It is apparently related to the fact that in the limit, for a very large chemical potential, an ideal gas must appear. This is true, of course, but, for an ideal gas, not only μ → ∞, but the pressure is small. Therefore, this puncture in the Bose– Einstein distribution must also be repaired, and this we did in the papers [23], [24]. Suppose that Vmax is the volume for which the number-theoretic entropy generalizing the Bose– Einstein entropy coincides, to a prescribed accuracy, with Shannon’s entropy. Set  Vmax  2 Vmax − x2 dx + Vmax . (23) ϕ(V ) = V

It is this function that we use for multiplying the Bose distribution instead of multiplying by V . Using the formulas for the Zeno lines Z = 1 and Z = 1 − ε, where ε → 0, one can find the intersection angles of the isochores with the Zeno line. Using this together with the alternatives of the Bose–Einstein condensate in the form of decreasing degrees of freedom, one can uniquely and precisely determine the function ϕ(V ) by which one must multiply the number-theoretical distribution in order to obtain the distribution of classical particles. The new distribution in momenta and coordinates depends on temperature and density as parameters. The Zeno line 2 corresponds

to a focal point on the Lagrangian manifold. In order to determine this point, the following precision o V1 suffices. This means that the Zeno line 2 is determined by two terms of the thermal (dressed, screened) potential with respect to the small parameter V1 . Therefore, the dependence of the distribution in momenta of particles with values on the twodimensional Lagrangian manifold, corresponding to the thermodynamical parameters T, P, S, ρ, is

2 determined by only two terms of the viral decomposition of the thermal potential Ψ arV . The dressed potential, as we know, appears as an additive extra term in the chemical potential μ. Hence the final distribution in momenta and coordinates is given by the formula presented in the paper [23], in which one must must perform the following change of notation  2 ar , μ=μ +Ψ V where μ is the new chemical potential. Thus, our distribution depends on a classical Hamiltonian of the form  2 ar p2 +Ψ . H= 2m V

(24)

If H = H1 + H2 and {H1 , H2 } = 0 ({·, ·} is the Poisson bracket), then the distributions for H = H1 and H = H2 correspond to two independent events. In contrast to the quantum case, the so-called “exchange interaction" does not take place here. The distribution corresponding to H = H1 + H2 is not equal to the product of the distributions corresponding to H1 and H2 . Therefore, the additivity condition for the energies of noninteracting systems fails here. MATHEMATICAL NOTES Vol. 88 No. 3 2010

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6. RELATIONSHIP WITH ECONOMICS In the theory of assets in economics, a condensate of a new kind arises. This happens when the spectators unite into groups (first in pairs, then in large clusters), as the result of which each group (cluster) acquires at least one bill. In the first case, we shall say, for expressiveness, that N − Ncr spectators die of hunger (in the ordinary Bose condensate, they collapse to the lowest energy level, i.e., they cannot move or give any contribution to the total energy), while, in the second case, everyone survives, and so the number of spectators (the number of particles in the statistical physics setting) does not change. This new condensate is mainly related to the appearance of pairs, i.e., dimers. Irving Fisher (a pupil of the greatest specialist in thermodynamics, Gibbs) noticed the remarkable correspondence between the volume V in thermodynamics and the volume Q of goods in economics. If the chemical potential in the Bose–Einstein distribution is large, then it becomes the Maxwell distribution, and this leads to the main equation for ideal gases (25)

P V = N T.

The correspondence V → Q, P V is the price of goods, N → M , M is the total amount of money, T → v, v is the rate of turnover, leads us to the main law of economics (the Irving Fisher law): or

PQ = Mv

v=

PQ . M

(26)

As applied to pre-revolutionary Russia, this relation adequately reflects the economic situation up to 1910, and the rate of turnover v oscillates slightly around a monotone increasing quantity. Here P never appears separately from Q and together P Q (the analog of energy) is an indivisible quantity. However, formula (15) does not take into account the possible occurrence of a crisis—the social crisis of overproduction. The rate of turnover is, naturally, bounded by a constant, say, v0 . Definition. We shall say that, for Q ≥ Q0 , an overproduction crisis occurs for the maximum rate of turnover v = vmax . If we take this into account, we must modify the Fisher law slightly: or

P Qnew = M v

Qnew =

Q . 1 − v/vcr

(27)

For a sufficiently large vmax , this becomes Fisher’s law. It is necessary, however, to take into account some more delicate considerations that we have mentioned previously: the fact that the bills are undistinguishable [1]. Although debts are always intertwined with assets [9], in this article, we only considered assets. Here the condensate, as we have seen in the examples from the Korov’ev interlude, leads to the formation of clusters, thus reducing the dimension. We have established the relationship between the dimension and the total volume of goods Q on the basis of the number-theory distribution, i.e., according to the principle of maximum number of variants (or the choice of the most probable state). Since the binary logarithm of the number of variants is the entropy, we construct the dependence of P Q on the rate of turnover v, using the principle of maximum entropy. We have patched up the “hole" in the Bose–Einstein distribution by using Ryabushinskii’s objection to Rayleigh. It is possible that Ryabushinskii, brought up in a family of pragmatic starovery-dvoedany (incidentally, so was the author), intuitively felt the connections with economics. In contrast with Adam Smith’s equilibrium model of the economy, here we are saying that equilibrium means death to the economy (i.e., leads to a primitive barter economy), while its survival is the metastable state of the juggler balancing a vertical pole on his nose or chin. Let i be the ith region, and let vi be the rate of turnover of goods in the ith region. Let vi < vi+1 , i = 1, 2, . . . , k, k 1. MATHEMATICAL NOTES Vol. 88

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MASLOV

The money Mi is distributed in a natural way. Thus, k 



Mi = M,

(28)

vi Mi = P Q.

(29)

Suppose that the critical dimension of problem (28)–(29) is d0 . Then, as the crisis of overproduction is approached, the distribution Mi is of the form similar to the distribution (32) from [27]. It seems that, for v = vcr , the crisis of overproduction is just around the corner. How to vary slightly the volume of goods Q and the prices P so that the crisis “goes to infinity” and it will be hard to catch up with it just as, according to Zeno, for Achilles to catch up with the tortoise. According to [27], [28], a slight change in the “prices–volume of goods” relation yields Zeno line 2, i.e., the situation in which it is impossible to catch up with the crisis no matter how near to it we may be. (see graph 5 in [28]). This is the same transformation that is performed in the distribution (32) from [27], where ϕ(V ) was defined in (23) for the case in which, instead of the volume V and the pressure P , the volume of goods Q and the prices P are considered according to the correspondence principle. ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Research (grant no. 08-01-00601). REFERENCES 1. V. P. Maslov, Threshold Levels in Economics, arXiv:0903.4783v2 [q-fin.ST], 3 Apr 2009. ´ About Science (Nauka Publ., Moscow, 1983) [in Russian].(Translation from the French 2. Henri Poincare, ´ books “Science et Hypothese," ` ´ ` of Poincare’s “Valeur de la Science," “Science et Methode," “Dernieres ´ Pensees") 3. V. P. Maslov, “Dequantization, statistical mechanics, and econophysics,” in Contemporary Mathematics, (Amer. Math. Soc., Providence, RI, 2009), Vol. 495, pp. 239–279. 4. V. P. Maslov, “Zeno–line, Binodal, T –ρ Diagram and Clusters as a new Bose-Condensate Based on New Global Distributions in Number Theory,” arXiv 1007.4182v1 [math-ph] 23 July 2010. ˝ “On some asymptotic formulas in the theory of partitions,” Bull. Amer. Math. Soc. 52, 185–188 5. P. Erdos, (1946). 6. H. Temperley, “Statistical mechanics and the partition of numbers.I. The transition of liquid helium,” Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 199 (1058), 361–375 (1949), http://www.jstor.org/pss/98351. 7. B. Ya. Frenkel, Yakov Il’ich Frenkel (Nauka Publ., Moscow–Leningrad, 1966) [in Russian]. 8. E. M. Lifshits and L. P. Pitaevskii, Statistical Physics, Part 2: Theory of Condensed State, (Fizmatlit, Moscow 2004) [in Russian]. 9. V. P. Maslov, Economic law of increase of Kolmogorov complexity. Transition from financial crisis 2008 to the zero-order phase transition (social explosion), arXiv:0812.4737 29 Dec 2008 ¨ 10. E. Schrodinger, What is Life? The Physical Aspect of the Living Cell (Cambridge Univ. Press, Cambridge, 1944; Atomizdat, Moscow, 1972). 11. V. P. Maslov, “Quasithermodynamics and a correction to the Stefan–Boltzmann law,” Mat. Zametki 83 (1), 77–85 (2008) [Math. Notes 83 (1–2) 72–79 (2008)]. 12. L. D. Landau and E. M. Lifshits, Theoretical Physics, Vol. 5: Statistical Physics (Nauka, Moscow, 1964) [in Russian]. 13. V. P. Maslov and V. E. Nazaikinskii, “On the distribution of integer random variables related by a certain linear inequality: I,” Mat. Zametki 83 (2), 232–263 (2008) [Math. Notes 83 (2), 211–237 (2008)]. 14. V. P. Maslov, “Quasithermodynamic correction to the Stefan–Boltzmann law,” Teoret. Mat. Fiz. 154 (1), 207–208 (2008) [Theoret. and Math. Phys. 154 (1), 175–176 (2008)]. 15. V. V. Kozlov, Thermal Equilibrium according to Gibbs and Poincare, ´ in Contemporary Mathematics (Institute of Computer Studies, Moscow–Izhevsk, 2002) [in Russian]. 16. V. I. Lenin, Materialism and Empiriocriticism (Politizdat, Moscow, 1984) [in Russian]. 17. V.P.Maslov, Solzhenitsin and the February Revolution, in In Defense of Science, Bull. 3 (Nauka, Moscow, 2008), pp. 125–130 [in Russian]. ¨ 18. A.Einstein, “Zur Elektrodynamik bewegter Korper,” Ann. d. Phys. 17, 891, (1905). ´ 19. H.Poincar ´e, “Sur la dynamique de l’electron,” Rendiconti del Circolo Matematico di Palermo XXI, 129, (1906). MATHEMATICAL NOTES Vol. 88 No. 3 2010

NUMBER THEORY, DIMENSION THEORY, AND THE CRISIS OF OVERPRODUCTION 413 20. V. P. Maslov, “Thermodynamics of fluids: Qualitative study,” Teoret. Mat. Fiz. 161 (2), 224–242 (2009) [Theoret. and Math. Phys. 161 (2), 1513–1528 (2002)]. 21. J. Rayleigh, “The principle of similitude,” Nature 95, 66–68 (1915); D. Riabouchinsky, “Letters to Editor,” Nature 95, 591 (1915); J. Rayleigh, “Letters to Editor,” Nature 95, 644 (1915). 22. W. Frish, Turbulence: A. N. Kolmogorov’s Heritage (Fazis, Moscow, 1998) [in Russian]. 23. V. P. Maslov, “Thermodynamics of fluids: The law of redestribution of energy, two-dimensional condensate, and T-mapping,” Teoret. Mat. Fiz. 161 (3), 422–456 (2009) [Theoret. and Math. Phys. 161 (3), 1681–1713 (2002)]. 24. V. P. Maslov and O. Yu. Shvedov, The Complex Germ Method in Many-Particle Problems and in Quantum Field Theory (Editorial URSS, Moscow, 2000) [in Russian]. 25. E. M. Apfelbaum and V. S. Vorob’ev, “Correspondence between the critical and the Zeno-Line parameters for classical and quantum liquids,” J. Phys. Chem. B 113, 3521–3526 (2009). 26. V. P. Maslov, “Correspondence principle between T-ρ diagrams and interaction potentials and a distribution of Bose–Einstein type,” Math. Notes 88 (1–2), 57–66 (2010). 27. V. P. Maslov, “Solution of the Gibbs Paradox using the Notion of Entropy as a Function of the Fractal Dimension,” Russian J. Math. Phys. 17 (3), 251–261 (2010). 28. V. P. Maslov, “Thermodynamic equations of state with three defining constants,” Math. Notes, 87 (5), 728737 (2010). 29. V. P. Maslov, Quantization of Thermodynamics and Ultrasecond Quantization (Inst. Komp’yuternykh Issledovanii, Moscow, 2001) [in Russian].

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