COMMENTS TO K. MARTINÁS' #1 LECTURE

on 27th April 2001

held at RMKI CRIP, Budapest, Hungary

written by B. Lukács

regular member of the Research Group of the Hungarian

OTKA T/029542

"Entropic Constraints for Possible Futures"

with the obligatory reverence

To My Honourable Team Leader Martinás

(together with my protests)

If one is surprised about the Baroque title, or does not understand the reason of these notes, I cannot help. The situation is nontrivial and irrelevant for outsiders. It is enough to state that the team leader organised a boycott against me in the team. Definitely: she wrote to all members they must decide if they would work with me or with her. Then the other members do not answer my letters and e-mails, and the team leader stopped discussions with me about the topics. However now the April 27 lecture was made at my Institute (for some purpose or other), so I attended to the lecture.

The team leader stated some new results of her. I do not question that they are her results. However in some cases I have earlier results, similar or related. (This is parallel evolution; our thermodynamic background is rather similar, and, as everybody can see, we have a lot of common publications.) In other cases I doubt the result. Of course, the simplest way would be to discuss the points with her; but because of the boycott technical problems would arise.

And anyway, I have to demonstrate that I am still working in the aforementioned topics and that I want to participate in the work of the financially supported group, I made the following Comments to the lecture of the team leader. Now comes the proper presentation. Also, look another site of me http://www.rmki.kfki.hu/~lukacs/reindeer.html.

ON ECONOMIC THERMODYNAMICS: DOES UTILITY FUNCTION MEAN FREE ENTERPRISE?

B. Lukács

"Entropic Constraints for Possible Futures", OTKA T/029542

lukacs@rmki.kfki.hu

mail: B. Lukács, KFKI RMKI, H-1525 Bp. 114, Pf. 49, Budapest, Hungary

ABSTRACT

As the refrain of a duet of the famous musical "Gunner Annie" tells: "Anything you can do I can do better...". This is the morale of my comments.

1. THERMODYNAMICS: THE AXIOMATIC FORMALISM

This is not a study about history of science. I simply suggest some good book. For the subsequent argumentation the best is Landsberg [1]. Axiomatic thermodynamics first states that it is a macroscopic or average theory. There are micro processes behind, the number of degrees of freedom is enormous, but most average away. A moderate number must remain, e.g. for conservation laws &c., and they are used as macroscopic variables.

The set of these variables must be defined at the beginning. The way of selecting the proper set is given in [1]. For simplicity's sake we start with the *extensive* set X^{i}. The extensives are some integrals or sums over the whole system, as total particle number, volume, energy and so on. Uniting 2 subsystems *in* *equilibrium* *with* *each* *other* the extensives are additive; but of course "equilibrium" is not always a trivial notion. However this additivity is not too sensitive on the definitions.

Now we need a thermodynamic potential. Avoiding lengthy logical excursions let us tell that it is an extensive function of the extensives, P(X^{i}). At this moment let P remain still quite general #2, but homogeneous linear in X's:

(1) P = X^{r}P,_{r}

where, according to the Einstein convention, there is automatic summation for indices occurring twice, above and below #3.

Now let us form the intensive set Y_{i}, canonically conjugate to Xi (see also: symplectic structures). Having selected a P(X), it yields the intensive set as

(2) Y_{i} = P,_{i}

It is easy to see that Y's are homogeneous of zeroth order. For simplicity's sake, take a system in equilibrium, divide it virtually into N equal parts and then reunite. Extensives change as

(3) X -> NX

From eq. (1) P changes as

(4) P -> NP

Then Y's change as

(5) Y -> Y

Hence the set {Y_{i}} does not feel the *extension* of the system.

I stop here with axioms; that is not my goal. However: how to get the *proper* potential?

There is *equilibrium*, at least in limiting sense. In equilibrium Y's do not change in uniting systems (above). Out of equilibrium the systems change, currents flow, &c., when uniting them, and after some changes equilibrium is formed. Then we may require that Y's be equal for different systems, with different functions P_{(a)}, in equilibrium. We can regulate the states via X's (which, we assume, can be measured somehow), and observe what different states of different systems are in equilibrium:

P_{(a)}, P_{(b)}

Y(a)_{i}(X)=P(a),_{i}(X), Y(b)_{i}(X)=P(b),_{i}(X)

Y(a)_{i}(X(a))=Y(b)_{i}(X(b))

and then, hence P(a)(X(a)) and P(b)(X(b)) are measured simultaneously.

Surprisingly enough, three different systems are almost enough to realise the above program, and further systems are no help except for diminish the measurement errors, to check, &c. [5]. Some freedom of convention always remain in P [6]. But that is not a problem if we are careful [7], [8].

Now we must start somewhere. There are 2 conventions at the beginning: the "Urpotenzial" may be energy or entropy. For many reasons here we choose entropy S. Then energy is one of the independent extensive variables. Via the simplectic structure we can substitute some extensives with intensives; I will not do this. Note that entropic intensives differ from energic ones by a division with T, signs &c., so do not confuse the two sets. We look, however, at the Gibbs-Duhem Relation.

Make an infinitesimal change in the state, X->X+dX. Then of course

(6) dS = S,_{r}dX^{r}

But from (1) & (2):

(7) S = S,_{r}X^{r} = Y_{r}X^{r}

whence

(8) X^{r}dY_{r} = 0

So the n dY's are not independent in vectorial sense. If (8) does not hold, we are not in Thermodynamics, at least not in the usual sense. Not surprisingly, e.g., the so called "thermodynamics of black holes" [9] does not obey the Gibbs-Duhem Relation; the reason is the strong self-gravitation, when homogeneous linearity is not natural at all anyway.

Until now the treatment is quite general. Now let us see the fundamental Laws. The *Zeroth Law* already has been used: there is a set of variables Yi, whose equality for different systems is sufficient and necessary condition for equilibrium. In the present algorithm the *First* *Law* is some combination of (6) and (7), e.g.

dS = Y_{r}dX^{r}

or

S = Y_{r}X^{r}

Hence one can rewrite it for an energy balance of course; but energy is only one of the extensive variables #4. That is a balance equation.

Now would come Second Law, but we jump over it for a moment. Third Law tells something about limiting behaviour of potentials, and also Fourth (not well known) and so on [8], [10]. We can skip them here, although they are interesting. Their necessity is connected with the remaining freedom in the potential [6].

Now we are coming to the mysterious *Second* *Law*. Here I follow more or less the construction of [1].

There is an irreversibility in nature. In ancient times it was formulated in the way: macroscopic/mechanical work can be completely dissipated away into heat/microscopic motion; but not backwards. "Mechanical motion is inevitably degrading." And so on. To catch the notion mathematically, first it is necessary to single out some "natural" or "spontaneous" processes. Usually they are the *adiabatic* processes, although it is not always clear what does "adiabatic" exactly mean. More or less a system is considered *adiabatically* isolated if surrounded by walls permitting only the action of mechanical work #5. Very rapid processes, e.g., are generally adiabatic.

Then the simplest way to formulate an irreversibility is a linear inequality. And indeed,

(9a) dS >/= 0 for adiabatic processes

seems good. This is a form of Second Law. There is another. Separate somehow heat transfer and mechanical work, irreversible (~~d~~Q) and reversible (dW) energy transfers (see again Note #5), and then

(9b) ~~d~~Q >/= 0 for adiabatic processes

Of course the two forms would be equivalent were ~~d~~Q to be TdS. *But I will not assume this.*

Now back to Ref. [1]. We have systems. We have performed a lot of measurements. The irreversible processes have been learnt in details. We have got

(10) ~~d~~Q = Z_{r}(X^{k})dX^{r}

Now we want to write ~~d~~Q with the minimal number of variables. So we introduce appropriate functions instead of the X's:

J^{I}(X^{k}), I=1...N

so that the formula

(11) ~~d~~Q = G_{R}(J^{K})dJ^{R}

has the minimal possible number of terms. (11) is the canonical Pfaff form of the system.

Now, in usual physical thermodynamics the rank of the Pfaffian is
2, and ~~d~~Q can be reduced to TdS but not to dQ. However
that is not necessary even in physics and does not follow from any
fundamental law. One can indeed incorporate different thermodynamic
formalisms with different irreversibilities into physics [12]. There is an
experience behind the Pfaffian rank N=2: the nonexistence of perpetua mobilia. For N>2 perpetua mobilia of second kind (i.e. even with fixed energy) are possible, even with irreversibility of the familiar form ~~d~~Q>/=0! Such devices did not succeed, and there are measurements suggesting that *now* N=2 in physics. This is not certain for the Planck epoch, which either lasted cca. Planck time 10^{-44} s, or any long time in a state N>2 [12] (and then our N=2 physics is a limiting case). However that is a question not decided just now. But, for any case, we do not have any argument against perpetua mobilia in economy, *and we have some for them.* So if one wants to use a thermodynamic formalism for economy, she must consider other values of N too.

2. IS ECONOMY THERMODYNAMICAL?

I do not know if Economy is thermodynamic for structure or not. I do not have a copy of Laws of Nature; and even if I had, Economy is not a natural science. The Laws of Economy are formed by human society (albeit not by parliaments). However I do know that Thermodynamics is a good analogy to Economy, so thermodynamics-like formalisms are promising there. And that is so because of the irreversibility/rationality of economic processes.

The key notion about economic irreversibility was formulated by Bródy/Martinás #6 and is written (somewhat briefly) in [13]. The idea is that you always know what is raw material and what is product. (It may depend, of course, on actual conditions, cf. cannibalization of machines in shortage &c.) Production always has a natural direction, and in rational economies it does not go backwards; Commerce can go in both directions so it is reversible.

Then Ref. [13] suggested a formalism. You can get the economic thermodynamics if you first take the physical one and then exchange

entropy to utility

energy to money

particle numbers to quantities of products

(and maybe) volume to population.

The resulted structure is elegant, clear, and gives some deeper insight into economy, although up to now I do not know any practical truncation of the set of extensives leading to a handleable model system. I feel it necessary only to discuss the utility function.

Utility function S is used in economy generally as a sublinear function of income, GDP, &c. [14], [15]. The sublinearity expresses the proverb that "the second dish is less needed". Now, the utility function of Ref. [13] depends on money + other goods, but that is a naturalgeneralisation; and it is slower than homogeneous linear. Namely, in a stable state the second derivative matrix of the entropy density is negative definite.

Thus now the utility function S correctly expresses the decreasing desire for goods, and also expresses that the desirability of a state depends on all the goods including money.

It is difficult to imagine an economy without utility function; it would mean that the society does not compare the desirability of different states. So I was rather surprised to hear that Dr. Martinás accepted that she does not have now utility function and this is the reason that desires are not transitive/comparable. I must confess that I verbally lured her into this statement; maybe she simply admitted my formulation. I am expert in verbal tricks, specially in Hungarian. There is the possibility that she wanted only to tell that the increase of S is not the principle behind the system.

However, she did not tell this and I wrote earlier, 3 times, twice in economic contexts: [12], [16] and [17], first 12 years ago. The possibility will be discussed in Chap. 4. For any case, if somebody is doing something new, keeps it in secret from me, her subject in the common research, and then makes a lecture where I am present, it is better to formulate either very clearly or very mystic.

Another statement was that the opinion that the Kyoto CO2 agreement is against USA's economic competitivity is somehow erroneous. I think she is right *in a broader sense*. But to get such a result one would need the unification of present Economy and Environs. Until this unification is not at hand, the original statement is correct and scientific; and attempts to attack an incomplete theory with intuition is not enough. It is possible that some results are behind, but they were not mentioned at the Lecture. I state this only because ) I wrote about the problem twice [17], [18], some 7 years ago, although not in really first-class places (Martinás knows at least the second whose scientific lector she was); and 2) it belongs to the research topic where I am a regular member but under boycott. The consequences of increasing CO2 level, if they exist #7, may compensate indeed away the advantages of industrial growth; or they may not, which is just the question. For any case, they do not arise at the same place. USA emits many CO2, but her coastline is not too vulnerable; while southern islander communities are vulnerable while not producing too much CO2. And then what?

Now let us see the N=2 case in more physical details.

3. IS SECOND LAW EQUIVALENT WITH TOTALLY FREE ENTERPRISE AND NIGHT-WATCHMAN STATE?

Ref. [13] gave a working model description for an economic system. The description works; it is sure because Thermodynamics works, and the model is isomorphic with Thermodynamics #8. The only (!) question is: *which* economy is described by [13]? In [16], 12 years ago, I stated that fully free enterprise is compatible with [13] (namely with N=2), and nobody has disproven this so far.

Uneq. (9b) is a differential or local irreversibility principle. But for N<3 this generates global irreversibility as well. See first N=1.

If N=1, the Pfaffian (11) reads as

(12) ~~d~~Q = dQ(X^{i})

So there is a *function* Q whose increase is the irreversibility principle. *(The difference between dQ and dQ is that dQ is only symbolic, not a change of a function Q.)* Then there are surfaces in the state space: reversible processes move along these surfaces, and irreversible ones going up and never down.

For N=2 we get ~~d~~Q=T(X^{k})dS(X^{k}). Then, still, until T>0 the situation is the same as for N=1. For negative temperatures, pressures, &c. many people believe that there are "problems"; there are not [19], but such regions do not appear for general systems and for which they appear they appear for "exotic" parts. In economy T<0 would mean negative value of money [13]. Let us not speak about such states and then there is still global irreversibility from local one. Higher entropy states are "spontaneously" generated, and the utility of the states of economy is not decreasing if not forced. If one thinks that this is an absurd character of the model, let me know it.

However for N>2 global irreversibility is not a consequence of the local, ~~d~~Q>/=0. Ref. [1] states that all points of the state space are available from a fixed point, not only half of them. This will be discussed in Chap. 4.

Case N=1 was the idea of Merchantilist economists from mid-1600 when they catched the ear of French Kings. (Their first leader was Colbert; the last Necker, the Swiss banker in the time of the sad philosopher Rousseau, not a French but a Savoiard, more definitely a Swiss Calvinist which is important to understand the ideas of the time and their effect on merry and Catholic France.) The idea of Rousseau was to live simply, very simply. On the same time, that of the Merchantilists, from finance minister to tax administators, was that Economy's goal is to increase the amount of gold (in the treasury). That is an N=1 Pfaffian: there is a quantity whose increase is the extremum principle. The model works; however did not describe end-1700 French society (or anything else), the consequence was the massacre known vulgarly as Great Revolution, consolidated by the military massacres of Napoleone di Buonaparte, a Corsican artillery officer, the Member of the Mathematical Academy of France, inventor of the Napoleon Theorem #9. After this tremendous sentence written to demonstrate my wide knowledge, I summarize: N=1 economies are no good at all.

Case N=2 is rational, optimal and so on. Now, it does not contain state intervention. It does not contain anything concurring with "profit" principles. It may contain State. State may collect tax; it may take it to pay police, militia and such things which are necessary conditions for industry; State may collect simple head taxes, but nothing to direct economy. And that is the idea of XIXth century Manchester liberals. State is the night-watchman of Economy, and Economy works about its single maximum principle. (The Invisible Hand and such things.)

This is a possible model. Maybe it is a good model. However since the New Deal not operating anywhere. Martinás and I are physicists enough not to confuse our preferences with the system under observation. (In addition I do not know her economic preferences.) I wrote in 1989 that her N=2 model is Free Enterprise, and I wrote in 1989 and in 1995 that N>2 would be needed to describe contemporary economies. I do not think that immediately the utility function S should be thrown away. Maybe N=3 will be enough.

And I have chosen the Internet to communicate this because I am banished from the discussions in OTKA T/029542. I think the other 4 fellows will read this sooner or later. Also, in this way I am documenting my work to the OTKA Bureau.

4. KHEOPS' PYRAMID, ZWENTENDORF POWER PLANT, BÖS (GABCIKOVO)-NAGYMAROS DAM AND OTHER PERPETUA MOBILIA OF SECOND KIND

Dr. Martinás told something that there is no irreversibility in economy. I did not hear this
sentence (her voice was very low) but another told that he heard. (Hearsay!) However I remember the
argumentation. One can make exact cyclic processes and at the end he gets back to the starting situation.
This is true, and I will tell examples as I did already 6 years ago. However I tell that *you can make
exact cyclic processes satisfying the local irreversibility condition (9b) everywhere.* Of course not for
N<3. Then a part of the state space is inaccessible [1].

Now take the simplest thermodynamics where all points are accessible, N=3. The equations and the physics are discussed in [12].

Take 2 close points. You can connect them with infinitely many paths. In the Merchantile and Manchesterian economies they are either on the same hypersurface (zero measure), or Point 2 belongs to higher utility and then sooner or later the economy will reach that hypersurface and then simple commerce will bring us there, or Point 2 belongs to a lower utility and is behind us anymore. For a subsystem of course it is possible to be in that state; in physics subsystems can decrease their entropy if they put away "heat" into the surroundings. (E.g. living organisms keep down their specific entropy.)

However for N=3 on some paths Point 2 can be reached from Point 1 so that ~~d~~Q>/=0 everywhere, and on some paths Point 1 can be reached from Point 2. We can go to somewhere and then back; only not on the same path.

How is this possible? Ref. [1] gives the construction, only that is boring. Using a half-intuitional argumentation, now

(13) O </= ~~d~~Q = dZ + TdS

all 3 terms on the rhs. functions of X^{i}. Now assume that T>0. Still dS and dZ can balance each other. Between 1 and 2 dZ and dS are given, but T is changing, so indeed the sum can be anything. I repeat: full attainability is proven in [1]. I also mention an analogy from General Relativity. The light cone structure may exist in any point of the space-time (so local causality ds^{2}<0 holds) and still closed timelike curves may be possible, which violates global causality. If so, temporal paradoxes appear, but that is another matter. Cosmic Censorship is a principle to prevent the paradoxes but the Cosmic Censor may or may not exist.

Now: are economies working in this way? Yes, there are. Egyptian Pharaohs of the 3rd and 4th Dynasties made great pyramids built. It was organised in the following way. Peasants could not work for 1/3 of the year since the farms were inundated by the Nile. Therefore the Pharaoh, Life, Health, Strength for Him, took 1/3 of the crops as tax. Then, from the tax crop, he (Life, Health, Strength!) feeded the populace for 4 months. Of course they built the pyramid. Maybe they were not overly happy when paying the tax; but you pay tax anyway. But they were very happy when saved from utter starvation. The final balance was positive. This went so for several centuries and the system was stable enough. At the end it collapsed, but we still do not know why; and the founders of the 5th Dynasty were very popular for being democratic. They were not Great Gods, only the Sons of Ra, the Sun God.

OK, that is a pre-industrial story. Now comes 1977. Hungary and Slovakia agreed to build the Bös (Gabcikovo) - Nagymaros (Velky Marys?) Dam. Hidroelectricity is cheap and clean. The dam was built up for 90 % when turned out that electricity is negligible and freshwater accumulation is in danger. Then Hungary stopped the construction, some part was taken apart and now Hungary is happy. Slovakia finished her part, cannot use it properly, but uses it and is happy.

OK, that was an absurd plan before democracy. But democratic Austria constructed a nuclear power plant at Zwentendorf and was being happy. When it was ready for 95%, a referendum decided that they will not use it. It is so indeed; the building is empty. Maybe they will take it apart later, and then they will finish the full cycle: back to the initial stage while all steps went into a preferred direction. As Asimov called something "The Road that is Downhill Both Ways" [20]. Present physics do not know about such nice things; perpetua mobilia of second kind would be such, but it seems they do not exist #10 in physics. But they exist in economy and look quite interesting for me.

5. CONSERVATION LAWS

This Chapter will be short because here I do not reflect to the present Lecture. There are both in Thermodynamics and in economy some conservation laws or balance equations. If you produce castiron, you are using up iron ore and coal, and you are producing carbondioxide, sulphurdioxide and acidic rain. If you are producing steel, you are using up castiron and coal. And so on.

Some such equations reflect natural laws, some reflect present technologic limits. But some may reflect politics too.

Until 1990 in Hungary a unique money did not exist. Some money could be used for investment, some to buy raw material, some to pay employees. In a softer way, via taxation preferences &c. the practice is worldwide. And then, what about Economic Thermodynamics? Energy is unique, is it not?

Sometimes yes, sometimes not. First, I remember that it happened during such a discussion in 1986 that Dr. Martinás told me: "If your worthy comments will be needed, we explicitly will ask for it!" Which meant: "Shut up!" Now I did not accept her right to tell what may I say and what must not; so I stopped to discuss economy with her at all. Then she applied for grant together with me in 1998; interesting.

However the physical comment is not this. I can tell at least two physical situations for *two independent energies* in the system.

In LiF crystals oscillation energy of the lattice points is very weakly coupled to potential energy of nuclear spins in external magnetic field. For a long time equipartition happens; but then, changing the external field, two temperatures appear, conjugates to the two energies, and the two temperatures remain different for minutes [21]. This is a way to demonstrate negative temperatures.

More than 10 years ago it is known that in low energy heavy ion physics 2 temperatures appear in the system. I do not give references here; I do not think that anybody except Dr. Martinás is really interested and I remember I gave her a reference 10 years ago. The mechanism is the following. Since the temperature is not far above the binding energy, the nucleus is decomposed but not into nucleons but into fragments, which are excited small nuclei. One temperature belongs to kinetic motions of fragments, the other to the internal excitations. Two intensives mean two extensives, so there are two disjoint energies in the system.

The Austro-Hungarian Dual Monarchy was on double monetary system up to 1873: not Austrian and Hungarian but gold *and* silver. It worked somehow. Economic Thermodynamic may be applied even to such situations. At least I can do; but I have other tasks to do.

6. VARIOUS CONCLUSIONS

I would first conclude that Dr. Martinás, although she is the rightful leader of T/029542, cannot prove the nonexistence of utility function, also she cannot prove the lack of substitutions in consumption. However 1) she spoke so low at times that it was hard to hear all the arguments; 2) it is possible that she did not tell all.

If her point is that the Bródy-Martinás-Sajó model does not describe contemporary economies because of N=2, I agree, but I wrote this already many times. If her point is that some environment variables should be incorporated also, I agree but I have it written too and it will be complicated. If her point is that some Theoretical Economics is performed in the ivory tower and she can do better, I agree, but I can do even better, as I quoted a musical in the Abstract: "Anything you can do I can do better...".

Do not misunderstand me. I think, Dr. Martinás is a great thermodynamic expert. Generally I used to believe she knows more than I in Thermodynamics. I still think so (even then I could do better…) and am confused by the Lecture.

Thermodynamics, I think, still has some future in economy. Let us use it rather than to try to invent something new whose structure is not yet known. But if I misinterpreted the Lecture, my opponent can reply. Maybe the readers of the web will like the arguments; maybe not.

Finally I again protest against the ban and boycott I meet in a research group whose membership I applied and won. I hope that if nasty consequences will arise then they will fall back on the boycotters and specially the team leader. However now I demonstrated my ability to work and I produced something the OTKA Bureau can consider if wants so.

********

This work belongs to the topics of OTKA T/029542 but of course was not supported by it or helped by anybody in it.

NOTES

#1 Here rules of written and spoken English contradict each other. I cannot write Saxon genitive's "s" after "s". However Hungarian "s" means "sh". But most people will not pronounce.

#2 Symbol P is used for many related goals. In an earlier stage of evolution, e.g., P stood for *ekaentropy* of inherently unequilibrium states [2], [3]. In that time rumor told that the symbol had been selected from the initial of Martinás' husband. However, I think not; ekaentropy would require E, that letter is occupied, so remains P as *potential*. Now P is definitely only a potential; we have not yet defined it.

#3 Thermodynamics has a Riemannian structure [4]. So at the end I could apply even a General Relativity-like formalism on economy? Of course I could. But, first, can I publish the results? And, also, the role of Equivalence Principle would still need some clarification. But, indeed, any set of parameters ("coordinates") could be used, which may be useful.

#4 We are well past of the Age of Enlightment, when the Conservation of Energy was a war cry of Science vs. Religion. In General Relativity energy is generally not conserved because the Universe is generally not stationary. And still there is no problem with Thermodynamics.

#5 Then, of course, one should first uniquely separate mechanical work and heat transfer. This is highly nontrivial, for example, because ideal isolating walls should be infinitely thick and then they would have infinite heat capacity. In 1986 we started a project with K. Martinás to clear up this question, and we have guessed the answer. However that work stopped two times for unknown reasons. I tried to formulate the problem in [11] but even myself am not satisfied. I guess it is beyond a single author.

#6 I was not there in the moment of formulation but I heard it soon enough. So I cannot tell who of the two suggested it.

#7 The matter is obscure. The CO2 level is indeed increasing, CO2 is a greenhouse gas because of its absorption lines in near infrared, so one would expect more significant correlations. Maybe still there is too much politics in a physical question.

#8 On a more Byzantine language: homousion or homoiousion? Ref. [13] is homousious to thermodynamics, N>2 economies are only homoiousious.

#9 To be definite: Take a triangle. Draw equilateral triangles on each sides. Connect the far vertices with the opposite vertices of the original triangle. Then 1) the 3 lines meet in one point; 2) the enclosed circles of the equilaterals all cross this point; 3) the centers of the enclosing circles are the vertices of a triangle which is again equilateral. Nice, isn't it? For more literature see [19].

#10 Note that for particles in thermal equilibrium Planck and Hausen [22], [23] were able to prove ~~d~~Q=TdS *if* there are no perpetua mobilia of second kind. So, using also the argument for closed cycles too, if they exist, N>2.

REFERENCES

*(I do not think readers will read the cited papers. OK, but read the author names. Even that may be interesting. And also, you can find *http://www.rmki.kfki.hu/~lukacs/reindeer.html* online.)*

[1] P. T. Landsberg: Thermodynamics. Interscience, New York, 1961

[2] B. Lukács & K. Martinás: Annln. Phys. **45**, 102 (1988)

[3] B. Lukács, K. Martinás & T. Pacher: Astron. Nachr. **307**, 171 (1986)

[4] L. Diósi & B. Lukács: Phys. Rev. **A31**, 3415 (1985)

[5] K. Martinás: Acta Phys. Hung. **50**, 121 (1981)

[6] B. Lukács & K. Martinás: Phys. Lett. **114A**, 306 (1986)

[7] B. Lukács, K. Martinás & G. Paál: in Gravity Today, ed. Z. Perjés, World Scientific,

Singapore, 1988, p. 247

[8] B. Lukács: KFKI-1994-14

[9] J. D. Bekenstein: Phys. Rev. **D7**, 2333 (1973)

[10] B. Lukács & K. Martinás: Acta Phys. Pol. **21B**, 177 (1990)

[11] B. Lukács: Acta Climat. **XXIV**, 3 (1992)

[12] B. Lukács & G. Paál: Acta Phys. Hung. **66**, 321 (1989)

[13] A. Bródy, K. Martinás & K. Sajó: Acta Oec. **35**, 337 (1985)

[14] F. R. Chang: Econometrica **56**, 147 (1988)

[15] M. Banai & B. Lukács: KFKI-1989-68

[16] B. Lukács: Acta Oec. **41**, 181 (1989)

[17] B. Lukács: in Proc. ERÖFI II, 1994, ed. A. Rácz, KFKI-1995-11, p. 175

[18] Lukács B.: Környezetgazdálkodás fizikusszemmel. TKTE, Budapest, 1994

[19] H. S. M. Coxeter: Introduction to Geometry. John Wiley & Sons, New York, 1969

[20] I. Asimov: The Gods Themselves. Fawcett Crest, Greenwich, Ct, 1973

[21] E. M. Purcell & R. V. Pound: Phys. Rev. **81**, 279 (1951)

[22] M. Planck: Annln. Phys. **19**, 759 (1934)

[23] N. Hausen: Z. Phys. **35**, 517 (1934)

**My HomePage, with some other studies**, if you are curious.