This is a paper about economical thermodynamics (or thermodynamical economics) from the Proceedings of the Second Autumn School on Reactor Physics, ERÖFI II (ed. A. Rácz), Lillafred, 7-10 Nov., 1994. Since in recent years limitless and ill-defined argumentations are going on about sustainable development/consumption, and this paper is at least mathematically rigorous, but the Volume was distributed mainly amongst reactor physicists, now I show the article to the whole World. If World is not interested, it is its own responsibility henceforth.

Hungary is a small country of cca. ten million. So you must not be surprised by the fact that organiser of the school and editor of the Volume had been earlier the graduate student of Dr. K. Martinás, whose name will repeatedly occur in the paper. Hungarian colleagues may guess even a correlation.

Trivial misprints have been removed and equations restructured according HTML needs, otherwise the text is unchanged, edited from the original MSDOS Word 4.0 file.




KFKI-1995-11, p. 175



B. Lukács

Central Research Institte for Physics RMKI, H-1525 Bp. 114. Pf. 49., Budapest, Hungary


MultiRatio Cooperative, H-1117 Szerémi u. 39, Budapest, Hungary


We discuss the problem of contradicting economic and ecologic principles. The discussion will start from the utility function of economy, which possesses entropy-like properties, on which the so called economical thermodynamics is based. We show that no thermodynamics of economy is possible, but that of economy and ecology together is not out of question. In such a description the controversies may be absent.


In recent years ecologic thinking, green mentality &c. are fashionable. People seem to accept a viewpoint that economic rationality is not everything, or must not be dominant or must be suppressed if contradicts to ecology. E.g. the Hungarian Green Alternative just have formulated a set of requirements that Hungary should promise not to build or use nuclear power plants, and not to export or import electricity produced by nuclear power plants. Now, some of these requirements are obviously not connected by any radioactive hazard in Hungary, since, even if one accepts that a nuclear power plant causes pollution (which is questionable), electric currents generated in nuclear power plants are physically undistinguishable from electric currents of other power plants. Therefore export and import of electricity of nuclear power plants are completely irrelevant for ecology of Hungary, therefore this point does not belong to ecology at all.

However the use of fissionables may be an ecologic and economic question. Arguments and counterarguments range widely and often contradict. The standpoinds can be classified fourwise:

1) NP is not dangerous and economically advantageous.

2) NP is not dangerous but economically not profitable.

3) NP is dangerous but economically advantageous.

4) NP is dangerous and economiocally not profitable.

Here NP stands for "electric energy production in nuclear power plants".

Physicists are not afraid of nuclear energy if fissionables are properly handled. Many data show that NP causes less radioactive emission than the use of coal to the same purpose. The New York Central Railway Station has higher radiation inside than permitted for a nuclear power plant (from the granite slabs), so could not work as a nuclear reactor. However let us permit all four possibilities for the sake of argumentation.

In Cases 1) and 4) economic and ecologic criteria point into the same direction and no conflict is expected. In Case 2) strong lobbies still may force such energy production but there is no ecologic problem with this. However in Case 3) a conflict arises: something is advantageous from one viewpoint and disadvantageous from the other. Then what to do?

The present paper is not an analysis of radiation hazards, reactor safety, disarmament agreements &c.; nuclear reactors or even U235 or Pu239 will not be mentioned henceforth at all. We formulate a general problem in a general situation. Consider a situation when the states of the environment and economy are fixed. Now imagine a change which is advantageous economically but disadvantageous for the environment (or vice versa). Then is the change advantageous for us or not? Are the "qualitatively different" gains or losses comparable? If they are, how? If we can answer this question, the advisability of the change can be decided. If we cannot actually answer the specific question but can show how the answer could be find, still we can start to look for it.


A usual answer is that only such changes should be permitted which are at least not disadvantageous in both aspects. However often such change is not possible at all. Consider coal power plants. More electricity is economic advantage, but it needs more coal burning, resulting in CO2 contamination (global warming) and SO2 emission (more asthmatics and acid rains). Gas emissions can be reduced by expensive filters &c., in which case the more electricity is no more economically profitable.

In the generic case two extremum principles for the same variables result in two different equations of motion, often contradicting, with no solution at all.

The present qualitative example shows that one source of the controversy is that production processes of economy are often sources of pollution. This cannot be helped. In a strict sense almost everything is pollution. Even natural components of the atmosphere are pollution if they are emitted in a quantity comparable to the original one. Oxygen is a vital component of air, but oxygen emission raising to the concentration from the present 21% to 30% would cause imbalanced metabolism with various illnesses and degenerations. Fresh water is a very valuable treasure, but global warming would melt much Antarctic ice and the resulting fresh water would dilute the seawater. Then the result would be extinction of some salt-sensitive fishes plus inundation of lowlands.

So we need the net results of gains and losses. But for this we must quantify the effects of both the environmental and the economical changes, in dimensionally identical quantities. Then the sum is the net effect.


In economy quantification is theoretically clear. Let us try to measure the whealth of a nation. In first approximation consider the material goods available at a moment (or consummed during a definite period). They are the money M and the goods Ni. A change is a production process

(3.1) (M,Ni) -> (M+dM,Ni+dNi)

with some conservation laws expressing that production eats up raw material and money.

Now a quite general experience is that in a normal society in normal situations it is clear what is raw material and what is endproduct (Bródy, Martinás & Sajó 1985). By other words, there is a kind of irreversibility in production, while pure commerce is of course reversible. (In this moment we ignore special taxes on rum, tobacco &c.)

This statement does not mean that the direction of the process is always the same. Generally planks and nails are used to build tables. In great shortage of planks tables can be dismantled to get planks and nails. However at fixed (M,Ni) the change is profitable production in one direction and irrational wasting in the other.

Let us postulate this observation as a general rule. Then follows that the degree of the Pfaffian form of the problem is 1 or 2 (Landsberg 1961). This is the case postulated by Bródy, Martinás & Sajó (1985); then i) economy has its own thermodynamics; and ii) that is isomorphic to physical thermodynamics. Both statements are questionable, so let us start from the beginning. (For detailed discussions see Lukács (1989); Lukács & Paál (1989).)

Pfaffian forms are defined in differential geometry and thermodynamics. Here we start from thermodynamics, and then, mutatis mutandis, try to apply the formulae to irreversible economy. Consider the energy E as potential, i.e. function of the independent extensives Xi. It is a homogeneous linear function:

(3.2) E(ßXi) = ßE(Xi)

Now consider general small changes of the extensives. Some are irreversible, some not. Even irreversible changes can be reversed if the system is open. Therefore let us restrict ourselves to circumstances when irreversible changes are really irreversible, say, to "adiabatic" changes. Now, something is growing during an irreversible change, constant in reversible one, and would decrease if an irreversible change could go backwards. (The sign is a matter of definition.) This more general fact is behind Second Law. Therefore one can decompose dE as

(3.3) dE = (E/Xr)dXr = dW + dQ

(henceforth throughout the paper there is an automatic summation for indices occurring twice, above and below; this is the Einstein convention). Here dW belongs to the reversible changes and dQ>0 for irreversible ones. The particular form of the differential d shows that dQ is not necessarily a differential of a function Q(Xi).

One can measure dQ in processes if E is operatively defined. Then we know dQ's and the corresponding dXi's. Obviously

(3.4) dQ = yr(Xk)dXr

and then the functions yi(Xk) can be determined from measurements.

The rank of the matrix

(3.5) Gik º yi/Xk (3.5)

is £n, and if r(G)=n then there is no reversible change in the system at all. On the other hand r(G)³1 since there are irreversibilities. (This has been postulated.) Now by means of transformation of the variables one can reach one of the canonical forms

dQ = dQ(Xi) (K=1) or

dQ = T(Xi)dS(Xi) (K=2) or

(3.6) dQ = dZ(Xi) + T(Xi)dS(Xi) (K=3) or

dQ = U(Xi)dZ(Xi) + T(Xi)dS(Xi) (K=4)

and so on.

Here S, Z, &c. are the new, canonical variables instead of Xi. For a given system such a form is not unique, but the available minimal K is. Let us call this minimal K the rank of the Pfaffian form of the increment of the non-compensated heat and proceed.

According to general belief, in physical thermodynamics K=2, hence comes the familiar dQ=TdS, where S is the entropy. This quite can be true (for a discussion see Lukács & Paál (1989)); however we will go over to economy, so let us see first, how necessary is K=2.

The simplest thermodynamic systems of education contain a single conserved kind of particles. Then seemingly the number of independent variables is 2 (N being a constant parameter); and for n=2 with reversible changes (which do exist) K=2 is the possible maximum.

The problem is that N is an independent variable even if it is constant. As an example consider a matter with some simple interaction resulting in an interaction energy density quadratic in particle density. Then

(3.7) E = E1 + aV(N/V)

where E1 is some nice thermodynamic energy function. Then still the second term is homogeneous linear in N and V, so Cond. (3.2) is fulfilled; but solely in V it is not homogeneous linear and then E would not be a thermodynamic potential.

For n=3 and with some reversible changes K is 1, 2 or 3. K=3 can be excluded with some extra postulates of which the most usual is the nonexistence of perpetua mobilia of second kind (Planck 1934; Hausen 1934). It is quite possible that perpetua mobilia of second kind do not exist in the physical world, but economic perpetua mobilia may exist. Therefore we do not restrict ourselves to K=2 and proceed.

There is a theorem (Landsberg 1961) whose proof is transparent but long and boring that local irreversibilities in individual processes result in global irreversibility in the state space if K£2, but not for K³3. Namely choose a point somewhere. This point is i (inaccessible) if it is inaccessible from at least one neighbouring point by dQ=0 curves. Now, for K³3 the space does not contain i points (Landsberg 1961). Then let us postulate that irreversibility exists so in any actual small change dQ³0. Then draw different curves between two nearby points 1 and 2. These curves can be classified into 3 classes. On the first dQ>0 from 1 to 2, on the second dQ<0 from 1 to 2 and on the third dQ=0. Then on a curve of type 1 one can go from 1 to 2 but not backward, on a type 2 one can go from 2 to 1 but not backward, and on a type 3 one can go to and fro. Now from the above theorem it follows that in a space K³3 either all the 3 kinds of curves exist, or at least type 3 curves. In spite of irreversibilities of actual processes one can go round and round with any two endpoints.

As a comparison consider the case K=2. To point 1 there exists at least one nearby point whence it is inaccessible with dQ=0 (see the theorem); let us choose this other point 2. Since dQ<0 processes are not permitted (irreversibility), this means that either dQ>0 on all paths from 1 to 2 or dQ<0 on all. For simplicity assume the first case (if not, reverse points 1 and 2; then due to linearity in the form of dQ the sign change to opposite). Then, by processes obeying irreversibility one can go from 1 to 2 but cannot go from 2 to 1.

On the other hand, for K=2 dQ=TdS. Therefore, if the function T(Xi) is positive (which is true at least in a domain of the state space), then dQ>0 implies dS>0. But S is a function on the state space. Therefore there exist some hypersurfaces of S(Xi)=const., according to the theorem they are not crossing each other, and one can go "up" in S but not "down". By other words, if the present S value is So, then (for K£2) the state space is divided for me into two halves. I can go in the direction of higher S's (but cannot come back); I cannot go towards lower S's, and can freely move on the hypersurface separating the half-spaces, where S(Xi)=So.

But this restriction is absent in spaces K³3. There one can go in any direction, only special curves are needed. One can go from 1 to 2 on one curve, and can return later to 1, only another path must be chosen. Every state is accessible from every state on devious paths, and the original state can be restored, in spite of the existing irreversibility laws. In such a system, then, perpetua mobilia can exist.

Now leave thermodynamics and go to economy. The economic literature has not classified national economies according to the value of K so we must start with this here.

In an economy K=1 permitted processes do not carry into one half of the state space, and perpetua mobilia do not exist. Now there is a function Q(Xi) and obviously it is homogeneous linear in Xi. But then we can introduce Q as a state variable. The simplest example is the economic system of monetarians in the XVIIIth century France (resulting in bankrupcy, revolution &c.) where the money M was postulated to be the variable governing irreversibility. The state tried to maximize M itself as the governing principle of economy. Obviously the society of France did not want to maximize M and revolted.

In an economy K=2 still one half of the state space is unavailable: progression goes irreversibly into the remaining half. Some states are more and more desirable for the society. Production causes a growth of S, while commerce leaves it unchanged. Such an economy was postulated by Bródy, Martinás & Sajó (1985), and such an economy might have existed in the age of free enterprise capitalism but hardly in our century. In a society whose economy is of Pfaffian K=2 the future is more or less prescribed although the speed of evolution can be increased or decreased.

Here we stop for a while. In an economy K=2 the function

(3.8) S = S(M,Ni)

measures the desirability of the given amount of goods for the society. Therefore S measures the actual utilitas of the goods; not of one of them, but of all of them in the given distribution together. It is possible that it is more desirable to possess much more iron with a little less butter, but not so to possess a little more iron with a much less butter; because iron can be utilized in some degree and butter can also in another degree. So S (which is the economic entropy) can be called utility function henceforth and denoted by U. It is intimately connected to the utility functions U of neoclassical growth models (Solow 1956; Chang 1988) depending on the consumption. Thenceforth the abbreviation S will be used for another role.

U must have the same mathematical properties as the entropy since now economic and physical thermodynamic formalisms are mathematically equivalent. Then introduce one more variable, the population P, constant, for example. So

(3.9) U = U(P,M,Ni)

and then the second derivative matrix of

(3.10) u º U/P = u(xi; xiº{M/P,Ni/P})

is negative semidefinite. This expresses the sublinearity of U (Chang 1988). (The second million ton of steel is not so important than the first one.)

Then, if and until K=2, U measures the economic gain from a change (building power plants, melting iron, &c.). It seems that half of our task in fulfilled (in principle; nobody ever measured the form of u(m,Ni). In the Appendix we list counterarguments and continue to discuss K; however here we accept U as a measure in a rough approximation and switch to environment.

The variables of U will be denoted by Bi (bona). We definitely do not claim that {Bi} would contain only P, M and material goods; later we return to this point.


Pollution deteriorates our environment for ourselves. This is just the definition for pollution. However, it is not easy do define what is deterioration and what is not.

First: the optimal environment cannot be defined "objectively". Oxygen is vital for us and lethal to anaerobic bacteria. The Big Oxygen Pollution some 1 Gy ago, caused by suicidal Cyanophytae, was catastrophic for the majority of the biosphere then; but was very appropriate for us in the far future. If environment-conscious blue mosses had avoided the oxygen pollution, we could have not come into existence.

Second: there is no environment "optimal for Man". Esquimaux and Amazon Indians need different temperatures, humidities &c.

Third: peoples often occupy territories which are unhealthy, dangerous or tiring. This they do for other gains.

Problem 1) is very important, but we are human. For us human interests are important, and the needs of tetanus bacteria are irrelevant; until they do not try to eat us. Problem 2) is important, but we may deal with our actual community on our actual territory. Problem 3) is a key point of this paper; let us keep it in mind.

Now, dirt deteriorates the environment. Biological individuals are open thermodynamic systems, keeping their entropies low and constant by consumming low-entropy food and excreting high-entropy refuse.

Now, if the entropy is high in the environment, then the food is not too low in entropy, and it is hard to get rid off the extra entropy too. It seems as if thermodynamics of open systems would suggest that the entropy, specific entropy or entropy density of the environment would measure the degree of pollution.

This suggestion is not exactly true. Zero entropy environment would be a zero-temperature world with all material components nicely separated and kept in compartments. The present atmosphere is a nitrogen-oxygen mixture with some mixing entropy: a pure oxygen atmosphere would be almost as lethal as a pure nitrogen one. However the measure of the pollution is intimately connected with entropy, so we call it S as sordes.

The state of the environment is characterized by a set of variables CI (circumstantiae), and

(4.1) S = S(CI)

The bigger S the less appropriate environment (for us and caeteris paribus).


During the history privies were being cleaned. Some by slaves, some by prisoners, but most by free persons. In Japan the collected material was used by the cleaner on the rice fields, but in Europe it was not. This work, according to the Le Chaterier-Brown Law, goes in an environment of high enough S, because just the endproduct of the open thermodynamic system is in the environment. Still, with a sufficient salary people performed this task. During the history it has been being usual for tribes or towns to occupy malarial marshes for fishing or blocking commercial routes, or to live in deep valleys without sun and with poisonous fumes for mining valuable metals. Therefore human communities generally do not maximize U(Bi) or minimize S(CI); they accept higher U as recompensation for higher S or vice versa. It seems as if they would like to maximize a combination of U and S which we call here F, florentia of the community.

Since both U and S are entropy-like quantities, the first guess is F = U-S. But we shall see in due course that this form is impossible.


According to the formalism of Bródy, Martinás & Sajó's economic thermodynamics (1985), market prices are the chemical potentials i derived from the economic entropy, which is now U. Then (Lukács, 1989)

(6.1) i = i(Nk/M,P) = i(Bk)

with some integrability conditions for the cross derivatives which will not be discussed here.

But spiritual goods as language ability, car driving ability, &c. have market prices when learning, teaching and using them. Then spiritual goods must be incorporated among Bi.

This is only a technical problem (serious). However the next problem is fundamental. Some market prices show serious seasonal changes, say tomato. This is so because in wintertime tomato must be grown in heated greenhouses and the fuel price goes into the tomato price. Summertime the greenhouse is heated by Sun, freely.

Then one may guess via eq. (6.1) that U=U(Bi,t). However this form is forbidden. We are in economic thermodynamics. U is a thermodynamic potential (isomorphic to entropy), and a thermodynamical potential must depend on thermodynamic extensives or intensives (U solely on extensives) and nothing else at all. If the potential is explicitly time-dependent, then the Gibbs-Duhem relation does not hold and the description is not thermodynamics at all (Diósi, Lukács, Martinás & Paál 1986). If one believes in economic thermodynamics as a description, then something instead of the solar energy flux must be introduced as an extensive variable.

This is possible, but the environmental conditions generally influence the prices. On fields polluted by acidic rains (or natron) yields are lower, so food prices go up. Therefore the variables CI do influence the prices i. But they should not, because in the thermodynamic formalism i builds up from derivatives of U(Bi). We have arrived at a fundamental contradiction with economic thermodynamics.

The problem exists backwards too. S(CI) measures our discomfort in an environment. Now, well fed and generally healthy individuals feel lower discomfort at the same pollution. Ad absurdum, whealty individuals can wear gas masks, and then they survive in the same environment where Rousseau's Noble Savage dies in ten minutes. Therefore S should depend on Bi, but according to the construction it does not.

Therefore economic thermodynamics does not describe the real economies. It may describe the separated economic sector of an ideal Free Enterprise society without taxes &c., with a continuously rejuvenating environment and with a planetary rotation axis orthogonal to the orbital plane.

However some shortcomings could be repaired by using F instead of U. Since F is a combination of U and S, obviously

(6.2) F = F(Bi,CI)

Now, by using F as thermodynamic potential, i will build up from derivatives of F; therefore

(6.3) i = i(Bk,CK)

Then prices of goods can depend on environmental conditions (as real market prices do).

One may (or may not) believe that F depends on its variables via the two partial potentials approximately, as

(6.4) F ~ F(U(Bi),S(CI))

to express the relative autonomies of economy and ecology. At the present stage of art this question is open. But we know that F~U-S cannot hold. An additive form would decompose in the derivatives, and then market prices would not depend on environmental characteristics.

Some appropriate F(Bi,CI) potential may describe the purposes of a Free Enterprise society (without taxes). But then the description is not thermodynamics of economy, but thermodynamics of something integrated from economy and ecology.


Assume that a team of congenial economists, ecologists and sociologists has found the F function of a society. We will not use any specific property of F (which are unknown anyway) except that generally it is increasing with U and decreasing with S. Then there remains to calculate the optimal path.

Now, there are some conservation laws for the quantities Bi and CI. For example physical conservation laws are valid, and chemistry sets constraints too until the technology is constant. For example, consider iron industry. We produce iron in a furnace from average iron ore. Then the process uses up:

money, M=B1;

iron ore, B2;

coal, B3;

oxygen, C101

and produces

iron, B4

carbon dioxide, C102

sulphur dioxide (+ trioxide), C103

slag, B5

(where we started to number CI from 101, for convenience). Among the rates the obvious proportionalities

dB2/dt = -(1/a)(dB4/dt)

dB3/dt = -b(dB4/dt)

dB5/dt = c(dB4/dt)

(7.1) dM/dt = -q(dB4/dt)

dC102/dt = -(dB2/dt)

dC103/dt = w(dC102/dt)

dC101/dt = -dC102/dt - dC103/dt

Here a is the Fe concentration of the ore, b, c and q are technology-dependent (b being in the order of unity) and w is the sulphur content of the coal. For moderate times the technology-dependent factors are constant, and tricks can be made by new technologies but then q often increases.

Now, F is an increasing function of B1 (more money helps), of B3 (more coal helps), of B4 (more iron helps), maybe of C101 (oxygen is necessary); a decreasing function of C103 (sulphuric acids are unhealthy) and maybe of C102 (if CO2 is really causing global warming); and depends slowly on B2 (iron ore is not used for anything else) and of B5 (slag is not too valuable). So during iron melting

(7.2) dF ~ {-qF,1 - bF,3 + F,4 - (1/a)(1+w)F,101 +

+ (1/a)F,102 + (1/a)wF,103}*(dB4/dt)

Therefore dF is proportional with the rate of the industrial process, and is a sum of 1 positive and 5 negative terms.

A priori, then, one would guess that iron industry is a general loss. Still iron possesses a finite, positive market price. So there are technologies producing iron with increase of F, i.e. technologies, by means of which the society is (feels to be) going to flower more and more. The only criterion is that the third term in the bracket must overbalance the others.

With bad technologies it cannot overbalance them. And remember that F is an entropy-type quantity, so its second derivative matrix must be negative semidefinite in a stable society in the same way as that of the entropy is negative semidefinite in a stable thermodynamic system (Kirschner 1970). So with all other quantities fixed F is slower than linear in B4. Therefore (with all the others fixed) F,4 is decreasing with B4 and from a value cannot overbalance the negative terms. Technologically developed countries generally decrease their iron production (and start to buy iron from abroad). But this is done only when they already have much iron.


The present treatment is still not quantitative because nobody measured a reliable F(Bi,CI) anywhere. (But not any U utility function, either.) Still in some parts it could be made quantitative. Observe that at the moment when a society stops with iron industry, on the right hand side of eq. (7.2) some "ecologic" derivatives just balance some "economic" ones which latter ones can be expressed via known market prices. So "ecologic prices" could be measured too, in a methodical observation process.

But even at the present state of art some conclusions can be drawn. They are listed as follows.

1) No thermodynamics of economy exists. The utility function U cannot be a thermodynamic potential within economy; it depends on ecologic variables too.

2) The utility function U is characteristic to the actual society and cannot be based on "objective" natural sciences.

3) The dirt function S of ecology cannot be defined "for Nature", only for a community using Nature. Then the most natural is to determine it for Man, and in details, just for us.

4) There is no thermodynamics of ecology either, because S depends on economic variables too.

5) Still unified economy+ecology (nameless) may possess a potential function F for which some extremum principle holds, i.e. which is irreversibly growing in rational societies. Then this F is roughly increasing with U and decreasing with S, but cannot be a pure difference.

6) Because in most processes source terms of economic and ecologic variables are coupled, it is generally impossible to act in such a way that both economic utility grow and the environment improve. (See eq. (7.2).)

7) The society feels such changes desirable in which economic gain overbalances ecologic deterioration or ecologic improvement overbalances economic losses. If the function F exists, the partial losses and gains are comparable, because they automatically add up in dF.

8) The society is stable if the second derivative matrix of F is negative semidefinite. Then F is a function slower than linear in its variables. Therefore a poor society (U is low) in a very good environment (S is low) will like industrial development of practically any type, while a rich community in a dirty environment will be very choosey. Hungary is somewhere in between.


The author acknowledges earlier illuminating discussions with Drs. K. Martinás and G. Paál. A remark of Beatrix Paál called the author's attention on the intimate connection of dirt and entropy, and on the negative contribution of any of them to utility, which in itself is isomorphic with entropy.


At the end of Sect. 3 we interrupted the discussion of the degree K of the Pfaffian form of our economy. That interruption had been wise, because afterwards it turned out that no thermodynamics is possible for the economy itself separated from ecology; and a nonexistent description has no Pfaffian form either.

Still we can ask, what would be K if thermodynamics of economy were to exist. To answer this let us see once more why it does not exist. There are at least two good reasons as follows.

1) Prices depend on ecologic variables (acidity of rains, CO2 content of air, &c.).

2) Prices depend on astronomical events. The main influence comes from the actual position of Earth on the ecliptic (simpler, the season). But this is not the only astronomic influence. English price index seems to correlate with the sunspot number above 2s (Lukács 1991), and according to the Hudson Bay Company records especially selling, and so prices of fox and lynx furs are significally correlated to that; most probably due to better nocturnal hunting conditions with polar light (Gamow, 1967).

Now, Problem 1) may be neglected for short times, and Problem 2) can be ignored by taking annual averages. It is hard to say what will be the validity of a description using annual data for a few years, but then we can discuss the situation. It remains obscure if that averaged economy has a thermodynamic description, but let us proceed. Local irreversibilities from raw material to endproduct are seen in each production process. Is global irreversibility seen as expected for K=2?

No; and 3 counterarguments will be mentioned very briefly below.

A) The number of goods Bi is not necessarily finite, newer and newer goods, material and spiritual, appear continuously. Thermodynamic formalism needs a fixed state space. This difficulty will be soon overcome because biology of early Earth needs a description when evolution meant competition of different DNA molecules, virtually infinite for degree of freedom (Eigen 1971).

B) Inflation means that the "natural" measure of B1ºM is time-dependent. We do not know if this is a wrong definition for M or an explicit time dependence in U; the latter would be incompatible with thermodynamics (Lukács 1989; Diósi, Lukács, Martinás & Paál 1986).

C) It seems that the tax system restricts the dimensionality of the subspace of reversible economic processes. We give the details only in this point.

Consider physical thermodynamics (K=2); all other K=2 thermodynamics would be isomorphic with it. There in the canonical Gibbsian form the energy is the thermodynamic potential, its variables are the extensives, of which entropy S is one:

E = E(S,X2,...) i = 2,...,N (A.1)

On the other hand

dQ = TdS (A.2)

So any change is reversible in which S is constant, which is a hypersurface of N-1 dimensions in the state space of N dimensions. If the difference between the dimensionality of the state space and the reversible subspace is greater than 1, then the form (A.2) is impossible and so K>2. Now we are going to count the dimensions.

1) The number of dimensions of the state space is N; we assume it to be finite and fixed to have a chance to have thermodynamics.

2) At least one dimension is needed for production which cannot be substituted by pure commerce (Bródy, Martinás & Sajó 1986). Then the maximal dimensionality of pure commerce is N-1; we assume that value otherwise K>2 and economic thermodynamics either does not exist or not isomorphic to physical thermodynamics at all.

3) Now, contemporary tax systems are not neutral. Some goods (rum, tobacco, &c.) are burdened by extra taxes. Consider two persons. A possesses raw iron, B rum. They sell each other and back. That is pure commerce. But some part of the rum price went to the treasury, not to B, so at the end the initial state is not reproduced. Some pure commerce is irreversible.

4) Then the dimensionality of the subspace of reversible processes is <(N-1), so dQ=TdS is impossible, therefore K>2.

Not all tax systems cause K>2. The "minimal state" of liberal theoreticians maintains order and law, needs money for it, but law and order is needed for production, so it is a normal cost of economic activity. A small VAT of universal percentage or a head tax does not change the direction of processes and therefore does not alter K. Taxes used to enforce scientific nourishment do alter it.

While specific taxes on windows, chimneys, tea or entertainment were use in the past as far as records exist, they mainly avoided the sphere of economy. Contemporary states organise, influence or orientate the economic activities by extra taxes and endowments. This influence pervades the economy, so can alter the irreversibility of processes.

Now consider a K=3 case. Then the increment of heat gets the canonical form

dQ = dZ(Bi) + T(Bi)dS(Bi) (A.3)

(For such a hypothetical physical system see Lukács & Paál (1989).) The irreversibility principle is dQ>0. But hence dZ>0 and dS>0 does not follow. There are two competing potentials of which at least one has to grow but anything is permitted if dZ+TdS³0. If competing "principles" (e.g. profit vs. equality) are acting then K>2.

And, reversing the proofs of Planck (1934) or Hausen (1934) in a K>2 economy economic perpetua mobilia are possible. So, ad absurdum, even closed cycles are possible in which every step is desirable.

It is quite possible that for the Pfaffian of coupled economy + ecology K>2 too (when cyclic building and dismantling of power plants is rational and desirable). However for this we have no evidence at the present status of knowledge.


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