I think yes. The study of the statistical weight of the energy of the atoms or that of the money allows to think so, see (Perturbations and Statistical Distribution of the Thermal Energy and Advance in Statistical thermodynamics)
I will be glad to have comment.
Dear Behnam: thanks for your well-argued answer. The aim of the question is to focus the attention on the fact that, with a mean value, in the both cases, as soon as an atom has a large energy or one inhabitant has a lot of money we need a counter part that is a lot of atoms with very few energy or people with very few money. This is simple to understand but it is not respected in the determination of the classical distribution of the thermal energy.
For systems in equilibrium, the probability P of occurrence of a macrostate comprised of N microstates and hence of entropy S = k ln N decreases exponentially with its negentropy delta S = Smax - S: P = e^[-(delta S)/k]. But the probability of occurrence of a macrostate ON EARTH is NOT NECESSARILY governed by P = e^[-(delta S)/k], because the disequilibrium between the heat source of the Sun and the heat sink of the 2.7 degree cosmic background radiation, and to a much smaller extent radioactive isotopes in Earth not yet decayed to nuclear equilibrium, continually forces Earth out of equilibrium. The economy, too, ultimately feeds on the these disequilibria --- otherwise there would be no economy. Thus we should not expect the probability of possessing money M to necessarily decrease exponentially with increasing M. In fact it decreases more slowly, typically a power-law decrease rather than an exponential one.
Dear Behnam: thanks for your instructive comments. I will try to go deeper in the understanding of the economy.
Dear Jack: Your comment with the entropy is important, but in this direction what I can underline is that mine approach leads to establish the important Boltzmann hypothesis S=klogW, see the perturbations and the Statistical Distribution of the thermal energy section 8.5. Kind Regards Xavier
Thermodynamic equilibrium implies that the probability that a small system in contact with a heat reservoir at temperature T is in an energy state E is the Boltzmann distribution P = e^(-E/kT), but the converse is not necessarily true. For example, a gas can manifest macroscopic circulatory motion and thus NOT be at thermodynamic equilibrium, yet the probability that any given gas molecule is in a state of energy E can still be given by the Boltzmann distribution. Likewise, economic thermodynamic equilibrium implies that the probability that a small system (an inhabitant) in contact with a money reservoir (the economy of a large country) will have money M decreases exponentially with increasing M, but the converse is not necessarily true. For the converse to be true in both the thermodynamic and economic cases, detailed balance must be obeyed. Detailed balance requires any motion of a gas molecule to be equally probable as its reverse and hence forbids macroscopic circulation in a gas. Detailed balance requires any economic transaction to be equally probable as its reverse. Clearly this doesn't occur in any economy: purchase of items from a store is much more probable than the reverse, selling items back to the store. The one-wayness of any economy is ultimately powered by, and exists only at the behest of, the one-wayness of energy flow from Sun to Earth to cosmic background radiation. Thus, while the distribution of M may more typically be a power-law decrease with increasing M, it can also be an exponential decrease.
Dear Jack: Many thanks for your recommendation of “Advances in Statistical Thermodynamics”. The introduction focus the attention on an important point concerning the heat “The heat is a notion for which a long time was needed to understand it well. One of the difficulties was our experience of with: the heat and the cold. I remember the emotion of a Indian friend of mine, explaining me his confusion discovering that a good welcome has to be warm in our country since in his country the Kerala region very hot in India has to be cold. In Europe the experience of different cultures leads to inverse the meaning of the same word cold which becomes chaleur in French, caldo in Italian but with the same origin. As a result, the understanding of this notion was a long travail of the researchers during the 19th century as underlined by Brush in his book « The Kind of Motion We Call Heat »”, Kind Regards Xavier
Thank you Xavier and Behnam for your very insightful comments. While many notions from statistical mechanics are not necessarily applicable in economics, economics is nonetheless of course related to thermodynamics and statistical mechanics, in that thermodynamics and statistical mechanics are more fundamental. Thermodynamic disequilibrium --- a system such as Earth maintained in an extremely improbable state statistical-mechanically --- is prerequisite for the existence of an economy --- but NOT vice versa.
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