- ENERGY, ENTROPY - Is Loschmidt's greatest discovery still waiting for its discovery?

Andreas Trupp

The previous version of this article including several figures (that differs substantially from the actual version you are looking at) was first published in Physics Essays , Vol. 12 No. 4, December 1999, pp. 614-628. A reprint of that article can be found in Infinite Energy Magazine, Issue 43 (May-June 2002), pp.13-25.

You might want to take a look at Mr. R. Graeff's page in which he discloses details on his successful experiments on the gradation of temperature in well-insulated columns of gas and, with especially impressive results, of liquids subject to earth's gravity. Similar experiments were performed by Chuanping Liao , “Temperature Gradient Caused by Gravitation”, International Journal of Modern Physics B, Vol. 23, No. 22 (2009), pp. 4685-4696 .

I had the pleasure of giving a short talk at the "First International Conference on the Quantum Limits to the Second Law" that took place at the University of San Diego (July 29-31, 2002). I discussed possible Second Law violations in the field of electrostatics , and also possible Second Law violations when it comes to gases that obey Van-der-Waals’s law (vapors).

Comments welcome !

Abstract: In 1868 J.C. Maxwell proved that a perpetual motion machine of the second kind would become possible, if the equilibrium temperature in a vertical column of gas subject to gravity were a function of height. However, Maxwell had claimed that the temperature had to be the same at all points of the column. So did Boltzmann. Their opponent was Loschmidt, who died more than 100 years ago in 1895. He claimed that the equilibrium temperature declined with height, and that a perpetual motion machine of the second kind operating by means of such column was compatible with the second law of thermodynamics. Thus he was convinced he had detected a never ending source of usable energy for mankind. In this article, proof is given for the hypothesis that the equilibrium temperature is indeed a function of height, based on the "hydrostatic" equation of equilibrium, and on the General Gas Law. Moreover, special attention is drawn to the atmosphere of Venus, the temperature gradation of which might be capable of corroborating Loschmidt's thesis in a much better way than lab experiments on planet Earth could.

1) Introduction

2) On the history of the second law of thermodynamics

3) Maxwell's (hypothetical) perpetual motion machine of the second kind

4) A proof of the stratification of temperature in gases subject to gravity

5) A late completion of Boltzmann's homage paid to Loschmidt; Consequences For The Nature of Time


1) Introduction

When taking a look at any textbook on general physics, one finds the second law of thermodynamics formulated in two equivalent ways: "The total entropy of an isolated system can never decrease", and "A perpetual motion machine of the second kind is impossible". Today, doubting the impossibility of such a machine is just as inconceivable as is the assertion that a perpetual motion machine of the FIRST kind might exist, which creates energy from nothing. A closer investigation, however, reveals that in the second half of the nineteenth century, a vivid debate was held among most reputable scientists on the possibility of a perpetual motion machine of the second kind. One of the most prominent propagators in favor of such a possibility was Josef Loschmidt, a name today well known even to school-kids through the celebrated Loschmidt's number. A perpetual motion machine of the second kind would be capable of permanently creating, in a cycle, mechanical energy from just one single reservoir of heat. Thus it would become possible, for instance, to convert the (dissipated) energy of heat contained in the air into mechanical energy without requiring a second, colder reservoir for the absorbtion of the refuse heat. Therefore mankind would have available a source of energy that practically cannot be exhausted.

Loschmidt and Maxwell asserted that, if there were a stratification of temperature in a column of gas subject to gravity, the construction of a perpetual motion machine of the second kind would be possible. Up to the present day, no one has ever challenged that assertion. Maxwell believed that the temperature of the gas subject to gravity could not be stratified, but had to be the same at all points. He did not provide a special proof; rather, he intuitively extended his formula of velocities of molecules (which had been derived without regarding gravity) to a gas subject to gravity. Boltzmann sided with Maxwell; in contrast to Maxwell, he attempted to prove that the homogeneous temperature of a gas subject to gravity was ensured by the kinetic theory of gases. Loschmidt, however, was convinced that a perpetual motion machine of the second kind was compatible with the second law of thermodynamics. In that point, he disagreed with Clausius, Thomson, Boltzmann, and Maxwell. In particular, he believed that a perpetual motion machine of the second kind could be operated by means of a vertical column of gas, the temperature of which he claimed to be stratified.

In the 20th century, Loschmidt's "revolutionary" assertion has hardly been paid any attention. After all, it was mentioned by Stephen G. Brush in his 1978 book: "The Temperature of History." However, Brush does not give more than a clue when telling his readers that the dispute over the stratification of temperature between Boltzmann and Loschmidt provided a contribution to the debate on the second law of thermodynamics. No further details are offered.

In recent times (1995), it was Claude Garrod 1)who tried to give a new proof of the uniformity of temperature. His arguments will be scrutinized further below.

It should be noted that the Second Law, when understood as the assertion that a perpetual motion machine of the second kind cannot be built, is not subject to possible falsification by observing nature, but by observing inventors. As a consequence, the "fruitlessness" of any efforts in building a perpetual motion machine of the second kind could either be due to the incompetence of inventors, or to nature itself not allowing the construction of such a machine. The "fruitlessness" alone does not provide any means for deciding which of these two alternatives is true.

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2) On the history of the second law of thermodynamics

Though the expression "perpetual motion machine of the second kind" was only introduced by Ostwald towards the end of the 19th century, the impossibility of such machine had been postulated as an axiom by Clausius already in 1849 and by Thomson in 1850. Both Clausius and Thomson are considered the discoverers of the second law of thermodynamics 1a) .

One may wonder how Clausius and Thomson could obtain their firm belief in the truth of their axiom. The fact alone that such a machine had not been invented until those days is not capable of explaining this conviction. In addition, one has to take into account that already in the 18th century the opinion of the impossibility of a perpetual motion machine prevailed, long before the theorem of the conservation of energy or the distinction between the first and the second law of thermodynamics were advanced. The idea of a system of movable parts that, having come to rest once, would still be able to get into motion on its own, was simply inconceivable 2) . Of course, the 18th century scientists were not yet familiar with the kinetic theory of heat and did not realize that apparent rest turns into motion in the microscopic perspective. Such knowledge would have impeded the formation of Clausius' and Thomson's axiom.

Picking up reflections previously published by Carnot, Thomson declared the impossibility of a perpetual motion machine the foundation of his further investigations in the field of thermodynamics: "It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest surrounding objects" 3) . From the impossibility of such a perpetual motion machine one can easily infer the second law in a very general form, stating that a system will not depart from an attained state of equilibrium without interference from outside. Loschmidt accepted the second law in such a general form only (more precisely: he believed that the second law could be derived from the mechanical principle of least action). However, as emphasized by Loschmidt several times, it is impossible to invert the order of inference, i.e. it is not permitted to infer the impossibility of a perpetual motion machine of the second kind from the second law in its very general form:

"From these reflections one can draw the conclusion that the second law of thermodynamics can be inferred from the axiom of Clausius 'It is impossible to transfer heat from a colder to a warmer body without compensation', or from the equivalent one of W. Thomson 'It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects', that the inversion of that inference, however, is not permissible, because the content of the second law is more general than that of those axioms." 4)

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3) Maxwell's (hypothetical) perpetual motion machine of the second kind

In the 1860's and 70's scientists dealt with the question of whether a gas, which is subject to gravity in an insulated column, had the same temperature at all points, or whether its temperature was a function of height.

Maxwell was convinced that the temperature of a gas subject to gravity had to be uniform at all heights. Moreover, like Thomson and Clausius, he regarded a perpetual motion machine of the second kind to be impossible. However, he was very sure that if there were a gradation of temperature and if that gradation were different for different substances, a perpetual motion machine would be possible:

"In fact, if the temperature of any substance, when in thermic equilibrium, is a function of the height, that of any other substance must be the same function of the height. For if not, let equal columns of the two substances be enclosed in cylinders impermeable to heat, and put in thermal communication at the bottom. If, when in thermal equilibrium, the tops of the two columns are at different temperatures, an engine might be worked by taking heat from the hotter and giving it up to the cooler, and the refuse heat would circulate round the system till it was all converted into mechanical energy, which is a contradiction to the second law of thermodynamics. The result as now given is, that temperature in gases, when in thermal equilibrium, is independent of height, and it follows from what has been said that temperature is independent of height in all other substances." 5)

Modifying the device introduced by Maxwell, we put up with one column (filled with gas) only, which is thermally isolated from its surroundings. A vertical pipe is arranged in a way that it penetrates the entire column from its bottom to its top. The material of the pipe consists of small sections permeable to heat; each of such sections is followed by a section that is not permeable to heat. So the number of permeable sections equals the number of impermeable ones. In the interior of that vertical pipe, a piece of metal (of cylinder shape) suspended by a rope is allowed to move up and down. To start with, that metal piece has the temperature of the upper part of the column. Initially (before the device begins its operation), that temperature is also the temperature of the ambient. When being slowly lowered to the floor, the metal piece adopts the temperature of the gas surrounding it; by gathering tiny bits of heat at different heights where it is in thermal contact with the permeable sections. Thus it becomes hotter (without disturbing the phenomenon of a temperature gradation of the gas as such).

Finally it has reached the bottom (and the maximum temperature). Now it is towed back to the top in such a fast way that the temperature of the piece of metal has no chance of declining. In order to move the piece of metal up and down, no net work has to be spent: The expenditure of work when making it move upwards it completely compensated by the gain in work when it is lowered to the bottom. The hot metal piece, however, can make water boil, and by the steam thus created a steam engine can be run. The refuse heat of that engine flows into the upper part of the column (not into the ambient).

One might be tempted to assume that the process of creating work has to come to a standstill as soon as the temperature of the column has reached uniformity (due to the extraction of heat at its bottom and the adding of refuse heat at its top). However, the starting point of our (and Maxwell's) reflections was the hypothetical assumption (which Maxwell did not believe to be true in reality) that a uniform temperature of a column of gas subject to gravity is NOT A STATE OF EQUILIBRIUM. Hence we have to conclude that the gas, left to itself during that break, will resume its state of temperature gradation. Then, the whole process can start again. The internal energy of the gas as a whole will thus be diminished and turned into mechanical work without a second heat reservoir.

Strictly speaking, Maxwell's original device only demonstrates that DIFFERING temperature gradients of two substances enable the construction of a perpetual motion machine of the second kind, whereas the modification gives proof of the possibility of such machine already in the event of a temperature gradation existent in one single substance, that is in case of a temperature gradation as such (see W. Dreyer, W. Müller, W. Weiss, Tales of Thermodynamics and Obscure Applications of the Second Law , for a brilliant report on the debate between Boltzmann and Loschmidt).

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4) A proof of the stratification of temperature in gases subject to gravity

a) Different from some simple gas models in which no collisions of molecules take place, the real molecules collide with each other every tiny fraction of a second. They also collide with the walls and the bottom of the container. Imagine a tall gas container in a gravity field. The gas pressure on the ceiling shall be vanishably weak due to the very low density at this altitude. The pressure on the bottom, muliplied by the surface area of the bottom, is identical with the weight of all gas molecules, though the vast majority of the molecules find themselves, at any time, well ABOVE the bottom in free “flight" or in collisions with other molecules. This identity of weight and pressure is due to the fact that the total weight of the container (including its contents) cannot depend on whether the molecules form a gas in equilibrium, or a frozen solid body instead (the weight of which would rest on the bottom of the container). Otherwise, energy could be gained from nothing, just by moving the container up and down in an elevator (at a constant velocity), with the heavier state made to materialize during the downward motion, and with the lighter state made to materialize in the upward motion.

Hence, in a state of equilibrium of the gas, the sum of all changes in vertical momentum of those molecules that are being reflected from the bottom of the container, must equal the weight of all molecules. The change in momentum can be brought about by elastic reflections, inelastic reflections, or by a complete trapping of a molecule, followed by its release thereafter. That is to say:


(Sum DELTA p)/DELTA t = Nmg

with m being the mass of a single molecule, p being the vertical momentum of a molecule (which is reversed by the elastic reflection on the bottom), N being the total number of gas molecules in the container, g being the gravitational acceleration.

b) A similar equality of weight and momentum change holds true for a single molecule in the gas when undergoing collisions with other molecules . Gains in vertical momentum of the molecule in an upward (positive) direction will, to a very large extent, be neutralized by gains in vertical momentum in a downward (negative) direction. Nevertheless, the positive gains prevail over the negative ones, with the total sum of all changes (positive and negative) in momentum during a longer period of time, divided by that time interval, approaching the weight mg of the molecule (the prevailing of the positive gains is accounted for by the existence of a vertical pressure gradient in the gas; “bumps" from below are thus slightly more frequent than “bumps" from above, given the diameter of the "suffering" molecule is not vanishing, which of course would also make collisions impossible). Otherwise, that is, if the sum of all changes in vertical momentum of the molecule, divided by the long temporal interval considered, were not equal with the weight mg of the molecule, the molecule - when looked at over a longer period of time - would find itself in an overall accelerated motion, either upward or downward. The same would then be true for all other molecules, since all the molecules of the gas behave in the same way, if only the intervals of time considered are long enough. However, such an overall motion of the gas would not be consistent with the presumed state of equilibrium, in which the gas is supposed to be. When a gas is in equilibrium, a molecule is levitated (performing small irregular displacements only), as if gravity did not exist.

c) Let us now take a closer look at our gas in the very tall column. Let the ground plate be in thermal equilibrium with the ambient. As a consequence, both the equilibrium pressure of the gas at the bottom (which is equal to the total mass of the gas times g divided by the sectional area of the column), and its equilibrium temperature at the bottom are given, whatever the state of equilibrium of the column as a whole might be like.

According to Dalton' law, the gas at the bottom, which we suppose to exert a pressure of 1 bar, is nothing but a mixture of ten different gases of the same kind, each of which is exerting a pressure of 0.1 bar. Each of these ten "sub-gases" is thus exerting a lifting force of 0.1 mg on an arbitrarily chosen, single molecule (located just above the ground plate), as the molecules of each sub-gas are undergoing random collisions with that molecule (with the net momentum transfer per second resulting in a permanent force exerted on that molecule), so that the total lifting force, exerted by all ten sub-gases together, amounts to mg .

Assuming that the temperature is uniform throughout the column, a molecule at a higher altitude - where the pressure is 0.1 bar - is faced with the following situation: The gas exerts a lifting force of just 0.1 mg on the molecule. It is as if only one sub-gas were present at this altitude, thus counteracting no more than one tenth of its weight. Hence, it cannot stay at its momentary location. This proves that the gas, when at uniform temperature, cannot be in a state of equilibrium. Quod erat demonstrandum .

d) For an equilibrium to exist, the pressure gradient would have to be linear throughout the gas, instead. Only then would a single molecule, whose "thickness" and size (as a "collision target" for other molecules) do not change with height, be levitated independently of its position as a result of the many collisions it is undergoing. Since a linear pressure gradient would be accompanied by a steep decline in temperature with height, intrinsic convection would, in general, set in as soon as the adiabatic gradient is reached, thus preventing a linear pressure gradient to materialize.

e) The decline in temperature with height in a hypothetical state of equilibrium is thus much steeper than the adiabatic temperature gradient, which a parcel of air rising in the atmosphere - being only slightly hotter than the ambient air at any time - is subject to. However, as mentioned already, convection would set in as soon as the gradient is tending to exceed the adiabatic lapse rate (not mentioning heat radiation which sets in as soon as there is gradient as such), and would hence not allow the materialization of a temperature gradient steeper than the adiabatic one.

The uniform density of the gas is hence a state that a gas will never achieve. Instead, the adiabatic lapse rate will be a limit. In this respect, there is some resemblance to the common view according to which convection processes are blamed for the fact that the state of uniform temperature, though supposed to be a state of equilibrium, is never reached in the atmosphere.

Hence, the atmosphere is in restless motion, even if not disturbed from outside. Convection processes thus play, at large scale, the part that Brownian motion plays at small scale.

It is important to recognize that the role of mutual collisions of the molecules is crucial for the phenomenon of temperature gradation. Without these collisions, no levitation of molecules could exist, and the temperature of the model gas would indeed be uniform, as has been shown by several authors
. This is how, in recent times, Walton 6) and also Garrod 1) proved the uniformity of temperature in a vertical column of gas subject to gravity, when, in their models, they assumed that no mutual collisions of the molecules would take place, so that the gain in height would have to be at the expense of the vertical component of velocity only (the two lateral components of motion are not affected by a collisionless rise of molecules taking place in those model gases). See also, for a similar proof, F.L. Roman, J.A. White, S.Velasco, Microcanonical single-particle distributions for an ideal gas in a gravitational field, Eur. J. Phys., vol. 16 -1995-, pp. 83-90, additional remarks in Eur.J. Phys., vol. 17 -1996-, pp. 43-44; Charles A. Coombes, Hans Laue, A paradox concerning the temperature distribution of a gas in a gravitational field, Am. J. Phys., vol. 53 -1985-, pp. 272-273.

The temperature gradation in the troposphere of Venus is an empirical proof of Loschmidt's thesis (besides the experimental results found by Graeff and Liao). Venus has an atmosphere mainly of carbon dioxide, with a pressure of 93 bar at the ground. The temperature at the ground is uniform at 740 K, day and night, in summer and in winter, at the equator, and in the polar regions. The uniformity of the temperature of the surface makes convection processes (other than those "internal" convections described in the foregoing) almost impossible. This is a striking difference compared to the lower atmosphere of planet Earth: As regards the troposphere of planet Earth, the almost adiabatic temperature gradient found there is commonly accounted for by the undisputed fact that temperature of the surface varies a lot. This is why “parcels" of hot air rise after being heated by thermal contact with warm spots on the ground, while other "parcels" of air sink downward where the ground is cooler. By this circulating convection process, the air in the lower atmosphere is commonly assumed to be prevented from reaching its equilibrium state which is believed to be of uniform temperature.

If this belief were true, the temperature of the lower atmosphere of Venus should be much closer to uniformity than the lower atmosphere of planet Earth. But this is not the case. The temperature gradient in the atmosphere of Venus is 80 percent of the adiabatic gradient, similiar to the situation in the troposphere of planet Earth (heat radiation may easily explain why the gradient does not reach 100 percent of the adiabatic gradient). Over 60 km in altitude, the lower atmosphere of Venus loses almost 500 degree K at a constant rate. The troposphere of Venus can be supposed to be closer to equilibrium than that of planet Earth is, but the actual state of the troposhere of Venus is not that of a uniform temperature, but a state of a stratification of temperature.

Moreover, it is commonly considered as highly puzzling how the atmosphere in the cloud region can rotate in about four earth days, whereas the solid planet Venus rotates in 243 earth days: " In spite of a great deal of theoretical effort and a number of specific suggestions, there is still no accepted mechanism for the basic motion of the Venus atmosphere, nor is it given convincingly in any numerical general circulation model. What is needed is to convert the slow apparent motion of the Sun (relative to a fixed point on Venus) into a much more rapid motion of the atmosphere ." (Encyclopedia of the Solar System, edited by P.R. Weissman et al, 1999, p. 158.) The internal convection described in the foregoing could provide the answer.

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5) A late completion of Boltzmann's homage paid to Loschmidt; Consequences For The Nature of Time

When Josef Loschmidt died in 1895, Boltzmann held a memorial speech addressed to the Chemical-Physical Society of Vienna on the 29th of October, 1895. On this occasion, he rated the computation of the number of molecules contained in a unit volume to be Loschmidt's greatest discovery 12) . Such a rating must be contradicted. A discovery at least equivalent to the one mentioned by Boltzmann is the compatibility of the second law of thermodynamics and the perpetual motion machine of the second kind. If Loschmidt's discovery of that compatibility had been widely accepted in those days, the evolution of energy technology might have been a different one. Unfortunately, Loschmidt's arguments in favor of the stratification of temperature in a gas subject to gravity do not provide a strict proof. With a strict proof at hand, he could have spread his thesis with a greater psychological effect.

Furthermore, Loschmidt pointed out that the second law could be derived from the principle of least action. To put it differently: he replaced the original foundation of the second law (that is the axiom of the impossibility of a perpetual motion machine of the second kind) by a different one. Doing so he referred to Boltzmann, who had already displayed such foundation in his article "Über die mechanische Bedeutung des 2. Hauptsatzes der Wärmetheorie" . It can be left undecided whether or not the derivation of the second law from the principle of least action is strictly convincing. In a recent article, G. Bierhalter, who has published several articles on the history of the second law, doubted the strictness of such reasoning. 13) . In any case, the second law and a perpetual motion machine of the second kind are compatible, as soon as we no longer define the second law as it has been usual. In that common form the second law asserts that for a quantity of heat to pass from one body to another, the temperature of the receiving body must be lower than that of the donating body. Rather, one should formulate the second law as the assertion that heat does not pass from one body to another if such a passage would destroy a state of equilibrium. A state of equilibrium, in turn, is a state in which the net flow of heat has become zero. To put it differently: If the flow of heat between two bodies has come to a standstill, that flow will not start again by itself, but only by interference. In most cases, the two bodies will then have the same temperature; in some cases, however, the temperatures of the two bodies will be different from each other. Of course, this will deprive entropy of its character as a variable of state, and the Second Law could no longer be expressed by applying the notion of entropy.

Defining the Second Law the way just proposed is almost identical with the definition of the word “equilibrium". Nevertheless, the consequences of an adoption of that definition are enormous: The fact that any difference in density and temperature within a quantity of gas (large or small) not subject to gravity will vanish after a while, has to be ascribed to the original state of matter (in the visible universe), to which our present state is causally linked, not to the Second Law. In that respect, there is a resemblance between temperature and gravity: The fact that water in a cup, when stirred, will climb the walls of the cup as a result of the centrifugal “force" at work, is -as General Relativity tells us- a result of the special way the distant stellar masses of the universe are distributed. In much the same way, the fact that cold water, when added to a cup containing hot coffee, will mix with the coffee to form a liquid of uniform temperature, is a result of the state of matter in the universe billions of years ago. Or as S.M. Caroll (“The Cosmic Origins of Time’s Arrow”, Scientific American, June 2008, page 26) puts it:

 “The universe started off orderly and has been getting increasingly disorderly ever since. The asymmetry of time ... plays an unmistakable role in our everyday lives: it accounts for why we cannot turn an omelet into an egg... And the origin of the asymmetry we experience can be traced all the way back to the orderliness of the universe near the big bang. Every time you break an egg, you are doing oberservational cosmology.”

The often discussed paradox, that is the question why temperature differences within an ideal gas will always vanish though all motions of the molecules are reversible (so that increases in temperature differences should be as frequent as reductions of these differences) is thereby resolved: The initial state of the gas to start with (which itself is causally dependent on prior states of things) isn't of the right kind for generating temperature differences.

Moreover, this recognition gives rise to revisit Boltzmann's famous dispute with Zermelo. In a universe endless both in time and in space, he argues, there must exist “islands" in which, by random processes, matter is organized, whereas the universe is barren and at uniform temperature elsewhere. Living beings (including intelligent machines) on such an island will define the arrow of time by saying that the future is the less organized state, while the past is the more organized state of their island (there is no physical definition of the arrow of time other than this one, since the laws of physics are time-symmetric). Later on, this concept of time was consolidated by Hans Reichenbach (“The Direction of Time"), who stressed that in those ordered states, the system is not free to find itself (as a result of random processes) in a much different state soon after or slightly prior to the moment in time considered (when the system is highly ordered), as laws of nature allow only slight changes within short periods of time. Different from dice or roulette balls, every gas has a memory.

The universe is hence in possession of different states, but is lacking of an intrinsic ordering of these states by the category earlier / later. Instead, such an ordering is ex trinsic. It seems that Boltzmann's view of the arrow of time is quite correct despite the fact that cosmology, by assuming the Big-Bang at the “origin" of the universe, may be dismissing the assumption of a universe endless both in space and time.

Finally, for reasons of clarity, the following reflection should be underlined: The dismissal of an intrinsic ordering of earlier / later follows from the statististical concept of the Second Law already. Reichenbach has clearly stressed that point. In other words: For such a view to be adopted, it is not necessary to allow the construction of a perpetual motion machine of the second kind. The presumed impossibility of a perpetual motion machine of the second kind could be accounted for by the assumption that the “initial" conditions of matter were such that its operation is excluded for all times, or at least for eons of time, as the causal chains, given those initial conditions, could never lead to such a phenomenon. The benefit of a perpetual motion machine of the second kind, in this context, lies in the fact that it makes the extrinsic character of an ordering earlier / later more evident.

It is hardly known that Ernst Mach, too, was skeptical as regards the reach of the Second Law. He objected to the generalization of the original, technical concept of entropy, that is the amount of heat received or given off by a body divided by its temperature, and was skeptical that an increase in technical entropy could be paralleled with an increase in disordered motions of particles. In chapter 102 of his "Principles of the Theory of Heat", he wrote (my own translation from German): "The mechanical view of the Second Law, which distinguishes ordered and disordered motions by paralleling the increase in entropy with the increase in disordered motions at the expense of ordered motions, appears to be quite artificial. Taking into account that a real analogue of the increase in entropy does not exist in a purely mechanical system made up of perfectly elastic atoms, one can hardly reject the idea that an infringement of the Second Law should be quite possible -even without any help from demons- , given such a mechanical system were indeed the basis of the heat phenomena."

Loschmidt, on his part, had the following vision for the future: "Thereby the terroristic nimbus of the second law is destroyed, a nimbus which makes that second law appear as the annihilating principle of all life in the universe, and at the same time we are confronted with the comforting perspective that, as far as the conversion of heat into work is concerned, mankind will not solely be dependent on the intervention of coal or of the sun, but will have available an inexhaustable resource of convertible heat at all times" 15) .

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1) Claude Garrod: Statistical mechanics and thermodynamics, Oxford University Press, 1995

1a) See also: R. Baierlein, "How Entropy got its Name", American Journal of Physics, 60, 1151.

2) See E. Pertigen: Der Teufel in der Physik - Eine Kulturgeschichte des Perpetuum Mobile, (Berlin: Verlag für Reisen und Wissenschaft 1988).

3) Transactions of the Royal Society of Edinburgh 20 (1851), 265.

4) J. Loschmidt, "Über den Zustand des Wärmegleichgewichts eines Systems von Körpern mit Rücksicht auf die Schwerkraft I", Sitzungsberichte der mathematisch - naturwissenschaftlichen Classe der Kaiserlichen Akademie der Wissenschaften zu Wien 73.2 (1876), 135.

5) J.C. Maxwell, "On the Dynamical Theory of Gases", The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 35 (1868), 215/216. Some years later, Maxwell repeated his argument. In his book "Theory of heat", published in London in 1877, he writes (p. 320): "...if two vertical columns of different substances stand on the same perfectly conducting horizontal plate, the temperature of the bottom of each column will be the same; and if each column is in thermal equilibrium of itself, the temperatures at all equal heights must be the same. In fact, if the temperatures of the tops of the two columns were different, we might drive an engine with this difference of temperature, and the refuse heat would pass down the colder column, through the conducting plate, and up the warmer column; and this would go on till all the heat was converted into work, contrary to the second law of thermodynamics. But we know that if one of the columns is gaseous, its temperature is uniform. Hence that of the other must be uniform, whatever its material." Thus Maxwell did not modify his assertion that if there were a temperature gradation in a column of gas subject to gravity, a perpetual motion machine of the second kind would become possible.

6) A.J. Walton, "Archimedes' Principle in Gases", in: Contemp. Phys., 1969, Vol. 10, No. 2

6) Loschmidt, "Über den Zustand des Wärmegleichgewichts...I ", p. 133.

7) See L. Boltzmann, "Über die Aufstellung und Integration von Gleichungen, welche die Molekularbewegung von Gasen bestimmen" in L. Boltzmann, Wissenschaftliche Abhandlungen, edited by F. Hasenöhrl, vol. 2 ( Leipzig: Barth 1909), p. 56ff.

8) J. Loschmidt, "Über den Zustand des Wärmegleichgewichts eines Systems von Körpern mit Rücksicht auf die Schwerkraft IV", Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften zu Wien 76.2 (1877), 225.

9) An argument similar to Boltzmann's can be found with S.H. Burbury, "Equilibrium of Temperature in a Vertical Column of Gas", Nature, Vol. 12 -1875-, p. 107.

10) L. Boltzmann: Lectures on Gas Theory, Dover Publ. 1964, par. 19, p.141

11) L. Boltzmann: Lectures on Gas Theory, Dover Publ., par. 15

12) See L. Boltzmann, "Zur Erinnerung an Josef Loschmidt", in L. Boltzmann, Populäre Schriften (Leipzig: Barth 1905).

13) G. Bierhalter, "Von L. Boltzmann bis J.J. Thomson: die Versuche einer mechanischen Grundlegung der Thermodynamik", Archive for the History of Exact Science 44 (1992), 25-72.

14) See Loschmidt, "Über den Zustand des Wärmegleichgewichts... I", p. 141.

15) Loschmidt, "Über den Zustand des Wärmegleichgewichts... I", p. 135.

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