Dear Sofia,

I think one can easily clarify the apparent conflict, but I have to write down some equations.

Please have a look into the appended pdf-file.

Reversing Loschmidt's argument, particles that have more kinetic energy tend to fly higher than particles with less kinetic energy. So they spend less time at lower heights. But by the time they reach higher altitudes, they lose kinetic energy. Intuitively, this is how I understood why the temperature is constant in a column of gas in thermal equilibrium in a gravitational field.

Here is a reasonably coherent discussion of the Maxwell-Boltzmann distribution in the presence of a gravitational potential: https://www.eng.fsu.edu/~dommelen/quantum/style_a/cboxmb.html

This is also a topic of interest for self-gravitating systems of stars; there is an extended discussion of the "isothermal sphere" in Binney and Tremaine's Galactic Dynamics, for instance, and how it emerges as a solution for dynamical self-gravitating systems.

No, Remi, even if the kinetic energy acquired by the center-of-mass movement of a molecule during the gravitational fall, is enough big to excite a jump between the internal energy levels of the molecule, I don't see how such a transition can happen. Please recall that Boltzmann supposed elastic collisions. Such collisions preserve the center-of-mass energy and linear momentum, s.t. no part of this energy goes to, or come from, the internal energy of the particle.

Remi,

you say : "Some disassociation would occur".

Recall that Maxwell-Boltzmann (M-B) distribution is for ideal gas - point-like particles with no internal structure? Neither does M-B distribution suppose presence of photons.

Dear V. T. Toth,

First of all thank you for trying to solve this question that I know that it is not simple.

A) Now, in the article that you recommended (and thank you for it), there is something not clear to me: what is EP - μ ? Is it a potential energy, or the kinetic energy? If it is a potential energy this is a problem, because in the Maxwell-Boltzmann distribution of velocities in the exponent appears the kinetic, not the potential energy.

b) But I see some non-clear thing in the model with the boxes. The article says:

"suppose that you make the particles in one of the boxes hotter. There will then be a flow of heat out of that box to the neighboring boxes until a single temperature has been reestablished."

How is implemented that flow? What are its carriers? The particles from the boxes. As long as the particles from the heated box move to boxes at the same height, their velocities remain constant, and the collisions redistribute the excess of energy among the boxes. However, if the particles from the heated box go to upper/lower boxes, their velocities decrease/increase. Then, what about the temperature of the upper/lower boxes? This is exactly Loschmidt's argument.

The question is, this argument is correct?

Remi, did you see my comment that the MB distribution of velocities was given for an ideal gas with no internal structure? You seem not to have read my comment. There are no "moieties".

Now please be kind, look also at my comment to V. T. Toth. You touched an issue, but in my opinion you didn't come to it correctly - so it seems to me. There are no "halves". Particles in a gas, have also a bigger or smaller vertical velocity component. This component gets modified due to the gravity, but because of the collisions, at each altitude and altitude this modification is redistributed and the horizontal component is also modified. The question is whether this modification leads to the distribution of the horizontal velocities equal to the distribution of the vertical velocities, and equal to the MB distribution. In that case we would have at each height thermodynamic equilibrium, with T depending on the height.

Dear Sofia: Never mind that site to which I linked, I knew I saw a better explanation somewhere... well, it's on my own darn Web site! Something I wrote eight years ago: https://www.vttoth.com/CMS/index.php/physics-notes/72

The gist of it is that even in the case of an arbitrary (smooth) potential, the temperature of a column of gas in thermal equilibrium remains constant due to Liouville's theorem (that is, the invariance of the volume element in phase space).

The paper that I cite in my write-up seems to be available for free at https://tallbloke.files.wordpress.com/2012/01/coombes-laue.pdf

Sofia,

if I consider a column of gas extending high in a gravitational field, which cannot emit radiations to the external world,  if the pressure of the gas is high, the gravitational field will be compensated, if the gas is rarifact (low pressure, few molecules), there is no way to compensate it and the top part of the tower will be colder.

It is obvious that in frequent collisions (high pressure and temperature PV=nRT), due to crossing trajectories, the particles do not preserve their energies in priviledged directions, so the overall energy will be diminished by the gravitational potential energy and the differences from top to bottom will be negligible.

Dear Sofia,

I think one can easily clarify the apparent conflict, but I have to write down some equations.

Please have a look into the appended pdf-file.

Hi Sofia.  I think this is a case where stat-mech arguments are full of subtlety but classical thermodynamics provides the answer in a straightforward manner.  If you had a vertical column of gas in a state of internal equilibrium, you could append a small thermal conduit running from the top of the column to the bottom.  If the gas had height-dependent temperatures (say hotter at the bottom and cooler at the top, as argued by Loschmidt), the temperature difference between the two ends of the conduit would cause a continuous flow of heat through the conduit, from the bottom of the column to the top.  You could then extract work from this heat flow as desired, thereby having constructed a “perpetuum mobile of the second kind” (extracting work from a single reservoir, with no other effect besides reducing the thermal energy of the reservoir) – which implies that you’ve violated the Second Law.  Hence, the temperature must be uniform throughout any system internally at thermal equilibrium, regardless of the presence or absence of a gravitational field.  The microscopic theory of how this equilibrium is reached and maintained evidently has some subtleties to it, but at least we know what the final answer must be.

Yes but rarefied gases do not behave as such.

This is a really interesting question. Loschmidt's argument is definitely correct (if we take temperature to be a measure of kinetic energy), if we consider the special case of a system composed of a single particle, experiencing elastic collisions with the walls of its container, in a gravitational field. This experiment can be carried out by simply letting a rubber ball bounce off the floor. The trajectory clearly has its highest average speed close to the floor.

However, this case does not correspond to the canonical ensemble, but to the microcanonical one. The conclusion is not necessarily warranted also for the canonical system (e. g. thermodynamic limit).

Maxwell, in his book Theory of Heat gives a proof by contradiction on p. 320 (Dover edition): if Loschmidt is correct, perpetual motion machines of the second kind are possible, something Loschmidt himself states explicitly in the paper Sofia mentions. Loschmidt considers it hope for mankind after coal, oil, and the sun run out. It is based on the consequence that different gases will have different temperatures at the top of the column, even if they have the same temperature at the bottom, so you could run a heat engine between them on the top and extract work from the temperature difference. Thus extracting work from a single heat reservoir. 

Since nobody has been able to construct such a machine in the 140 years or so since Loschmidt (or in all the years before that), one could consider this proof that Maxwell is correct. However, people have not stopped trying to make one, now usually with magnets, electricity, or quantum mechanics, so you could not find this proof satisfactory, and still hope it is possible and that Loschmidt could be right.

So let me give a pure thermodynamical argument. Loschmidt implicitly assumes that for a system in equilibrium the energy is the same everywhere. This is incorrect. A system in equilibrium has maximum entropy. So that should be taken as a starting point. It is a textbook exercise to show that if you have a system with different temperatures at different positions you can always increase the entropy by making the temperatures more equal. The highest entropy is obtained if the temperature is equal everywhere. There is a slight complication in this case compared to textbook exercises, since changing the local temperature also changes the local density. So the way to properly do it is as follows: write the total entropy as a functional of density and temperature, which themselves depend on height, or more general on position. Take the functional derivative with respect to these quantities, to find when the entropy is extremal, throwing in a Lagrange multiplier for the density since the total number of particles remains constant in the column. This immediately gives you two conditions: the first is that the temperature should be the same everywhere, and the second is the exponential dependence of the density on height. This is all there is to it. Taking the second derivative will show that the extremum is a maximum.

It should also be pointed out that dynamics has nothing to do with this. Equilibrium thermodynamics does not depend on the process that brought the system to equilibrium. You should always be able to calculate equilibrium conditions and properties without making reference to any dynamics, and dynamical constants should never enter into equilibrium expressions. I call this Logan's error in honor of the first paper I encountered where this mistake was made (S.R. Logan, Trans. Far. Soc. 63, (1967), 3004), albeit in a different context. Dynamics only tells you how long it takes or in what way equilibrium is attained, not what equilibrium looks like.

"Loschmidt considers it hope for mankind after coal, oil, and the sun run out. It is based on the consequence that different gases will have different temperatures at the top of the column, even if they have the same temperature at the bottom, so you could run a heat engine between them on the top and extract work from the temperature difference. Thus extracting work from a single heat reservoir. "

In other words he meant to extract gravitational energy.... very hard to believe...

Exactly, but it is. Here is a translation of the last two paragraphs on the bottom of p 139 of his paper:

These considerations have the remarkable consequence that it is indeed possible to derive the second law from Clausius Axiom: "It is impossible to move heat from a colder to a warmer body without performing work", or from Thomson's equivalent: "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 surrounding objects", whereas the reverse of this statement is not allowed, since the extent of the second law is broader than that of the above axioms.

With this the terrible shadow of the second law as the destructive principle of life in the universe is also diminished, and, at the same time, a consoling perspective becomes an option: mankind does not just need coal or sunlight to get work from heat, but will always have the unending supply of useful convertible heat at hand.

-----

It is unclear what he means exactly by the first paragraph, but the second is completely obvious.

I guess Maxwell saw the argument for what is was, and was a more strict believer in the second law than Loschmidt was. He does not spend more than two lines on his argument that the temperature must be the same everywhere. The consequence of Loschmidt's thinking is also that in a column of air the nitrogen at the top will have a different temperature than the oxygen, and  a different temperature than the container as well, since the atoms have all a different mass. He does not explain how it would affect a thermometer. 

I should also mention that there are still papers being written, and published, today where people try to extract work from one reservoir, but now usually with electric or magnetic  fields.  

Dear Gert,

maybe some General Relativists who think that conservation laws are not respected by Gravitation, may accept the argument of building a perpetual gravitational machine..

"The consequence of Loschmidt's thinking is also that in a column of air the nitrogen at the top will have a different temperature than the oxygen, and  a different temperature than the container as well, since the atoms have all a different mass."

Sure??? Hard to believe...

Again, indeed hard to believe but true. Although Loschmidt does not draw that conclusion explicitly himself. It is an immediate consequence from taking the total energy (kinetic + potential) everywhere the same, and equating the (average) kinetic energy with temperature (with Boltzmann's constant thrown in to get the units the same, and the numerical factor 3). This leads to a mass dependence of the temperature profile. I don't understand why he could not see the consequences, he is supposed to have been a smart guy, even within a molecule (and he does talk about molecules having parts with their own kinetic energy) the temperatures would be different for different modes of vibration, since those have different (reduced) masses. And vibrations were known and discussed, it was a problem why they did not contribute to the specific heat of gases. Quantum mechanics was needed to understand that, but classical vibrations should have given Loschmidt second thoughts about his idea as well. 

Another field where some people believe the second law can be beaten is living systems, in particular photosynthesis. The most recent paper I know of is from 2005 by Jennings: Biochim. Biophys. Acta, 1709, (2005), 251. But I'm pretty sure he won't be the last. He was wrong. I would almost say "of course", but we have to keep in mind that there is no proof of the second law. But I am also convinced that if living systems could beat the second law, every single organism would make use of that and there finally will be a free lunch for everybody. 

Very enlightening discussion, a pleasure to follow. Long live RG!

"Another field where some people believe the second law can be beaten is living systems, in particular photosynthesis. "

There are some differences between beating the second law and extracting vacuum energy or worse gravitational energy.

The energy balance is not violated necessarily by violating the II Thermdyn. law, it is violated instead the principle according to which the energy is distributed.

I don't know if it is possible to bid farewell to the II law of thermodynamics or to make even more interesting things: extract vacuum energy by means of the collective quantum physics (living things), finding limits for the Heisemberg Uncertainty principle.   

The only argument I'm aware of is that in the realm of known ensambles of gases or particles this is certainly not possible.

Loschmidt (and Jennings, and I in this case) is not talking about violating energy conservation, but second law violations. There is a big difference between the two. The ones I am talking about are related to so-called perpetual motion machines of the second kind, and involve turning equilibrium fluctuations into useful work. For the ones Stefano is talking about, I can recommend the book: "The search for free energy" by Keith Tutt. That search is also by no means over. 

Footnote: Actually I am not sure how Loschmidt views the second law. The paper basically establishes first (erroneously) how a system in a gravitational field has a temperature gradient, and the next part seems to be a suggestion for the construction of a heat engine based on that idea. The first paragraph I quoted could give some clue about his final ideas on the second law, but I don't get it. I also do not know if an attempt to make such a perpetual motion machine was ever made. If it had succeeded, we'd all be using one. And besides, his idea should also work for charged particles an an electric field, or spins in a magnetic field. There are a few recent papers about the latter, but as far as I remember they do not try to exploit temperature differences. A recent one is G. D’Abramo, Physica A , 390, (2011), 482. They are more closely related to constructs like Maxwell's demon or the Feynman ratchet.

1) In general, the idea that the gas in equilibrium will have the the same value for (dS/dE) everywhere is correct. This has been discussed several times in this set of responses.

2) However, there is a general-relativistic twist: what corresponds to (dS/dE) is not 1/T, the inverse temperature; but 1/T as modified by the factor sqrt(g44). This is shown in http://journals.aps.org/pr/abstract/10.1103/PhysRev.36.1791 .

(There is actually a much easier way to see this, but I don't have time to write it up today.)

This fact, which Loschmidt or Maxwell could not possibly have been aware of, still does not make perpetual motion machines of the second kind viable. 

Although it could be said that general relativists give higher priority to the second law of thermodynamics than general relativity: after all we now have radiating black holes with a finite temperature. 

Dear Gert,

GRT does not convince me in his main postulate...and  should handle properly the gravitational energy...but this is another thread...

Agreed. The question of the OP was if equilibrium is possible in a system in an external field and the answer must be an emphatic yes. 

I now have access to my library, and can give proper references. This note will argue that:

a) Sofia Wechsler's question about the validity of Loschmidt's argument can be answered: It is wrong. There must be, as Loschmidt would have agreed, thermal equilibrium within each level; but there must also be thermal equilibrium between each level.

b) Nonetheless, I will pursue my claim that it is not the local temperature T(z) which is the constant parameter throughout the gas, but the parameter T(z)*sqrt(g44)(z). Under non-astrophysical conditions, this is essentially indistinguishable from the local temperature.

c) For z1 > 2, noting that sqrt(g44)(z1)/sqrt(g44)(z2) = the blue-shift-factor, for light dropping from altitude z1 to z2, I will give an explanation for why the condition for thermal equilibrium is not T(z1) = T(z2), but rather is T(z1) = T(z2)/blue-shift-factor.

Part 1: Loschmidt is wrong

As has been argued many times already, there would be real problems with physics if there were not a well-defined and stable state of thermal equilibrium between the different layers of gas; this would break the 2nd Law of Thermodynamics (2LoT) and allow creation of a perpetual motion machine of the 2nd type. However pleasant that would be, we believe that it would be intellectually incompatible with the rest of physics.

1-a) Here is a formal thermodynamical argument as to why thermal equilibrium between two layers (which need not be immediate neighbors) seems to require that they have the same temperature:

If the layer of gas at height z1 passes energy dQ1 to the layer of gas at height z2, the respective changes of entropy are:

dS1 = dQ1/T1 = - dQ/T1  , and

dS2 = dQ2/T2 = + dQ/T2

This follows from the definition of entropy (dQ = T* dS) and from conservation of energy (dQ1= - dQ = dQ2)

The total change in system-wide entropy is thus:

dS_total = dS1 + dS2 = dQ*(- 1/T1 + 1/T2) = (T2 - T1)*dQ/(T1*T2)

In order for the system to be in a stable state, a small exchange like this must not change the system-wide entropy. Therefore, dS_total must = 0.

But that immediately implies that (T2 - T1) = 0 , so T2 = T1.

As z1 and z2 could have been any two layers of gas, and the argument holds equally for all pairs, we conclude that the local temperatures must all be the same:

T1 = T2 = T , everywhere throughout the system.

So, because we trust thermodynamics, and have no reason to trust Loschmidt, we conclude that the temperature is indeed the same everywhere.

1-b) But this still leaves us with the legitimate question, How can it be that the temperature at the top of the gas is the same as the temperature at the bottom, when gas molecules obviously lose kinetic energy (KE) climbing up and gain KE falling down? How can T2 not be > T1 ?

The answer is that, although it is true that the number of gas molecules at a given value of KE will be smaller at z1 than at z2, it is also true that the number of gas molecules at all values of KE will be smaller at z1 than at z2; and temperature parameterizes the percentage of higher vs. lower KE. The assumption that the gas populations at each level follow the Maxwell distribution with the same value of T turns out to be self-consistent. But rather than argue that point fully here, I will just point to the Feynman Lectures on Physics, Vol. I, Chapter 40, Section 4: The distribution of molecular speeds. It would take a lot of time to substantially improve on the clarity of that presentation.

Part 2: General Relativity steps in

Having tried to convince you that the temperature throughout the system is a constant, I will now turn around and claim that it is not the temperature that is constant, but the parameter T*sqrt(g44). I will cite valid authorities:

- Misner, Thorne & Wheeler, Gravitation (1970): Section 22.3, Hydrodynamics in curved spacetime; Exercise 22.7, Hydrodynamics with viscosity and heat flow.

- Tolman, Relativity, Thermodynamics and Cosmology (1934): Section 128, Thermal equilibrium in a gravitating sphere of fluid.

- Tolman and Ehrenfest, Physical Review 36,1791; 15 December 1930, Temperature equilibrium in a static gravitational field.

So why should you believe this?

Part 3: Why you should believe this

In this section, I will explain how general relativity (or, not to be too grand, considerations of gravitational effects) force us to modify the argument of Part 1-a) and result in the new equilibrium condition.

3-a) As a matter of experimental fact, a photon emitted at one altitude and received at a lower altitude has a higher locally-measured frequency when received than it had when emitted. This is known as the Pound-Rebka effect, and has been demonstrated with gamma rays. It can be shown to be a simple consequence of:

- conservation of energy

- Mass_gravitational = Energy/c^2

- photon_energy = h * photon_frequency

This is explained in the Feynman LoP, Chapter 42-6, The speed of clocks in a gravitational field. It's pretty clear there, but an article that makes it even more explicit is Stefano Quattrini's "Speed of clocks in a gravitational field; A Feynman's lecture revisited.":

https://www.researchgate.net/publication/264571216_SPEED_OF_CLOCKS_IN_A_GRAVITATIONAL_FIELD_A_FEYNMANS_LECTURE_REVISITED

(I have not read all of Quattrini's article and cannot claim to support all of it; but I think the presentation of Feynman's argument may expedite matters for some readers.)

The end result is that a photon emitted at altitude z1 at frequency f1 is received at lower altitude z2 at frequency f2, where f2/f1 = (1 + g(z1 - z2)/c^2) = gravitational-blueshift-factor = GBF.

How does that affect the argument of 1-a) ? It means that if

dQ1 = - dQ , then if dQ is turned into a photon of frequency f1 and sent down to z2, it is received as frequency f2 = f1 * GBF. When this is converted back into KE, what it adds is dQ2 = h*f2 = h*f1*GBF = dQ*GBF. Therefore,

dS_total = -dQ1/T1 + dQ2/T2 = -dQ/T1 + dQ*GBF/T2

.............. = dQ * (T2 - T1*GBF)/(T1*T2)

So dS_total = 0 only when T2 = T1 * GBF.

But GBF is, in the general relativistic framework, sqrt(g44(z1))/sqrt(g44(z2))

So the modified condition of thermal equilibrium is:

T(z1)*sqrt(g44(z1)) = T(z2)*sqrt(g44(z2))

The basic point is that falling from "higher potential" to "lower potential" boosts the energy of photons going down, whereas going upwards saps them of energy by the same factor. This means that an exchange of energy between a higher and a lower layer of gas will never be "fair": in thermal equilibrium, when the S_total is maximal, the lower level will always be dealing in bigger clumps of energy, so it must have a higher temperature to compensate for the blue-shift factor.

It is worth pointing out that the crux of the argument doesn't depend on general relativity: That only enters when we set the GBF = function of the ratio of g44 values. It really only depends on the fact of the Pound-Rebka blueshift, which can be demanded on the grounds of the three bullet-points above.

Article "SPEED OF CLOCKS IN A GRAVITATIONAL FIELD" A FEYNMAN’S LECTU...

It is possible to violate the second law of thermodynamics.

E.g. you can read my article "Diode rectifies thermal noise".

No it is not. I only believe it is possible if you do the experiment that shows the violation. My prediction is that it will not work. Since 2000 numerous papers, like yours,  have been published claiming the possibility of second law violations (see the picture). Here is a quote from one of them (D.P. Sheehan, A.R. Putnam, and J.H. Wright, A Solid-State Maxwell Demon,  Found. Physics,  32 (2002), 1557-1595) "Prospects are good for laboratory construction and testing of this solid state Maxwell demon in the near future."  We are now 15 years further in time and I still have not seen  a solid state Maxwell Demon built by Sheehan, or someone else for that matter. "Near future" is of course not very specific, but I would suggest not holding your breath.

The idea of using a diode has also been discussed numerous times. No one has been able to build a device that actually violates the second law.

Apart from that: if it were true, entropy would be a useless concept. Thermodynamic entropy can only be defined if perpetual motion machines do not exist. It is the first assumption Carnot makes. If pmm's do exist entropy is not a state function.

While it is a question whether a solid state Maxwell's demon will work, thermionic diodes and cold emission diodes will rectify thermal noise from resistors. You can read my article "Diode rectifies thermal noise".

If you can make a diode rectify thermal noise, you have a Maxwell demon. Well, actually more something like a Feynman ratchet, but that is basically the same thing. Equally impossible.

It turns out that it depends upon the nature of the spring which is used. I designed a needle valve with a gas spring (the gas spring was a nano-piston and a single Helium-3 atom). It will let atoms of Xenon (say Xenon-136) go in the easy direction about 100 000 times more frequently than in the opposite direction.

It is intuitively clear that if you have lighter gas particles (Helium-3) then such gas will have smaller pressure fluctuations than gas o heavier particles (Xenon-136).

First of all your intuition is not mine. Take an ideal gas. The compressibility, which is directly related to fluctuations in density (and thus pressure) is 1/p, independent of the mass of the particles. From a kinetic point of view it is the momentum of the particles, not the mass that on collision with a wall, or a valve, determines the pressure. Heavy particles have the same momentum as the light ones, they just move slower.

Secondly,  my answer to all people who claim they designed a perpetual motion machine (of the second kind) is: build it. My prediction: it won't work.

Heavy particles have bigger momentum p = SQRT(m*k*T). 

If you want an easy proof that the second law of thermodynamics can be violated, read my article "Diode rectifies thermal noise" in Research Gate. 

Sorry, I was a little too quick here, you are right of course about the momentum. But to get the force on the wall or valve you need the change in momentum (F=dp/dt) and any textbook on kinetic theory, or indeed https://en.wikipedia.org/wiki/Kinetic_theory_of_gases will tell you that this force is proportional to mv2, or on average which itself is proportional to kBT. As you so rightly point out. This of course leads to the ideal gas law for the pressure. p = nRT/V, which  does not depend on the mass of the atoms. The pressure of Xe or He on a wall or on a valve is exactly the same at a given temperature.

I have seen far too many easy proofs that the second law can be violated, and none of them ever led to a working device. If you build it I will come, and humbly apologize for my arrogance.

Attached is a nice story by Asimov about the age old question: "How can the entropy of the universe be massively decreased?"

I have a design of a perpetual motion machine of the second kind, based on rectification of thermal noise from resistors by cold emission diodes. It requires picotechnology - placing atoms and bigger parts with 10 pm accuracy. So it is a matter of future. The design promises power densities exceeding 1 kW/mm^3 of active volume.

Far future it is then  (see the Asimov paper).

Now 10 nm geometries are feasible. They become 2 times smaller every 2 years, so 20 years from now we should have 10 pm accuracies.

Regarding the Universe, one should notice that it is possible to violate the law of conservation of energy (one example is described in my article "Gravitational Self-interaction of a Moving Body"). I think that electrically positive and negative particles chasing on another in plasma of stars also violate the law of conservation of energy, and that at least some stars constitute perpetual motion machines of the first kind. 

I have to agree with Remi here. Although I often align myself with the "establishment" I am fully aware that it is an experimental fact that pmm's of the second kind (and of the first) so far have not been constructed or observed. This was true in Carnot's time and it is true now. As I already mentioned he takes it as the first essential ingredient of his derivation. The fact that his derivation is flawed since he thinks heat is conserved does not change this, in all later derivations it is also essential. If pmm's do exist you cannot prove that entropy is a state function and there is no second law.

In the two centuries after his work, there have been numerous attempts to see if the second law can be violated. So far nobody has been able to build a device that actually does that. Maxwell thought it should be possible. When he wrote about his demon, he stated: "Or in short if heat is the motion of finite portions of matter and if we can apply tools to such portions of matter so as to deal with them separately then we can take advantage of the different motion of different portions to restore a uniformly hot system to unequal temperatures or to motions of large masses. Only we can’t, not being clever enough".  (emphasis  mine).

When people started seeing how complex living systems are, it was suggested that they could violate the second law. James Jeans wrote for instance:

"In fact it would seem reasonable to define life as being characterized by a capacity for evading this law. It probably cannot evade the laws of atomic physics, which are believed to apply as much to the atoms of a brain as to the atoms of a brick, but it seems able to evade this statistical laws of probability."

Of course living systems don't evade this law at all, at least there is no evidence for it, the entropy of a system can decrease as long as the total entropy increases. And Freeman Dyson also saw options (along the lines of Asimov):

"Whether the details of my calculations turn out to be correct or not, I think I have shown that there are good scientific reasons for taking seriously the possibility that life and intelligence can succeed in molding this universe of ours to their own purposes"

More recently there was Jennings, who believes that photosynthesis is able to perform this feat:

"Thus, 1−T /Tr represents a kind of efficiency horizon beyond which negative entropy is produced and the second law is not obeyed. As this is impossible for a heat machine, it serves to underline the difference between photosynthetic photochemistry and a heat machine."

However, his analysis is severely flawed and widely considered wrong. And besides, as Carnot points out, any machine that is more efficient than his reversible engine would screw up his derivation.

In the past 10-15 years there have been numerous proposals for nano technology (since 10 pm is far smaller than atoms or molecules I don't think we'll ever get there let alone build structures on that scale), including the one I quoted earlier, and for which I am still waiting to see the experimental verification. 

For me there is a fundamental question to which I have no answer: Why isn't it possible to use equilibrium fluctuations to get useful work?

Sidenote: Statistically it is possible. Think of Boltzmann's remark on how the universe started:

"There must then be in the universe, which is in thermal equilibrium as a whole and therefore dead, here and there relatively small regions of the size of our galaxy (which we call worlds), which during the relatively short time of eons deviate significantly from thermal equilibrium. Among these worlds the state probability increases as often as it decreases."

Although for this to happen we would have to wait (quite a long time), he does not give suggestions on how to make it happen.

end sidenote

If it were possible it would overthrow all of thermodynamics, which is deeply ingrained in all our thinking. Planck would not have to worry about understanding light and deriving the expressions for black body radiation, and Bekenstein doesn't have to show that black holes have a temperature.  Suppose we could run such a machine in part of a heat bath. It would cool off that part (unless we also want first law violations, and I don't think that's going to happen, regardless of what H. Tomasz Grzybowski  states), so we would end with a warm and cool body, from which we could get even more useful work.

So, building such a machine would not only make you very rich, is also would overthrow  almost everything we know. And quite possibly screw up our universe if you can't contain it. Of course that should not stop you from thinking about it and trying.  But extraordinary claims need extraordinary evidence, and  I just don't consider calculations providing extraordinary evidence. However good they may look and however hard it may be to find the flaw in the reasoning. The only proof is to build a device based on your calculations.

PS I have not given references to the quotes in the text, but if you want them, just ask.

It is a question whether a CLASSICAL Perpetual Motion Machine of the Second Kind can work. But I clearly prove that thermal noise from a resistor can be rectified by diode at the same temperature as the resistor. 

Have you heard of constructive interference e.g. of electromagnetic waves? 

This question should be a sufficient hint regarding the law of energy conservation.

Thomasz, with this last contribution you gave me sufficient insight in your  understanding of fundamental physics to henceforward ignore your posts.

Remi suggested some reading and studying before you start to make outrageous claims, lest you end up as one of the considerable number of crackpots inhabiting RG.

http://physics.stackexchange.com/questions/23930/what-happens-to-the-energy-when-waves-perfectly-cancel-each-other

Remi, I will come back to your remark tomorrow. I think I disagree, but I have to phrase that carefully.

 Yeah, here's the funny thing. As part of  my consultancy work for a big oil company I of course immediately notified them of your invention, and we are currently drafting patent applications. That way we can keep this from ever coming on the market before the oil runs out, and then we can still clean up. Same as what we did with the engines running on water. Unfortunately you can no longer file patents, since you already published your work on RG, and we are way ahead of you.

 @Remi

I am not sure if I understand your remark properly, but you seem to imply that you already know that S is a state function. But actually that is something you have to prove. As far as I know there are at least three ways to do that. The first and most well known is using Carnot's approach (or more his followers) who basically showed that \oint_C dq/T = 0 for all closed contours C. As part of the proof you need the assumption that there are no engines more efficient than a reversible engine (which you can take with an ideal gas as working substance). A pmm would certainly be more efficient so this proof would fail.

A second possibility is  showing that there is an integrating factor for dq, which is then given as 1/T. I do not have that proof readily in my mind, but I assume you also have to show that T is the thermodynamic temperature. I'd have to look up which assumptions are involved exactly, so I can't immediately say that this proof also fails.

The third possibility is Caratheodory's approach, which starts with the assumption that not all states in an area around a certain given state are adiabatically accessible. It seems to me that this also fails if pmm's are possible.

I do not know of any paper, or section in a paper, of people claiming to be able to build a pmm if they only had the money, which addresses this, but it has always bothered my a little in all discussions about the possibility of pmm's.  I myself never worked out the precise details, but maybe it is time to do so.

@Tomasz, just out of curiosity: which university gives PhD's on the basis of one obscure never cited three page paper? I assume you know that the title Dr is protected, so if you call yourself that, it is fraud.

Interesting perspective, but in my mind that only leads to more questions. Mainly about the relation between thermodynamics and statistical mechanics. I'll have to think about that for a while.

Looking at the problem from a practical point of view I have three answers:

1. The well established standard version

The common knowledge points to the fact that the perfect gas in the external gravity field without the internal heat sources and without radiative effects will reach an equilibrium state with the constant temperature. Quite convincing arguments supporting this statement are presented in section 38 (chapter IV) of Landau and Lifshitz Statistical Physics I (note that the system is closed and that gravitational interaction of the particles in gas are neglected in all considerations; the horizontal motions and forces associated with the rotating frame of the reference are also excluded).

2. Methods based on application of the action principle

The interesting methodology to further analyze the original question is provided by the use of an action principle. All details are elaborated in the recent paper published in Entropy (Christian Frønsdal (2014), Entropy, 16, 1515-1546, Heat and Gravitation: The Action Principle)

http://www.mdpi.com/1099-4300/16/3/1515/pdf

3. Solution of the Boltzmann equation in the external gravity field

The best answer can be obtained after solving the Boltzmann Kinetic Equation in the external gravity field. This is the most difficult method, however, there are numerous examples; I like particularly the methodology outlined in

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.1019.3686&rep=rep1&type=pdf

Historical note

(with meteorological bias showing how important is the definition of a system under consideration)

Historically, the theorem governing the distribution of gas temperature in the external gravity field is due to Maxwell and Boltzmann. This theorem was wrongly questioned by Loschmidt and even by Lord Kelvin in his Baltimore lectures (1904). The main reason for criticism of the Maxwell-Boltzmann theorem was the lack of clear distinction between the closed and the open systems. Loschmidt withdrew his criticism under the pressure of authority and Lord Kelvin by the use of a pure reason.

Kelvin realized that the existence of the lapse rates observed in the actual atmosphere is associated with the fact that the system is not closed. In the simplest case of the open atmosphere, the vertical gradient of temperature is established as a result of the interaction of radiative cooling at the top and heating at the bottom with the restoring action of convection. The dry adiabatic gradient of temperature created as result of these interactions is equal to $g/c_p$ (about 9.8 degrees of Celsius per 1 km).

The situation is more complicated when we consider the "moist thermodynamics" leading to a smaller wet adiabatic gradient. The addition of aerosols and chemical constituents leads to even more complex problems which are still a subject of ongoing debates.

The Loschmidt-Maxwell polemic does not seem to have an easy solution. But I have an easy proof that the second law of thermodynamics can be violated - please read my article "Diode rectifies thermal noise".

My proof shows that: 1) noise in a nanoscale thermionic diode (all of which is at 1473 K) is negligible 2) the diode rectifies thermal noise from a resistor - the forward current is almost 2 orders of magnitude greater than the reverse current.

No, in this case the power is quite small - then the diode rectifies radiative thermal noise of the order of a millivolt. When noise from a resistor is rectified, power is bigger, but still very small 60 mV * 10^(-18) A = 6*10^(-20) W - it is not meant to be practically useful. It only serves proving that a diode can rectify noise from a resistor working at essentially the same temperature, in any case it converts heat from a single heat reservoir into electric energy. 

From the Richardson-Dushman law you can calculate the forward and reverse currents which will flow under the influence of thermal radiative noise (10^4 V/m * 10^-7m = 10^-3 V of thermal radiative noise). Since the work function of tungsten (cathode) is sufficiently smaller than the work function of platinum (inner surface of anode), the forward current will be greater than the reverse current.

Chemical potential differences result in contact potential difference, which is taken into account in my proof.

As I stated before, contact potential difference is taken into account in my proof. 

Technology producing useful power, which I mentioned, is based on one of the phenomena that violate the law of energy conservation. To see that there are such phenomena, you can consider constructive interference of EM waves.

I guess you just pretend not being intelligent enough.

As I already pointed out, the noise in a nano-scale thermionic diode is much smaller than the noise from a resistor. 

Also, when rectification is used, one can connect many such circuits in series or in parallel to get more power. With resistors alone in series you get more voltage but not more power. 

You just play a stupid scientist. 

Anyone interested in TRUTH can read my article "Diode rectifies thermal noise".

The noise in an all-hot thermionic diode, which I consider, is due to thermal radiation (described by the Stefan-Boltzmann law). The electric field intensity in the thermal radiation in my diode is about 10^4 V/m. It is essential that the diode is a NANOSCALE DIODE - with about 100 nm cathode-anode distance we get about 10^-3 V of noise, which is MUCH SMALLER than the voltage drop on the diode due to RESISTOR NOISE - which is about 60*10^-3 V. This is why CLEARLY A NANOSCALE DIODE WILL RECTIFY RESISTOR NOISE.