The issues Joe Born has raised are interesting in the way that pondering such questions reveals something about physics. In this case perhaps most about the nature of microcanonical ensembles, but also more generally about statistical physics.

What I don’t like in his texts is the criticism he presents on Brown’s comment. The comment is, unavoidably, simplifying, but not in a way that could be considered wrong in any sense.

]]>Assuming that this is the article to which Joe Born refers, an insulated silver wire thermally connects the top and bottom of the gas column. Any calculations related to a microcanonical ensemble are therefore irrelevant.

]]>In physics values that are not measurable even in principle are usually not considered worth any attention. Therefore the calculated very small difference in the average kinetic energy is not of physical significance. Thus it does not contradict anything Dr. Brown wrote.

]]>So what? The premise of Dr. Brown’s thought experiment is a non-zero equilibrium lapse rate. Is that physically meaningful? The issue is whether Dr. Brown’s logic is valid, and to test whether it is we have to deal with the world that his proof’s premise assumes, physically meaningful or not.

Pekka Pirilä: “Therefore all measurements of the nature you propose are impossible.”

But I didn’t propose any measurements. I took note of the value that theory gives an unmeasurable quantity, and I used it to test an assumption, on which Dr. Brown’s silver-wire proof is based, about what would happen in the perhaps physically unrealizable world of his proof’s ostensible premise.

Pekka Pirilä: “Therefore any measurement of the temperature disturbs the system much more than the difference to be measured.”

I have never contended otherwise. In fact, I said from the beginning that the incredible smallness of that gradient’s magnitude—which in most cases is probably orders of magnitude smaller than even the fluctuations in the gradient—shows that Velasco et al. essentially establish the ultimate conclusion of Dr. Brown’s proof. My problem is not his ultimate conclusion but his logic.

Please try to focus on the issue. The issue is not whether any physically measurable non-zero equilibrium lapse rate exists. The issue is whether Dr. Brown successfully refuted that proposition, i.e., whether he successfully established that non-zero equilibrium lapse rates necessarily imply that net heat could flow undriven perpetually.

I know you have contended that “The issue is not, whether a situation will result in perpetual motion or not.” But you’re wrong. Or, rather, the precise issue is perpetual heat flow; Dr. Brown’s whole proof depends on such a result’s following logically from the non-zero-equilibrium-lapse-rate premise to be refuted. Unless you can see that it does, we have nothing to discuss.

]]>Remember that the whole difference between the average total kinetic energies of a subvolume in a canonical and in a microcanonical ensemble is of the order of the variation in the potential energy of a single molecule. That minuscule energy is distributed among all molecules.

Any thermometer has a huge thermal capacity in comparison to a single molecule. Therefore any measurement of the temperature disturbs the system much more than the difference to be measured. Therefore all measurements of the nature you propose are impossible. This is not only a practical limitation, but it’s fundamental and cannot be circumvented even in theory – or in any physically meaningful thought experiment.

]]>In a proof by contradiction a premise is refuted by showing that it leads logically to a false conclusion. The proof is faulty if its reasoning is based on something inconsistent with the premise, or if the effective premise actually differs from that which is to be refuted.

My objection to Robert G. Brown’s silver-wire proof was that its actual premise differed from the one he attempted to refute. The one he attempted to refute was that at equilibrium an ideal-gas column disposed in a gravitational field would exhibit a non-zero lapse rate.

But his actual premise, I objected, was a combination of that proposition and the proposition that thermally coupling the gas column to another system would not affect the gas column’s equilibrium lapse rate. So, I implied, he actually proved only that a gas column’s equilibrium lapse rate couldn’t be non-zero IF it were unaffected by thermal coupling. However, I’m no longer sure he proved even that.

Dr. Brown assumed that thermal coupling would—as we infer from our observations of the zero-equilibrium-lapse-rate world—impose a common lapse rate on the coupled systems; he assumed that at equilibrium the two systems’ mean kinetic energies per molecule are required to be equal at each thermal-coupling point. Frankly, I assumed that, too. As a result of a discussion on another site, though, I realized that I wasn’t so sure that under the non-zero-equilibrium-lapse-rate premise—which must prevail throughout the proof—such an assumption is consistent with the proof’s premise.

To extrapolate to conditions that would prevail under the non-zero-equilibrium-lapse-rate premise of Dr. Brown’s proof, let’s drastically reduce the gas column’s number of molecules to two, and let’s give those two molecules different masses. Note that at any given altitude the mean kinetic energy of the more-massive molecule is less than that of the less-massive one; when they have equal kinetic energies and collide, the less-massive molecule always gains energy at the more-massive one’s expense. If we look at the two molecules as two thermally coupled systems, we see that at equilibrium such systems’ mean kinetic energies per molecule are not necessarily equal.

Yes, yes, I know. A single molecule is not a gas. What it’s undergoing is ballistic motion, not thermal motion. The effect I just pointed out is attenuated as the number of molecules increases. I’ve heard all those arguments. But they describe differences only in degree.

True, the effect is attenuated, but it is not extinguished entirely. So the fact remains that, to the extent that a non-zero equilibrium lapse rate does prevail—as it must according to the premise of Dr. Brown’s proof—thermal coupling may not necessarily imply equal mean kinetic energies per molecule. And Dr. Brown’s proof may therefore include another unfounded assumption, another difference between his proof’s ostensible premise and its actual premise.

Again, I don’t disagree with the essence of Dr. Brown’s overall equilibrium-isothermality conclusion; in fact, his conclusion was my reaction to Velasco et al. from the very start. But I remain convinced that proofs such as Dr. Brown’s based on finding perpetual undriven heat flow are invalid.

]]>Take a thermally isolated gas column and a negligible-thermal-mass thermometer that initially is thermally isolated from everything. Then bring the thermometer into thermal communication with the column at a high altitude so that the mean translational kinetic energy of the thermometer’s molecules equals that of the gas’s at that altitude.

Return the thermometer to isolation and, in that isolated state, bring it to a lower altitude, where, as Velasco et al. say, the mean molecular translational kinetic energy in the gas column is higher—but in accordance with the thermodynamic definition the temperature is the same. What happens to the thermometer’s mean molecular kinetic energy when the thermometer is brought into thermal communication with the column at the lower, higher-mean-kinetic-energy altitude?

Although I’ll leave you to draw your own conclusions, I’ll confess that the question is more complicated than it may seem to at first. True, Velasco et al. do say there’s a mean-kinetic-energy difference between altitudes. But that difference is an over-time mean that for any significant number of molecules is no doubt only a minuscule fraction of the that difference’s variance. At any time instant, that is, the mean molecular kinetic energy in one altitude range is only slightly more likely to be less than in a lower altitude range than it is to be higher.

And that doesn’t even take into account quantum mechanics, which for all I know so smears the “time instant” as to make that all meaningless; in this I’m venturing beyond what I understand.

But I do understand logic, and I remain convinced that Dr. Brown’s proof is invalid.

]]>“Two different columns of gas with different lapse rates. Place them in good thermal contact at the bottom, so that the bottoms remain at the same temperature. They must therefore be at different temperatures at the top. Run a heat engine between the two reservoirs at the top and it will run forever, because as fast as heat is transferred from one column to another, (warming the top) it warms the bottom of that column by an identical amount, causing heat to be transferred at the bottom to both cool the column back to its original temperature profile and re-warm the bottom of the other column. The heat simply circulates indefinitely, doing work as it does, until the gas in both columns approaches absolute zero in temperature, converting all of their mutual heat content into work.”

]]>To the extent that’s true, it’s irrelevant to Dr. Brown’s proof. And, to the extent that it’s relevant to Dr. Brown’s proof, it isn’t true.

It is true that an ideal-gas column whose possible microstates constitute a microcanonical ensemble when the column is thermally isolated will, upon thermal communication with another, erstwhile-isolated column, have a vastly expanded ensemble of microstates which is different from that microcanonical ensemble. But how does that fact validate Dr. Brown’s proof or establish the B-E Law, on which his proof is based?

On the other hand, the resultant composite system’s microstates themselves constitute a microcanonical ensemble—with its own non-zero kinetic-energy gradient in the presence of gravity and thus its own non-zero kinetic-theory lapse rate.

It is only if we use the kinetic-theory definition of temperature that Dr. Brown’s initial assumption—a non-zero lapse rate at equilibrium in an isolated gas column—makes any sense. And, under that definition, Velasco et al. show that coupling two different-lapse-rate columns (say, columns having different amounts of the same-molecular-weight ideal gas) would simply result in a new, (also isolated) composite system having a lower but still non-zero kinetic-theory-temperature-definition lapse rate.

Nothing requires the perpetual motion in which the B-E Law says such a coupling would result.

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