Where The Temperature Rules The Sun

Mr. Eschenbach: Okay, I’ll do your thought experiment. It’s really not too relevant to my theory, but I’ll do it if you’ll then do mine. I’ll preface my answer, though, with the observation that what I’m asking readers to consider is whether what they think they know of Fourier’s Law (which, remember, is heat flow down a temperature gradient) is completely consistent with what they know about the ballistic motion of an ideal-gas molecule in a gas column at equilibrium subject to gravity.

Now, your thought experiment. You say, “Ceteris paribus, will the temperature of the two sides ever equalize?” Actually, cetera aren’t paria in the situation in which my theory comes into play. Still, your thought experiment, your conditions.

Yes, I believe the temperatures at the silver bodies’ erstwhile-different-temperature ends will equalize. But—and here’s an interesting point—Fourier’s Law actually says they won’t. Not completely. If you solve the partial differential equation that Fourier inflicts upon you, you conclude that temperature difference will decay ever more slowly as the temperature difference decreases, so the difference never exactly reaches zero.

Yet you and I agree that the temperatures will equalize. Here’s what we don’t mean. Let’s say you precisely bisect the silver barbell at the handle into precisely equal-heat-capacity halves. If you could instantaneously measure heat energy in each half to infinite precision, do we mean that your heat-energy measurements would be exactly the same? No. Not exactly. Even at equilibrium the random energy transfers among molecules would make one of the halves ever so slightly hotter than the other. On the other hand, repeating the measurement might instead find the difference reversed. If you do the measurements repeatedly, you will almost always find differences, but some will be opposite the others.

Still, we say the two ends’ temperatures equalize. What we mean is that zero is the average value approached as the number of measurements gets large. Depending on how close to zero you want to find the average to be, though, the number of measurements you’d need to get there repeatably could be large indeed. If the precision to which you want to find zero is small enough with respect to the standard deviation of those individual-measurement differences, reliably finding an average close enough to zero may take more than a lifetime.

Now, as I said, my theory concerns a case in which cetera aren’t paria: there’s a gravitational field with respect to which temperature measurements occur at different altitudes. But what I’m saying that the tiny equilibrium lapse rate I claim gravity would impose upon an ideal gas would be so small with respect to the random variance that detecting it reliably would similarly be time prohibitive.

Okay, so now I’ve done your experiment. How about you do mine? Mine’s basically Dr. Brown’s without the silver wire, except we’ll make the column’s ceiling altitude exceed the molecules’ altitude limit, which is the ratio that the total gas-column (potential + kinetic) energy bears to the least-massive molecule’s weight.

Here’s the experiment. Take whatever time is required (i.e., presumably forever, but this is a thought experiment, so you can do it) to make a repeatable infinite-precision mean-molecular-kinetic-energy measurement of whatever molecules are found in a narrow altitude range that includes that maximum height we mentioned. (A molecule can reach that maximum only in those insanely unlikely but still possible cases in which all the other molecules sit at rest on the enclosure floor.)

Since we’re talking about equilibrium—saying that heat has stopped flowing—Fourier’s Law seems to say such an average measurement at that altitude would be the same as it would be everywhere else: total system kinetic energy (maybe 3/5 the total system energy for a monatomic ideal gas in three dimensions) divided by the number of molecules.

Given that the system’s entire total energy when a molecule reaches that altitude is soaked up by that molecule’s potential energy, though, wouldn’t your knowledge of ballistic motion tell you the average kinetic-energy value at that altitude would be near zero instead? Doesn’t the above interpretation of Fourier’s Law violate conservation of energy? Who you gonna believe, Joe Fourier or your lying concept of ballistic molecular motion and energy conservation?

Well, I know what I say. I say the measurement at that altitude would be near zero. I also say a measurement at zero altitude would be (just a skosh above) what Fourier says, so the lapse rate would be the Fourier equilibrium temperature divided by the molecules’ altitude limit. The four graphs I supplied upthread illustrate this. And the record length (a simulated week and a half) I needed repeatably to generate the last three graphs illustrates that measuring my claimed lapse rate would take too long as a practical matter.

Okay, your mind is rebelling. It seizes upon the fact that in the case of our atmosphere a molecule’s reaching the altitude limit is too improbable to occur before the universe ends. Or that the gravitational uniformity I’ve assumed isn’t even close on that distance scale. Or that I haven’t taken quantum mechanics into account. All those things are true, as are others.

But do those distinctions really make a fundamental difference? If you’re honest, doesn’t this thought experiment make you open to reconsidering how you look at Fourier’s Law?