Taming The Elephant

Suddenly they were all on the attack.  Anne got in the first lick.  “C’mon, Sy, you’re comparing apples and orange peel.  Your hydrogen sphere would be on the inside of the black hole’s event horizon, and Jeremy’s virtual particles are on the outside.”

[If you’ve not read my prior post, do that now and this’ll make more sense.  Go ahead, I’ll wait here.]white satin and 5 elephantsJennie’s turn — “Didn’t the chemists define away a whole lot of entropy when they said that pure elements have zero entropy at absolute zero temperature?”

Then Vinnie took a shot.  “If you’re counting maybe-particles per square whatever for the surface, shouldn’t you oughta count maybe-atoms or something per cubic whatever for the sphere?”

Jeremy posed the deepest questions. “But Mr Moire, aren’t those two different definitions for entropy?  What does heat capacity have to do with counting, anyhow?”

Al brought over mugs of coffee and a plate of scones.  “This I gotta hear.”

“Whew, but this is good ’cause we’re getting down to the nub.  First to Jennie’s point — Under the covers, Hawking’s evaluation is just as arbitrary as the chemists’.  Vinnie’s ‘whatever’ is the Planck length, lP=1.616×10-35 meter.  It’s the square root of such a simple combination of fundamental constants that many physicists think that lP2=2.611×10-70 m², is the ‘quantum of area.’  But that’s just a convenient assumption with no supporting evidence behind it.”

“Ah, so Hawking’s ABH=4πrs2 and SBH=ABH/4 formulation with rs measured in Planck-lengths, just counts the number of area-quanta on the event horizon’s surface.”

“Exactly, Jennie.  If there really is a least possible area, which a lot of physicists doubt, and if its size doesn’t happen to equal lP2, then the black hole entropy gets recalculated to match.”

“So what’s wrong with cubic those-things?”

“Nothing, Vinnie, except that volumes measured in lP3 don’t apply to a black hole because the interior’s really four-dimensional with time scrambled into the distance formulas.  Besides, Hawking proved that the entropy varies with half-diameter squared, not half-diameter cubed.”

“But you could still measure your hydrogen sphere with them and that’d get rid of that 1033 discrepancy between the two entropies.”

“Not really, Vinnie.  Old Reliable calculated solid hydrogen’s entropy for a certain mass, not a volume.”

“Hawking can make his arbitrary choice, Sy, he’s Hawking, but that doesn’t let the chemists off the scaffold.  How did they get away with arbitrarily defining a zero for entropy?”

“Because it worked, Jennie.  They were only concerned with changes — the difference between a system’s state at the end of a process, versus its state at the beginning.  It was only the entropy difference that counted, not its absolute value.”

“Hey, like altitude differences in potential energy.”

“Absolutely, Vinnie, and that’ll be important when we get to Jeremy’s question.  So, Jennie, if you’re only interested in chemical reactions and if it’s still in the 19th Century and the world doesn’t know about isotopes yet, is there a problem with defining zero entropy to be at a convenient set of conditions?”

“Well, but Vinnie’s Second Law says you can never get down to absolute zero so that’s not convenient.”

“Good point, but the Ideal Gas Law and other tools let scientists extrapolate experimentally measured properties down to extremely low temperatures.  In fact, the very notion of absolute zero temperature came from experiments where the volume of a  hydrogen or helium gas sample appears to decrease linearly towards zero at that temperature, at least until the sample condenses to a liquid.  With properly calibrated thermometers, physical chemists knocked themselves out measuring heat capacities and entropies at different temperatures for every substance they could lay hands on.”

“What about isotopes, Mr Moire?  Isn’t chlorine’s atomic weight something-and-a-half so there’s gotta be several of kinds of chlorine atoms so any sample you’ve got is a mixture and that’s random and that has to have a non-zero entropy even at absolute zero.”

“It’s 35.4, two stable isotopes, Jeremy, but we know how to account for entropy of mixing and anyway, the isotope mix rarely changes in chemical processes.”

“But my apples and orange peels, Sy — what does the entropy elephant do about them?”

~~ Rich Olcott

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Smack-dab in the middle

BridgeSee that little guy on the bridge, suspended halfway between all the way down and all the way up?  That’s us on the cosmic size scale.

I suspect there’s a lesson there on how to think about electrons and quantum mechanics.

Let’s start at the big end.  The physicists tell us that light travels at 300,000 km/s, and the astronomers tell us that the Universe is about 13.7 billion years old.  Allowing for leap years, the oldest photons must have taken about 4.3×1017 seconds to reach us, during which time they must have covered 1.3×1026 meters.  Double that to get the diameter of the visible Universe, 2.6×1026 meters.  The Universe probably is even bigger than that, but far as I can see that’s as far as we can see.

At the small end there’s the Planck length, which takes a little explaining.  Back in 1899, Max Planck published his epochal paper showing that light happens piecewise (we now call them photons).  In that paper, he combined several “universal constants” to derive a convenient (for him) universal unit of length: 1.6×10-35 meters.  It’s certainly an inconvenient number for day-to-day measurements (“Gracious, Junior, how you’ve grown!  You’re now 8×1034 Planck-lengths tall.”).  However, theoretical physicists have saved barrels of ink and hours of keyboarding by using Planck-lengths and other such “natural units” in their work instead of explicitly writing down all the constants.

Furthermore, there are theoretical reasons to believe that the smallest possible events in the Universe occur at the scale of Planck lengths.  For instance, some theories suggest that it’s impossible to measure the distance between two points that are closer than a Planck-length apart.  In a sense, then, the resolution limit of the Universe, the ultimate pixel size, is a Planck length.

sizelineSo that’s the size range of the Universe, from 1.6×10-35 up to 2.6×1026 meters. What’s a reasonable way to fix a half-way mark between them?

It makes no sense to just add the two numbers together and divide by two the way we’d do for an arithmetic average. That’d be like adding together the dime I owe my grandson and the US national debt — I could owe him 10¢ or $10, but either number just disappears into the trillions.

The best way is to take the geometrical average — multiply the two numbers and take the square root.  I did that.  It’s the X in the sizeline, at 6.5×10-5 meters, or about the diameter of a fairly large bacterium.  (In the diagram, VSC is the Vega Super Cluster, AG is the Andromeda Galaxy, and the numbers are those exponents of 10.)

That’s worth marveling at.  Sixty orders of magnitude between the size of the Universe and the size of the ultimate pixel.  Yet from blue whales to bacteria, Earth’s life just happens to occupy the half-dozen orders right in the middle of the range.  We think that’s it.

Could this be another case of the geocentric fallacy?  Humans were so certain that Earth was the center of the Universe, before Brahe and Galileo and Newton proved otherwise.  Is there life out there at scales much larger or much smaller than we imagine?

Who knows? But here’s an intriguing physics/quantum angle I’d like to promote.  We know a lot about structures bigger than us — solar systems and binary stars and galaxy clusters on up.  We know a few sizes and structures a bit smaller — viruses and molecules and atoms.  We’re aware of quarks and gluons that reside inside protons and atomic nuclei, but we don’t know their size or structure.

Even a proton is huge on the Planck-length scale.  At 1.8×10-15 meters the proton measures some 1020 Planck-lengths.  There’s as much scale-space between the Planck-length and the proton as there is between the Earth (1.3×107 meters) and the Universe.

It’s hard to believe that Terra infravita’s area has no structure whereas Terra supravita is so … busy.  The Standard Model’s “ultimate particles,” the electrons and photons and neutrinos and quarks and gluons, all operate down there somewhere.   It’s reasonable to suppose that they reflect a deeper architecture somewhere on the way down to the Planck-length foam.

Newton wrote (in Latin), “I do not make hypotheses.”  But golly, it’s tempting.

~~ Rich Olcott