“My apples and orange peels question, Sy, isn’t that the same as Jeremy’s? What’s the connection between heat capacity and counting?”
“You’re right, Anne. Hmm. Say, Al, all your coffee shop tables came with four chairs apiece, right?”
“Yup, four-tops every one, even in the back room.”
“You neaten them all up, four to a table, in the morning?”
“The night before. There’s never time in the morning, customers demand coffee first thing.”
“But look, we’ve got six people seated at this table. Where’d the extra chairs come from?”
“Other tables, of course. Is this going somewhere?”
“Almost there. So in fact the state of the room at any time will have some random distribution of chairs to tables. You know on the average there’ll be four at a table, but you don’t know the actual distribution until you look, right?”
“Hey, we’re counting again. You’re gonna say that’s about entropy ’cause the difference between four at a table and some other number is all random and there’s some formula to calculate entropy from that.”
“True, Vinnie, but we’re about to take the next step. How did these chairs wind up around this table?”
“We pulled them over, Mr. Moire.”
“My point is, Jeremy, we spent energy to get them here. The more chairs that are out of position — ”
“The higher the entropy, but also the more energy went into the chairs. It’s like that heat capacity thing we started with, the energy that got absorbed rather than driving the steam engine.”
“Awright, Anne!” from Jeremy <Jennie bristles a bit>, “and if all the chairs are in Al’s overnight position it’s like absolute zero. Hey, temperature is average kinetic energy per particle so can we say that the more often a chair gets moved it’s like hotter?”
Jennie breaks in. “Not a bit of it, Jeremy! The whole metaphor’s daft. We know temperature change times heat capacity equals the energy absorbed, right, and we’ve got a link between energy absorption and entropy, right, but what about if at the end of the day all the chairs accidentally wind up four at a table? Entropy change is zero, right, but customers expended energy moving chairs about all day and Al’s got naught to set straight.”
“Science in action, I love it! Anne and Jeremy, you two just bridged a gap it took Science a century to get across. Carnot started us on entropy’s trail in 1824 but scientists in those days weren’t aware of matter’s atomic structure. They knew that stuff can absorb heat but they had no inkling what did the absorbing or how that worked. Thirty years later they understood simple gases better and figured out that average kinetic energy per particle bit. But not until the 1920s did we have the quantum mechanics to show how parts of vibrating molecules can absorb heat energy stepwise like a table ‘absorbing’ chairs. Only then could we do Vinnie’s state-counting to calculate entropies.”
“Yeah, more energy, spread across more steps, hiding more details we don’t know behind an average, more entropy. But what about Jennie’s point?”
“Science is a stack of interconnected metaphors, Vinnie. Some are better than others. The trick is attending to the boundaries where they stop being valid. Jennie’s absolutely correct that my four-chair argument is only a cartoon for illustrating stepwise energy accumulation. If Al had a billion tables instead of a dozen or so, the odds on getting everything back to the zero state would disappear into rounding error.”
“How does black hole entropy play into this, Sy?”
“Not very well, actually. Oh, sure, the two systems have similar structures. They’ve each got three inter-related central quantities constrained by three laws. Here, I’ve charted them out on Old Reliable.”
“OK, their Second and Third Laws look pretty much the same, but their First Laws don’t match up.”
“Right, Al. And even Bekenstein pointed out inconsistencies between classic thermodynamic temperature and what’s come to be called Hawking temperature. Hawking didn’t agree. The theoreticians are still arguing. Here’s a funny one — if you dig deep enough, both versions of the First Law are the same, but the Universe doesn’t obey it.”
“That’s it, closing time. Everybody out.”
~~ Rich Olcott
Jennie’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?”
“That’s not quite what I said, Jennie. Old Reliable’s software and and I worked up a hollow-shell model and to my surprise it’s consistent with one of Stephen Hawking’s results. That’s a long way from saying that’s what a black hole is.”
My notes say D is the black hole’s diameter and d is another object’s distance from its center. One second in the falling object’s frame would look like f seconds to us. But one mile would look like 1/f miles. The event horizon is where d equals the half-diameter and f goes infinite. The formula only works where the object stays outside the horizon.”
“Wow, Old Reliable looks up stuff and takes care of unit conversions automatically?”
Hard Science Ain't Hard 