# A Beetled Brow

Vinnie’s brow was wrinkling so hard I could hear it over the phone. “Boltzmann, Boltzmann, where’d I hear that name before? … Got it! That’s one of those constants, ain’t it, Sy? Molecules or temperature or something?”

“The second one, Vinnie. Avagadro was the molecule counter. Good memory. Come to think of it, both Boltzmann and Avagadro bridged gaps that Loschmidt worked on.”

“Loschmidt’s thing was the paradox, right, between Newton saying events can back up and thermodynamics saying no, they can’t. You said Boltzmann’s Statistical Mechanics solved that, but I’m still not clear how.”

“Let me think of an example. … Ah, you’ve got those rose bushes in front of your place. I’ll bet you’ve also put up a Japanese beetle trap to protect them.”

“Absolutely. Those bugs would demolish my flowers. The trap’s lure draws them away to my back yard. Most of them stay there ’cause they fall into the trap’s bag and can’t get out.”

“Glad it works so well for you. OK, Newton would look at individual beetles. He’d see right off that they fly mostly in straight lines. He’d measure the force of the wind and write down an equation for how the wind affects a beetle’s flight path. If the wind suddenly blew in the opposite direction, that’d be like the clock running backwards. His same equation would predict the beetle’s new flight path under the changed conditions. You with me?”

“Yeah, no problem.”

“Boltzmann would look at the whole swarm. He’d start by evaluating the average point‑to‑point beetle flight, which he’d call ‘mean free path.’ He’d probably focus on the flight speed and in‑the‑air time fraction. With those, if you tell him how many beetles you’ve got he could generate predictions like inter‑beetle separation and how long it’d take an incoming batch of beetles to cross your yard. However, predicting where a specific beetle will land next? Can’t do that.”

“Well, another beetle might. …
Just thought of a way that Statistical Mechanics could actually be useful in this application. Once Boltzmann has his numbers for an untreated area, you could put in a series of checkpoints with different lures. Then he could develop efficiency parameters just by watching the beetle flying patterns. No need to empty traps. Anyhow, you get the idea.”

“Hey, I feel good emptying that trap, I’m like standing up for my roses. Anyway, so how does Avagadro play into this?”

“Indirectly and he was half a century earlier. In 1805 Gay‑Lussac showed that if you keep the pressure and temperature constant, it tales two volumes of hydrogen to react with one volume of oxygen to produce one volume of water vapor. Better, the whole‑number‑ratio rule seemed to hold generally. Avagadro concluded that the only way Gay‑Lussac’s rule could be general is if at any temperature and pressure, equal volumes of every kind of gas held the same number of molecules. He didn’t know what that number was, though.”

“HAW! Avagadro’s number wasn’t a number yet.”

“Yeah, it took a while to figure out. Then in 1865, Loschmidt and a couple of others started asking, “How big is a gas molecule?” Some gases can be compressed to the liquid state. The liquids have a definite volume, so the scientists knew molecules couldn’t be infinitely small. Loschmidt put numbers to it. Visualize a huge box of beetles flying around, bumping into each other. Each beetle, or molecule, ‘occupies’ a cylinder one beetle wide and the length of its mean free path between collisions. So you’ve got three volumes — the beetles, the total of all the cylinders, and the much larger box. Loschmidt used ratios between the volumes, plus density data, to conclude that air molecules are about a nanometer wide. Good within a factor of three. As a side result he calculated the number of gas molecules per unit volume at any temperature and pressure. That’s now called Loschmidt’s Number. If you know the molecular weight of the gas, then arithmetic gives you Avagadro’s number.”

“Thinking about a big box of flying, rose‑eating beetles creeps me out.”

• Thanks to Oriole Hart for the story‑line suggestion.

~~ Rich Olcott

<chirp, chirp> “Moire here.”

“Hiya, Sy, it’s me again.”

“Hi, Eddie. I thought you were done with your deliveries tonight. That was a good stromboli, by the way, just the right amount of zing and sauce.”

“Thanks. Yeah, I’m done for the day, but I was thinking while I drove home. We said that the Feds and the banks together can tinker with the money supply so there’s no Conservation of Money like we got Conservation of Energy. But then we said that it matters to keep money in local businesses instead of letting it drain away somewhere else. That says there’s only so much to go around like the amount doesn’t change. So which is it?”

“Good point. You’ve touched on another contrasting parallel between Physics and Economics. In Physics we mostly understand how atoms work and we’ve got a pretty good handle on the forces that control objects big enough to see. J Willard Gibbs, probably the foremost physicist of the late 1800s, devised Statistical Mechanics to bridge the gap between the two levels. The idea is to start with the atoms or molecules. They’re quantum objects, of course, so we can’t have much precise information at that level. What we can get, though, is averages and spreads on one object’s properties — speed, internal energy levels, things like that. Imagine we have an ensemble of those guys, mostly identical but each with their own personal set of properties. Gibbs showed us how to apply low-level averages and spreads across the whole ensemble to calculate upper-level properties like magnetic strength and heat capacity.”

“Ensemble. Fancy word.”

“Not my word, blame Gibbs. He invented the field so we go with his terminology. Atoms weren’t quite a respectable topic of conversation at the time so he kept things general and talked about ‘macroscopic properties‘ which we can measure directly and ‘microscopic properties‘ which were mysterious at the time. Think of three flasks holding samples of some kind of gas, OK?”

“No problem.”

“The first flask is stoppered, no gas can get in or out but energy can pass through the flask’s wall. Gibbs would call the confined collection of molecules a ‘canonical ensemble‘. Because the wall transmits energy we can use an external thermometer to measure the ensemble’s temperature. Other than that, all we know about the contents is the number of particles and the volume the particles can access.”

“Canonical?”

“In Gibbs’ usage it means that he’s pared things down to an abstract essence. It doesn’t matter whether what’s inside is atoms or fruitflies, his logic still holds. Now for flask number two. It’s heavily insulated so whatever energy it had inside originally, that’s what it’s got now. We can’t measure the temperature in this one. Gibbs would consider the particles in there to be a ‘microcanonical ensemble,’ with the ‘micro’ indicating the energy restriction.”

“Where there’s a microcanonical there’s gotta be a macrocanonical.”

“You’d think, but Gibbs used the term ‘grand canonical ensemble‘ instead. That’s flask number three, which has neither insulation nor stopper. Both energy and matter are free to enter or leave the ensemble. Gibbs’ notion of canonical ensembles and the math that grows out of them have been used in every kind of analysis from solid state physics to cybersecurity.”

“OK, I think I see where you’re going here. Money acts sorta like energy so you’re gonna lay out three kinds of economy restriction.”

“You’re way ahead of me and the economists, Eddie. They’ve only got two levels, though they do use reasonable names for them — microeconomics and macroeconomics. For them the micro level is about individuals, businesses, the markets they play in and how they spend their incomes. Supply-demand thinking gets used a lot.”

“Macro level is about regions and countries and the world. Supply‑demand plays here, too, except the macroeconomists worry about how demand for money itself affects its value compared to everything else.”

“They got bridges like Gibbs built?”

“Nope. Atoms are simple, people are complicated. The economists are still arguing about the basics. Anyway, the economists’ micro level assumes local money stays local and has a stable value.”

“Keeping my business stable is good.”

~~ Rich Olcott