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.”

“Who cares about one beetle?”

“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.”

Japanese Beetle, photo by David Cappaert, Bugwood.org
under Creative Commons BY 3.0

“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

Bridging A Paradox

<chirp, chirp> “Moire here.”

“Hi, Sy. Vinnie. Hey, I’ve been reading through some of your old stuff—”

“That bored, eh?”

“You know it. Anyhow, something just don’t jibe, ya know?”

“I’m not surprised but I don’t know. Tell me about it.”

“OK, let’s start with your Einstein’s Bubble piece. You got this electron goes up‑and‑down in some other galaxy and sends out a photon and it hits my eye and an atom in there absorbs it and I see the speck of light, right?”

“That’s about the size of it. What’s the problem?”

“I ain’t done yet. OK, the photon can’t give away any energy on the way here ’cause it’s quantum and quantum energy comes in packages. And when it hits my eye I get the whole package, right?”

“Yes, and?”

“And so there’s no energy loss and that means 100% efficient and I thought thermodynamics says you can’t do that.”

“Ah, good point. You’ve just described one version of Loschmidt’s Paradox. A lot of ink has gone into the conflict between quantum mechanics and relativity theory, but Herr Johann Loschmidt found a fundamental conflict between Newtonian mechanics, which is fundamental, and thermodynamics, which is also fundamental. He wasn’t talking photons, of course — it’d be another quarter-century before Planck and Einstein came up with that notion — but his challenge stood on your central issue.”

“Goody for me, so what’s the central issue?”

“Whether or not things can run in reverse. A pendulum that swings from A to B also swings from B to A. Planets go around our Sun counterclockwise, but Newton’s math would be just as accurate if they went clockwise. In all his equations and everything derived from them, you can replace +t with ‑t to make run time backwards and everything looks dandy. That even carries over to quantum mechanics — an excited atom relaxes by emitting a photon that eventually excites another atom, but then the second atom can play the same game by tossing a photon back the other way. That works because photons don’t dissipate their energy.”

“I get your point, Newton-style physics likes things that can back up. So what’s Loschmidt’s beef?”

“Ever see a fire unburn? Down at the microscopic level where atoms and photons live, processes run backwards all the time. Melting and freezing and chemical equilibria depend upon that. Things are different up at the macroscopic level, though — once heat energy gets out or randomness creeps in, processes can’t undo by themselves as Newton would like. That’s why Loschmidt stood the Laws of Thermodynamics up against Newton’s Laws. The paradox isn’t Newton’s fault — the very idea of energy was just being invented in his time and of course atoms and molecules and randomness were still centuries away.”

“Micro, macro, who cares about the difference?”

“The difference is that the micro level is usually a lot simpler than the macro level. We can often use measured or calculated micro‑level properties to predict macro‑level properties. Boltzmann started a whole branch of Physics, Statistical Mechanics, devoted to carrying out that strategy. For instance, if we know enough about what happens when two gas molecules collide we can predict the speed of sound through the gas. Our solid‑state devices depend on macro‑level electric and optical phenomena that depend on micro‑level electron‑atom interactions.”

“Statistical?”

“As in, ‘we don’t know exactly how it’ll go but we can figure the odds…‘ Suppose we’re looking at air molecules and the micro process is a molecule moving. It could go left, right, up, down, towards or away from you like the six sides of a die. Once it’s gone left, what are the odds it’ll reverse course?”

“About 16%, like rolling a die to get a one.”

“You know your odds. Now roll that die again. What’s the odds of snake‑eyes?”

“16% of 16%, that’s like 3 outa 100.”

“There’s a kajillion molecules in the room. Roll the die a kajillion times. What are the odds all the air goes to one wall?”

“So close to zero it ain’t gonna happen.”

“And Boltzmann’s Statistical Mechanics explained why not.”

“Knowing about one molecule predicts a kajillion. Pretty good.”

San Francisco’s Golden Gate Bridge, looking South
Photo by Rich Niewiroski Jr. / CC BY 2.5

~~ Rich Olcott