Tag: Hysteresis

The Latte Connection

On By rolcottIn Physics: Entropy and Thermodynamics, UncategorizedLeave a comment

An early taste of Spring’s in the air so Al’s set out tables in front of his coffee shop. I’m enjoying my usual black mud when the Chemistry Department’s Susan Kim passes by carrying her usual mocha latte. “Hi, Sy, mind if I take the socially distant chair at your table?”

“Be my guest, Susan. What’s going on in your world?”

“I’ve been enjoying your hysteresis series. It took me back to Physical Chemistry class. I’m intrigued by how you connected it to entropy.”

“How so?”

“I think of hysteresis as a process, but entropy is a fixed property of matter. If I’m holding twelve grams of carbon at room temperature, I know what its entropy is.”

“Mmm, sorta. Doesn’t it make a difference whether the carbon’s a 60‑carat diamond or just a pile of soot?”

“OK, I’ll give you that, the soot’s a lot more random than the diamond so its entropy is higher. The point remains, I could in principle measure a soot sample’s heat capacity at some convenient temperature and divide that by the temperature. I could repeat that at lower and lower temperatures down to near absolute zero. When I sum all those measurements I’ll have the entropy content of the sample at my starting temperature.”

“A classical definition, just what I’d expect from a chemist. But suppose your soot spills out of its test tube and the breeze spreads it all over the neighborhood. More randomness, higher entropy than what you measured, right?”

“Well, yes. I wouldn’t have a clue how to calculate it, but that goes way beyond Carnot’s and Clausius’ original concept.”

“So entropy has at least a thin linkage with history and hysteresis. To you chemists, though, an element or compound is timeless — lead or water have always been lead or water, and their physical constants are, well, constant.”

“Not quite true, Sy. Not with really big molecules like proteins and DNA and rubber and some plastics. Squirt a huge protein like catalase through a small orifice and its properties change drastically. It might not promote any reaction, much less the one Nature designed it for. Which makes me think — Chemistry is all about reactions and they take time and studying what makes reactions run fast or slow is a big part of the field. So we do pay attention to time.”

“Nice play, Susan! You’re saying small molecules aren’t complex enough to retain memories but big ones are. I’ll bet big molecules probably exhibit hysteresis.”

“Sure they do. Rubber molecules are long-chain polymers. Quickly stretch a rubber band to its limit, hold it there a few seconds then let go. Some of the molecular strands lock into the stretched configuration so the band won’t immediately shrink all the way down to its original size. There’s your molecular memory.”

“And a good example it is — classic linear Physics. How much force you exert, times the distance you applied it through, equals the energy you expended. Energy’s stored in the rubber’s elasticity when you stretch it, and the energy comes back out on release.”

“Mostly right, Sy. You actually have to put in more energy than you get out — Second Law of Thermodynamics, of course — and the relationship’s not linear. <rummaging into purse> Thought I had a good fat rubber band somewhere … ah‑hah! Here, stretch this out while you hold it against your forehead. Feel it heat up briefly? Now keep checking for heat while you relax the band.”

“Hey, it got cold for a second!”

“Yep. The stretched-out configuration is less random so its entropy and heat capacity are lower than the relaxed configuration’s. The stretched band had the same amount of heat energy but with less heat required per degree of temperature, that amount of energy made the band hotter. Relaxing the band let its molecules get less orderly. Heat capacity went back up. temperature went back down.”

“Mmm-HM. My hysteresis diagram’s upward branch is stretch energy input and the downward branch is elastic energy output. The energy difference is the area inside the hysteresis curve, which is what’s lost to entropy in each cycle and there we have your intriguing entropy‑hysteresis connection. Still intrigued?”

“Enough for another latte.”

~~ Rich Olcott

Hysteresis Everywhere

On By rolcottIn Physics: Gaps, Physics: Money, UncategorizedLeave a comment

“We’ve known each other for a long time, ain’t we, Sy?”

“That we have, Vinnie.”

“So I get suspicious when we’ve specific been talking about a magnetic field making something else magnetic and you keep using general words like ‘driver‘ and ‘deviation‘. You playing games?”

“You caught me. The hysteresis idea spreads a lot farther than magnetism. It addresses an entire dimension Newton was too busy to think about — time.”

“Wait a minute. Newton was all about velocity and acceleration and both of them are something‑per‑time. It’s right there in the units. Twice for acceleration.”

“True, but each is really about brief time intervals. Say you’re riding a roller‑coaster. Your velocity and acceleration change second‑by‑second as forces come at you. Every force changes your net acceleration immediately, not ten minutes from now. Hysteresis is about change that happens because of a cause some time in the past. Newton didn’t tackle time‑offset problems, I suppose mostly because the effects weren’t detectable with the technology of his time.”

“They had magnets.”

“Permanent ones, not electromagnets they could control and measure the effects of. Electromagnetic hysteresis generates effects that Newton couldn’t have known about. Fahrenheit didn’t invent temperature measurement until two years before Newton died, so science hadn’t yet discovered temperature‑dependent hysteresis effects. The microscope had been around for a half‑century or so but in Newton’s day people were still arguing about whether cells were a necessary part of a living organism. Newton’s world didn’t have an inkling of cellular biophysics, much less biophysical hysteresis. At human scale, country‑level economic data if it existed at all was a military secret — not a good environment for studying cases of economic hysteresis.”

“So what you’re saying is that Newton couldn’t have tackled those even if he’d wanted to. Got it. But that’s a pretty broad list of situations. How can you say they’re all hystereseseses, … loopy things?”

“They’ve all got a set of characteristics that you can fit into similar mathematical models. They’re all about some statistical summary of a complex system. The system is under the influence of some outside driver, could be a physical force or something more abstract. The driver can work in either of two opposing directions, and the system can respond to the driver to change in either of two opposing ways. Oh, and a crucial characteristic is that the system has a buffer of some sort that saves a memory of what the driver did and serves it up some time later.”

“Wait, lemme see if I can match those pieces to my magnetic nail. OK, the driver is the outside magnetic field, that’s easy, the system is the magnetic iron atoms, and the summary is the nail’s field. The driver can point north‑to‑south or south‑to‑north and the atoms can, too. Ah, and the memory is the domains ’cause the big ones hold onto the direction the field pointed last. How’d I do?”

“Perfect.”

“Goody for me. So why are those guys on the radio saying the economy is hysterical, ‘scuse, has hysteresis? What’s which part?”

“Economies are complex beasts, with a lot of separate but interacting hysteresis loops. These guys, what were they discussing at the time?”

“Unemployment, if I remember right. They said the job market is sticky, whatever that means.”

“Good example. Here’s our basic hysteresis loop with some relabeling. Running across we’ve got our driver, the velocity of money, which claims to measure all the buying and selling. Up‑and‑down we’ve got total employment. The red dot is the initial equilibrium, some intermediate level where there’s just enough cash flowing around that some but not all people have jobs. Then a new industry, say cellphones, comes in. Suddenly there’s people making cellphones, selling cellphones, repairing cellphones –“

“I get the idea. More activity, money flows faster, more jobs and people are happy. OK, then the pandemic comes along, money slows down, jobs cut back and around we go. But where’s the stickiness?”

“In people’s heads. If they get into Depression thinking, everyone holds onto cash even if there’s a wonderful new cellphone out there. People have to start thinking that conditions will improve before conditions can improve. That’s the delay factor.”

“Hysterical, all right.”

~~ Rich Olcott

Elephant And Pengy

On By rolcottIn Physics: Entropy and Thermodynamics, Physics: GapsLeave a comment

(a hat-tip to Mo Willems, whose Elephant and Piggy books helped my grandkids discover reading)

“Hey, Sy, how come my magnetized nail’s hysteresis loop is so wide? It makes sense that the end‑case magnetizing happens because all the iron atoms get lined up in one direction or the other. But why ain’t the blue up‑curve right on top of the down‑curve?”

“Why do you think it should be, Vinnie?”

“Well, the red curve’s different because you got the outside field herding the iron atoms into domains where they all point the same way and that makes the nail’s magnetism grow from zero, and then the domains that agree with the outside magnetic field eat up the other domains until like I said they saturate. But on both sides of the blue loop the domains already exist, right, so the herding’s all done. Up or down it’s only domains growing and shrinking. Seems to me that the curves oughta be the same.”

“They are, near as I could draw them. You’re just not looking at them right. Rotate it 180°, see how they match up.”

“How ’bout that, they do, mostly. What’s going on?”

“You picked up that the vertical axis represents strength and direction, but you missed that the horizontal axis also represents strength and direction. Neither axis starts at zero, they’re both centered on zero. The driver is the outside magnetic field. No strength in the middle, increasing north‑bound strength to the right, south‑bound strength to the left. Start at the head‑end‑north corner and go down branch 2. The north‑bound driver strength decreases. That relaxes some of those north‑pointing domains and the nail’s net magnetism decreases just a bit. When the outside field’s strength gets down to neutral, about at the upper arrow, the nail’s still strongly magnetized. Most of the domains remember which way they were pointing. That’s the history that makes this hysteresis. The domains stay there until the outside field gets strong enough south‑bound to make a difference. That grows the south‑bound domains at the expense of northbound ones. All that goes on until we get to saturation at head‑end‑south corner and then we run exactly the reverse sequence. For most materials, the two extreme fields have the same strength, just opposite directions.”

“Wait, you said ‘for most materials.’ Different materials have different widths on that picture?”

“Good catch. Yes, there’s ‘hard‘ ones like rare earth magnets. They have a really wide hysteresis loop you can’t demagnetize without a really strong field. That’s good for where you want a permanent magnet that you don’t want to have to recalibrate, like on a spacecraft bound for Jupiter. You’d want a ‘soft magnet‘ with a narrow hysteresis loop for something like a transformer core that has to switch polarity sixty times a second.”

<longish contemplative silence> “Sy, I just got a great idea! And it uses that entropy elephant stuff you wrote about.”

“All right, out with it.”

“OK. When the nail is magnetized, it’s got all or at least most of its iron atoms pointing in the same direction, right? And when the outside field demagnetizes it, the atoms point all over the place, right? So the not‑magnetized nail has randomness, that’s entropy, and the magnetized one doesn’t. Where did the entropy come from? Gotta be from the outside, right? Can we use this to like suck entropy out of things?”

“Right, right and sorta right. I’m not happy with the idea of pumping entropy around. What’s really in play is energy, sometimes as magnetic field energy and sometimes as heat. You’ve got the core idea for a magnetic refrigerator. Put a field‑magnetized transfer material in contact with what you want to cool, then turn off the outside field. Heat from the target flows into the material, jiggles the atoms and scrambles the magnetization. Break the contact, cool and re‑magnetize the material and repeat. The idea’s been around since the late 1800s. The problem has been finding the right material to make it work. The best stuff has a tall, narrow hysteresis loop so it can be strongly magnetized yet forget it easily. Researchers have finally found some good candidates.”

“Too late to the party, huh?”

“Sorry.”

~~ Rich Olcott

The Tale of A Nail

On By rolcottIn Physics: Electromagnetism, Physics: Gaps, UncategorizedLeave a comment

“Wait, Sy, let me get my head around that hysteresis loop diagram. You got my iron nail starting at that red dot because it’s not magnetized yet so that’s zero on the up‑down magnetism deviation scale, right? And it’s also zero on the left‑right driver scale because we’re not laying a magnetic field on it.”

“Yup, that’s the starting point, Vinnie.”

“OK, then we turn on the outside field and if it’s strong enough the nail gets magnetic, too, and so we travel up the red line. But the line’s not straight, it’s bendy. Why ain’t it straight?”

“To keep this specific, I’ll stick to the current theory for magnetization of iron. At point zero the individual iron atoms have their personal magnetic fields in completely random orientations. What we measure outside the nail is the average of all of that, which nets out to zero. Now we turn on the external magnetic field a little bit at a time so we can measure the effect. You remember we said that the iron atoms in a magnet are organized in domains.”

“Sure. I don’t forget easy.”

“I’ve noticed. OK, that upward bend at the beginning is slow increase in the nail’s magnetization while those domains are forming up. First a few atoms in one small area orient their local fields relative to the external field. Their combined field influences neighboring atoms to join in. The process is called nucleation because those first few atoms form the nucleus of a domain. The nucleus gains strength by recruiting more atoms, making it an even stronger recruiter. The red line rises exponentially until there aren’t any more unrecruited atoms.”

“That’s the end of the upward bend, huh?”

“Mm-hm, now we enter the linear phase and a different magnetization process. Energy in the external field feeds the domains pointed parallel to it at the expense of domains at a different angle. Domain growth is roughly linear with applied field strength. That line would like to stay straight but nothing goes on forever except maybe the Universe. Sooner or later the domains start running out of room to grow into. Increasing the driver strength doesn’t produce any further effect and we say that the nail’s magnetic field is saturated.”

“That makes sense. Let’s see if I can figure the blue loop from where the head end is north. The number 2 arrow says that if we dial down the driver, that’s the outside field and we’re moving to the left, when we get to zero the deviation, that’s the nail’s field, is still going strong and we got a permanent magnet. If we adjust the outside field leftward beyond zero that kills off the nail’s field … Hey, so the backward domains are eating the forward ones, right?”

“Probably. Depends on the material. Not good to ride the theory too far without checking the experimental data but that’d be my guess.”

“OK, so we drive those little domains until they saturate with the head end south. When we dial down the driver’s field backward strength we move to the right and the nail climbs the number 3 curve. The driver field returns to zero but the nail’s still a backward permanent magnet. We push the driver and the nail to forward saturation again and we can go loop‑de‑loop. But we never go through the red dot again — either the nail’s a permanent magnet when the driver’s zero or it not a magnet while the driver’s strong but they’re never both zero again.”

“Unless we scramble all the domains by heating the nail white-hot and letting it cool away from any external fields.”

“You know what’s missing from that picture, Sy?”

I’d wondered if he’d spot it. “I’ll bite. What?”

“Numbers. Up‑down is how strong the magnet is, right, but I know my knife‑holder magnets are a lot stronger than my calendar marker magnets. And the side‑to‑side part is about how well the stuff holds its magnetism. What’s the theory that puts numbers on the graph?”

“Sorry to tell you this given your math aversion, Vinnie, but the numbers are buried in big, thick books with equations in them. Pictures can only get you so far.”

~~ Rich Olcott

The Hysterical Penguin

On By rolcottIn Mathematics, Physics: Gaps, Physics: Newton and Gravity, UncategorizedLeave a comment

“Sy, you said that hysteresis researchers filled in two of Newton’s Physics gaps. OK, I get that he couldn’t do atomic stuff ’cause atoms hadn’t been discovered yet. What’s the other one?”

Proposition XI, Problem VI
from Book I of Newton’s Principia

“Non‑linearity.”

“You’re gonna have to explain that.”

“It’s a math thing. I know you don’t go for equations, so here’s a picture to get you started on how Newton solved problems. Look at all familiar?”

“Whoa, looks like something toward the end of my Geometry class.”

“Exactly. Newton was trained as a geometer and he was good at it. His general strategy was to translate a physical system to a geometrical structure and then work out its properties as a series of geometric proofs. The good news was that he proved a lot of things that started us on the way to quantitative science. The bad news was that his proofs were hard to extend to situations where the geometry wasn’t so easy.”

“That’s easy?”

“For Newton, maybe it was. Who knows? Anyway, the toolkit they gave you in Geometry class was what Newton had to work with — logic, straight lines and some special curves like ellipses and parabolas whose properties had been studied since Euclid, all on a flat plane. Nearly everything depended on finding proportionalities between different distances or areas — this line is twice that one but equal to a third, that sort of thing. Proportionality like that is built into equations like here+(velocity×time)=there. See how distance traveled is proportional to time? The equation plots as a straight line, which is why it’s called a linear equation.”

“So what’s non‑linear look like — all wiggle‑waggle?”

“Not necessarily. Things can vary smoothly along curves that aren’t those classical ones. Newton’s methods are blocked on those but Leibniz’s algebra‑based calculus isn’t. That’s why it won out with people who needed answers. What’s important here is that Newton’s lines can’t describe everything. Mmm… where does a straight line end?”

“Either at a T or never. Same thing for a parabola. Hey, ellipses don’t really end, either.”

“Mm-hm. Newton’s lines either stop abruptly or they continue forever. They don’t grow or peter out exponentially like things in real life do. Suppose something’s velocity changes, for instance.”

“That’s acceleration. I like accelerating.”

“So true, I’ve experienced your driving. But even you don’t accelerate at a constant rate. You go heavy or light or maybe brake, whatever, and our speed goes up or down depending. The only way Newton’s geometry can handle variable acceleration is to break it into mostly‑constant pieces and work one piece at a time. Come to think of it, that may be where he got the idea for his fluxions method for calculus. Fortunately for him, some things like planets and artillery shells move pretty close to what his methods predict. Unfortunately, things like disease epidemics and economies don’t, which is why people are interested in non‑linearity.”

“So what do these hysteresis guys do about it?”

“Mostly algebraic calculus or computer approximations. But there wasn’t just one group of hysteresis guys, there was a bunch of groups, each looking at different phenomena where history makes a difference. Each group had their own method of attack.”

“Like your elephant thing with Anne, lots of notions about entropy.”

Typical hysteresis loop
Red — initial evolution
Blue — subsequent changes

“How’d you find out about that?”

“You wrote those posts, Sy, about three years ago.”

“Oh, that’s right. Talk about history. Anyway, it took decades for the ecologists, epidemiologists, civil engineers and several kinds of physicist to realize that they all have systems that behave similarly when driven by a stressor. Starting at some neutral situation, the system evolves in the driver’s direction to some maximum deviation where increased stress has no further effect. When the stress is relieved, the system may stick temporarily at the strained position. When it does evolve away from there, maybe a reverse driver is needed to force a return to the starting situation. In fact, if the forward and reverse drivers are applied repeatedly the system may never get back to the initial unstressed position.”

“Like that iron nail. Not magnetic, then magnetic, then reversed.”

~~ Rich Olcott

Hysteria

On By rolcottIn Physics: Gaps, Physics: Money, UncategorizedLeave a comment

<chirp, chirp> “Moire here.”

“Hi, Sy, it’s Vinnie again. Hey, I just heard something on NPR I wanted to check with you on.”

“What’s that?”

“They said that even with the vaccine and all, it’s gonna take years for us to get back to normal ’cause the economy’s hysterical. Does that mean it’s cryin’‑funny or just cryin’? Neither one seems to fit.”

“You’re right about the no‑fit. Hmm… Ah! Could the word have been ‘hysteresis‘?”

“Somethin’ like that. What’s it about?”

“It’s an old Physics word that’s been picked up by other fields. Not misused as badly as ‘quantum,’ thank goodness, but still. The word itself gives you a clue. Do you hear the ‘history‘ in there?”

“Hysteresis, history … cute. So it’s about history?”

“Yup. The classic case is magnetism. Take an iron nail, for instance. The nail might already be magnetized strongly enough to pick up a paper clip. If it can, you can erase the magnetism by heating the nail white‑hot. If the nail’s not magnetic you may be able to magnetize it by giving it a few hammer‑whacks while it’s pointed north‑south, parallel to Earth’s magnetic field. Things get more interesting if we get quantitative. A strong‑enough magnetic field will induce magnetism in that nail no matter what direction it’s pointed. Reverse that field’s direction and the nail stays magnetized, only less so. It takes a stronger reverse field to demagnetize the nail than it took to magnetize it in the first place. See how the history makes a difference?”

“Yeah, for some things.”

“And that’s the point. Some of a system’s properties are as fixed as the nail’s weight or chemical composition. However, it may have other properties we can’t understand without knowing the history. Usually we can’t even predict them without looking at deeper structures. Hysteresis highlights two more gaps in Newton’s Physics. As usual he’s got a good excuse because many history‑dependent phenomena couldn’t even be detected with 17th‑Century technology. We couldn’t produce controllable magnetic fields until the 19th Century, when Oersted and Ampere studied magnetism and electricity. We didn’t understand magnetic hysteresis until the 20th Century.”

“Haw! You’re talking history of history. Anyway, to me it looks like what’s going on is that the strong field gets the magnetic atoms in there to all point the same way and heat undoes that by shaking them up to point random‑like.”

“What about the reversing field?”

“Maybe it points some of the atoms in the other direction and that makes the nail less and less magnetic until the field is strong enough to point everything backwards.”

“Close enough. The real story is that the atoms, iron in this case, are organized in groups called domains. The direction‑switching happens at the domain level — battalions of magnetically aligned atoms — but we had no way to know that until 20th‑Century microscopy came along.”

“So it takes ’em a while to get rearranged, huh?”

“Mmm, that’d be rate-dependent hysteresis, where the difference between forward and backward virtually disappears if you go slow enough. Think about putting your hand slowly into a tub of water versus splashing in there. Slow in, slow out reverses pretty well, but if you splash the water’s in turmoil for quite a long time. Magnetic hysteresis, though, doesn’t care about speed except in the extreme case. It’s purely controlled by the strength of the applied field.”

“I’m thinking about that poor frog.”

‘You would go there, wouldn’t you? Yeah, the legendary frog in slowly heating water would be another history dependency but it’s a different kind. The nail’s magnetism only depends on atoms standing in alignment. A frog is a highly organized system, lots of subsystems that all have to work together. Warming water adds energy that will speed up some subsystems more than others. If Froggy exits the pot before things desynchronize too far then it can recover its original lively state. If it’s trapped in there you’ve got frog soup. By the way, it’s a myth that the frog won’t try to hop out if you warm the water slowly. Frogs move to someplace cool if they get hotter than their personal threshold temperature.”

“Frogs are smarter than legends, huh?”

~~ Rich Olcott