My All 2020 Mathematics A to Z: J Willard Gibbs

Charles Merritt sugested a biographical subject for G. (There are often running themes in an A-to-Z and this year’s seems to be “biography”.) I don’t know of a web site or other project that Merritt has that’s worth sharing, but if I learn of it, I’ll pass it along.

J Willard Gibbs.

My love and I, like many people, tried last week to see the comet NEOWISE. It took several attempts. When finally we had binoculars and dark enough sky we still had the challenge of where to look. Finally determined searching and peripheral vision (which is more sensitive to faint objects) found the comet. But how to guide the other to a thing barely visible except with binoculars? Between the silhouettes of trees and a convenient pair of guide stars we were able to put the comet’s approximate location in words. Soon we were experts at finding it. We could turn a head, hold up the binoculars, and see a blue-ish puff of something.

To perceive a thing is not to see it. Astronomy is full of things seen but not recognized as important. There is a great need for people who can describe to us how to see a thing. And this is part of the significance of J Willard Gibbs.

American science, in the 19th century, had an inferiority complex compared to European science. Fairly, to an extent: what great thinkers did the United States have to compare to William Thompson or Joseph Fourier or James Clerk Maxwell? The United States tried to argue that its thinkers were more practical minded, with Joseph Henry as example. Without downplaying Henry’s work, though? The stories of his meeting the great minds of Europe are about how he could fix gear that Michael Faraday could not. There is a genius in this, yes. But we are more impressed by magnetic fields than by any electromagnet.

Gibbs is the era’s exception, a mathematical physicist of rare insight and creativity. In his ability to understand problems, yes. But also in organizing ways to look at problems so others can understand them better. A good comparison is to Richard Feynman, who understood a great variety of problems, and organized them for other people to understand. No one, then or now, doubted Gibbs compared well to the best European minds.

Gibbs’s life story is almost the type case for a quiet academic life. He was born into an academic/ministerial family. Attended Yale. Earned what appears to be the first PhD in engineering granted in the United States, and only the fifth non-honorary PhD in the country. Went to Europe for three years, then came back home, got a position teaching at Yale, and never left again. He was appointed Professor of Mathematical Physics, the first such in the country, at age 32 and before he had even published anything. This speaks of how well-connected his family was. Also that he was well-off enough not to need a salary. He wouldn’t take one until 1880, when Yale offered him two thousand per year against Johns Hopkins’s three.

Between taking his job and taking his salary, Gibbs took time to remake physics. This was in thermodynamics, possibly the most vibrant field of 19th century physics. The wonder and excitement we see in quantum mechanics resided in thermodynamics back then. Though with the difference that people with a lot of money were quite interested in the field’s results. These were people who owned railroads, or factories, or traction companies. Extremely practical fields.

What Gibbs offered was space, particularly, phase space. Phase space describes the state of a system as a point in … space. The evolution of a system is typically a path winding through space. Constraints, like the conservation of energy, we can usually understand as fixing the system to a surface in phase space. Phase space can be as simple as “the positions and momentums of every particle”, and that often is what we use. It doesn’t need to be, though. Gibbs put out diagrams where the coordinates were things like temperature or pressure or entropy or energy. Looking at these can let one understand a thermodynamic system. They use our geometric sense much the same way that charts of high- and low-pressure fronts let one understand the weather. James Clerk Maxwell, famous for electromagnetism, was so taken by this he created plaster models of the described surface.

This is, you might imagine, pretty serious, heady stuff. So you get why Gibbs published it in the Transactions of the Connecticut Academy: his brother-in-law was the editor. It did not give the journal lasting fame. It gave his brother-in-law a heightened typesetting bill, and Yale faculty and New Haven businessmen donated funds.

Which gets to the less-happy parts of Gibbs’s career. (I started out with ‘less pleasant’ but it’s hard to spot an actually unpleasant part of his career.) This work sank without a trace, despite Maxwell’s enthusiasm. It emerged only in the middle of the 20th century, as physicists came to understand their field as an expression of geometry.

That’s all right. Chemists understood the value of Gibbs’s thermodynamics work. He introduced the enthalpy, an important thing that nobody with less than a Master’s degree in Physics feels they understand. Changes of enthalpy describe how heat transfers. And the Gibbs Free Energy, which measures how much reversible work a system can do if the temperature and pressure stay constant. A chemical reaction where the Gibbs free energy is negative will happen spontaneously. If the system’s in equilibrium, the Gibbs free energy won’t change. (I need to say the Gibbs free energy as there’s a different quantity, the Helmholtz free energy, that’s also important but not the same thing.) And, from this, the phase rule. That describes how many independently-controllable variables you can see in mixing substances.

In the 1880s Gibbs worked on something which exploded through physics and mathematics. This was vectors. He didn’t create them from nothing. Hermann Günter Grassmann — whose fascinating and frustrating career I hadn’t known of before this — laid much of the foundation. Building on Grassman and W K Clifford, though, let Gibbs present vectors as we now use them in physics. How to define dot products and cross products. How to use them to simplify physics problems. How they’re less work than quaternions are. Gibbs was not the only person to recast physics in vector form. Oliver Heaviside is another important mathematical physicist of the time who did. But Gibbs identified the tools extremely well. You can read his Elements of Vector Analysis. It’s not very different from what a modern author would write on the subject. It’s terser than I would write, but terse is also respectful of someone’s time and ability to reason out explanations of small points.

There are more pieces. They don’t all fit in a neat linear timeline; nobody’s life really does. Gibbs’s thermodynamics work, leading into statistical mechanics, foreshadows much of quantum mechanics. He’s famous for the Gibbs Paradox, which concerns the entropy of mixing together two different kinds of gas. Why is this different from mixing together two containers of the same kind of gas? And the answer is that we have to think more carefully about what we mean by entropy, and about the differences between containers.

There is a Gibbs phenomenon, known to anyone studying Fourier series. The Fourier series is a sum of sine and cosine functions. It approximates an arbitrary original function. The series is a continuous function; you could draw it without lifting your pen. If the original function has a jump, though? A spot where you have to lift your pen? The Fourier series for that represents the jump with a region where its quite-good approximation suddenly turns bad. It wobbles around the ‘correct’ values near the jump. Using more terms in the series doesn’t make the wobbling shrink. Gibbs described it, in studying sawtooth waves. As it happens, Henry Wilbraham first noticed and described this in 1848. But Wilbraham’s work went unnoticed until after Gibbs’s rediscovery.

And then there was a bit in which Gibbs was intrigued by a comet that prolific comet-spotter Lewis Swift observed in 1880. Finding the orbit of a thing from a handful of observations is one of the great problems of astronomical mathematics. Karl Friedrich Gauss started the 19th century with his work projecting the orbit of the newly-discovered and rapidly-lost asteroid Ceres. Gibbs put his vector notation to the work of calculating orbits. His technique, I am told by people who seem to know, is less difficult and more numerically stable than was earlier used.

Swift’s comet of 1880, it turns out, was spotted in 1869 by Wilhelm Tempel. It was lost after its 1908 perihelion. Comets have a nasty habit of changing their orbits on us. But it was rediscovered in 2001 by the Lincoln Near-Earth Asteroid Research program. It’s next to reach perihelion the 26th of November, 2020. You might get to see this, another thing touched by J Willard Gibbs.

This and the other other A-to-Z topics for 2020 should be at this link. All my essays for this and past A-to-Z sequences are at this link. I’ll soon be opening f or topics for J, K, and L, essays also. Thanks for reading.

JH van ‘t Hoff and the Gaseous Theory of Solutions; also, Pricing Games

Do you ever think about why stuff dissolves? Like, why a spoon of sugar in a glass of water should seem to disappear instead of turning into a slight change in the water’s clarity? Well, sure, in those moods when you look at the world as a child does, not accepting that life is just like that and instead can imagine it being otherwise. Take that sort of question and put it to adult inquiry and you get great science.

Peter Mander of the Carnot Cycle blog this month writes a tale about Jacobus Henricus van ‘t Hoff, the first winner of a Nobel Prize for Chemistry. In 1883, on hearing of an interesting experiment with semipermeable membranes, van ‘t Hoff had a brilliant insight about why things go into solution, and how. The insight had only one little problem. It makes for fine reading about the history of chemistry and of its mathematical study.

In other, television-related news, the United States edition of The Price Is Right included a mention of “square root day” yesterday, 4/4/16. It was in the game “Cover-Up”, in which the contestant tries making successively better guesses at the price of a car. This they do by covering up wrong digits with new guesses. For the start of the game, before the contestant’s made any guesses, they need something irrelevant to the game to be on the board. So, they put up mock calendar pages for 1/1/2001, 2/2/2004, 3/3/2009, 4/4/2016, and finally a card reading $\sqrt{DAY}$. The game show also had a round devoted to Pi Day a few weeks back. So I suppose they’re trying to reach out to people into pop mathematics. It’s cute.

Knot.

It’s a common joke that mathematicians shun things that have anything to do with the real world. You can see where the impression comes from, though. Even common mathematical constructs, such as “functions”, are otherworldly abstractions once a mathematician is done defining them precisely. It can look like mathematicians find real stuff to be too dull to study.

Knot theory goes against the stereotype. A mathematician’s knot is just about what you would imagine: threads of something that get folded and twisted back around themselves. Every now and then a knot theorist will get a bit of human-interest news going for the department by announcing a new way to tie a tie, or to tie a shoelace, or maybe something about why the Christmas tree lights get so tangled up. These are really parts of the field, and applications that almost leap off the page as one studies. It’s a bit silly, admittedly. The only way anybody needs to tie a tie is go see my father and have him do it for you, and then just loosen and tighten the knot for the two or three times you’ll need it. And there’s at most two ways of tying a shoelace anybody needs. Christmas tree lights are a bigger problem but nobody can really help with getting them untangled. But studying the field encourages a lot of sketches of knots, and they almost cry out to be done out of some real material.

One amazing thing about knots is that they can be described as mathematical expressions. There are multiple ways to encode a description for how a knot looks as a polynomial. An expression like $t + t^3 - t^4$ contains enough information to draw one knot as opposed to all the others that might exist. (In this case it’s a very simple knot, one known as the right-hand trefoil knot. A trefoil knot is a knot with a trefoil-like pattern.) Indeed, it’s possible to describe knots with polynomials that let you distinguish between a knot and its mirror-image reflection.

Biology, life, is knots. The DNA molecules that carry and transmit genes tangle up on themselves, creating knots. The molecules that DNA encodes, proteins and enzymes and all the other basic tools of cells, can be represented as knots. Since at this level the field is about how molecules interact you probably would expect that much of chemistry can be seen as the ways knots interact. Statistical mechanics, the study of unspeakably large number of particles, do as well. A field you can be introduced to by studying your sneaker runs through the most useful arteries of science.

That said, mathematicians do make their knots of unreal stuff. The mathematical knot is, normally, a one-dimensional thread rather than a cylinder of stuff like a string or rope or shoelace. No matter; just imagine you’ve got a very thin string. And we assume that it’s frictionless; the knot doesn’t get stuck on itself. As a result a mathematician just learning knot theory would snootily point out that however tightly wound up your extension cord is, it’s not actually knotted. You could in principle push one of the ends of the cord all the way through the knot and so loosen it into an untangled string, if you could push the cord from one end and if the cord didn’t get stuck on itself. So, yes, real-world knots are mathematically not knots. After all, something that just falls apart with a little push hardly seems worth the name “knot”.

My point is that mathematically a knot has to be a closed loop. And it’s got to wrap around itself in some sufficiently complicated way. A simple circle of string is not a knot. If “not a knot” sounds a bit childish you might use instead the Lewis Carrollian term “unknot”.

We can fix that, though, using a surprisingly common mathematical trick. Take the shoelace or rope or extension cord you want to study. And extend it: draw lines from either end of the cord out to the edge of your paper. (This is a great field for doodlers.) And then pretend that the lines go out and loop around, touching each other somewhere off the sheet of paper, as simply as possible. What had been an unknot is now not an unknot. Study wisely.

The Liquefaction of Gases – Part II

The CarnotCycle blog has a continuation of last month’s The Liquefaction of Gases, as you might expect, named The Liquefaction of Gases, Part II, and it’s another intriguing piece. The story here is about how the theory of cooling, and of phase changes — under what conditions gases will turn into liquids — was developed. There’s a fair bit of mathematics involved, although most of the important work is in in polynomials. If you remember in algebra (or in pre-algebra) drawing curves for functions that had x3 in them, and in finding how they sometimes had one and sometimes had three real roots, then you’re well on your way to understanding the work which earned Johannes van der Waals the 1910 Nobel Prize in Physics.

Future Nobel Prize winners both. Kamerlingh Onnes and Johannes van der Waals in 1908.

On Friday 10 July 1908, at Leiden in the Netherlands, Kamerlingh Onnes succeeded in liquefying the one remaining gas previously thought to be non-condensable – helium – using a sequential Joule-Thomson cooling technique to drive the temperature down to just 4 degrees above absolute zero. The event brought to a conclusion the race to liquefy the so-called permanent gases, following the revelation that all gases have a critical temperature below which they must be cooled before liquefaction is possible.

This crucial fact was established by Dr. Thomas Andrews, professor of chemistry at Queen’s College Belfast, in his groundbreaking study of the liquefaction of carbon dioxide, “On the Continuity of the Gaseous and Liquid States of Matter”, published in the Philosophical Transactions of the Royal Society of London in 1869.

As described in Part I of…

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