Fluffy the Cat, Image Credit: The Snell Group

A lot of my blog posts are about entropy. So hey, what’s the big deal about entropy?

As I’ve discussed in previous posts, it connects thermodynamics to computation and information theory, it’s intimately involved in causality, and it appears to be responsible for the arrow of time (the reason why you can remember the past but not the future). It was used by Lord Kelvin — the man who coined the term thermodynamics — to argue that Darwin’s theory of evolution couldn’t possibly be true. (He was later proven wrong about that — something I may get to in another post.) And it will eventually be responsible for the heat death of the universe.

But just in case you needed another reason to think entropy is cool… please allow me to give you another good one. Before information theory was invented, before computers, and before the phrase “arrow of time” was first uttered by Sir Arthur Eddington, there was still something pretty important about entropy: it was the gateway drug that got Max Planck so hooked on physics, it led him straight to the hard stuff: quantum mechanics. He never would have been able to discover that energy was quantized in our universe if he hadn’t become one of the world’s leading experts on entropy first.

When 16 year old Max Planck started college in the 1870’s, he told a professor at the University of Munich that he wanted to major in physics. The professor replied by warning him that it was probably a bad idea: “In this field, almost everything is already discovered, and all that remains is to fill a few holes.” Young Planck ignored the advice, saying he didn’t care if he discovered anything new, he just wanted to understand the known laws of physics at a deeper level.

Max Planck, 1878

When Planck began studying physics, thermodynamics was still a fairly new field — Clausius had only recently coined the term entropy in 1865. In 1877, he visited Friedrich Wilhelm University in Berlin for a year to take classes from two great German physicists who had each contributed important work to the development of thermodynamics: Kirchhoff and Helmholtz. He became close friends with Helmholtz during the year, and decided to do an independent study on Clausius’s work.

Planck finished his PhD at the age of 21, titling his doctoral dissertation On the Second Law of Thermodynamics. (You know, the one that says entropy always increases.)

During the early 1880’s, he developed an elaborate theoretical framework for thermodynamics, all on his own… only to realize afterwards that most of the same framework had already been developed independently by Gibbs.

The more research I do into the history of physics for my book, the more amazed I am at how many myths there are about how different discoveries happened in physics. For some reason, the stories you hear about how great insights into the laws of nature were achieved by famous scientists often end up being greatly exaggerated or simply made up.

Image Credit: Getty images

Did Archimedes ever exclaim “Eureka!” while taking a bath? Did Galileo ever drop 2 balls from the leaning tower of Pisa to see if they hit the ground at the same time? Did Isaac Newton suddenly figure out the law of gravity when an apple fell on his head? No, historians believe that probably none of these events ever happened — but people love to tell the stories anyway.

A similar story is often told about how Planck had his great insight (that energy is quantized) which gave rise to the birth of quantum mechanics. There are many variants of the tale, but curiously they all have to do with lightbulbs — something Planck never worked on at all. The lightbulb has long been a symbol of sudden insight, which is probably why this particular myth about Planck has spread so easily. For example, here is how Gizmodo tells it (linking to a Minute Physics Youtube video which is just as bogus):

Your typical generic lightbulb image

“Back in the early 1890s the German Bureau of Standards asked Max Planck to design a lightbulb that produced the maximum amount of light with the minimal amount of energy. And his research into saving the bureau a few dollars on its energy bills led to the study of quantum mechanics which could one day let us unlock the secrets of the universe.” — Gizmodo (spreading an urban legend about Planck)

In some versions of the legend, Planck was hired by an energy company or a consortium of energy companies. In other versions it’s the German government; in this case it’s the “German Bureau of Standards”. No organization by that name ever existed, but it likely refers to the Physikalisch-Technische Reichsanstalt (the Imperial Physical Technical Institute, or PTR) which was founded by Helmholtz and where some physicists closely affiliated with Planck worked.

Having read through a lot of historical biographies and accounts of Planck’s work recently, I have found no good evidence that Planck himself ever worked for the PTR (despite that mysteriously showing up on the PTR’s Wikipedia page in one place). However, it’s true that it was essentially a bureau of standards — their mission was to establish standard measurements for things like the meter or the kilogram, similar to NIST in the USA.

But even if the Wikipedia page which claims he did work there at some point (without any details or a source) is right, Planck was certainly never involved with designing any lightbulbs (and nor was anyone at the PTR for that matter). He was a theoretical physicist interested in very abstract mathematical ideas like entropy; he would have been useless as a lightbulb engineer. And even the more experimental physicists who worked there in the 1890’s with whom Planck was in close contact with (Wien, Lummer, Kurlbaum, Rubens, Paschen, etc.) were working with black-body ovens, not lightbulbs.

Black-body ovens were a special kind of oven (designed by Wien & Lummer) to provide data that might help meet a challenge Kirchhoff had offered to all physicists in 1859: find the universal radiation law for black-bodies (which he had proved existed, but had no idea how to find). Kirchhoff coined the term “black body” in the same year, to mean any material body which has a surface that absorbs all of the light that hits it, reflecting none of it. Kirchhoff described this as the most urgent open problem in theoretical physics, and yet it would be over 40 years until Max Planck finally solved it (in 1900).

Thanks to Helmholtz’s influence, Planck took over Kirchhoff’s position at Friedrich Wilhelm University in 1889 and remained there as a professor throughout the 1890’s (during the entire period when the urban legend claims he was supposedly working at PTR or at an energy company).

The reason why people like Kirchhoff and Planck saw black-body radiation as such an interesting problem had nothing to do with lightbulbs. Black-body radiation was interesting in part because it served as the link between matter and aether. There was a theory of heat, developed by people like Clausius and Kelvin, which was thought to have to do with the vibrations or motion of matter; and there was a theory of light, developed primarily by James Clerk Maxwell, which was thought to have to do with vibrations of aether. But nobody fully understood the connection between heat and light, matter and aether. How did the two interact?

Image Credit: Lucid Learning

It was interesting because it was the missing link between matter and aether, but it was especially interesting to Planck because it was a fertile testing ground for his own ideas about entropy and the 2nd law of thermodynamics. Maxwell and Boltzmann had argued that the 2nd law was a statistical law, having to do with the random motion of molecules, whereas Planck believed it was an absolute law similar to the 1st law of thermodynamics (conservation of energy). It turned out, Planck was wrong and Maxwell and Boltzmann were right, but Planck’s realization of that in 1900 was exactly what opened up the rabbit hole, through which he stumbled down into the strange land of quantum mechanics.

Image Credit: Chronicle / Paul Chinn

The connection between heat and light was that all warm bodies radiate light— this is called thermal radiation. As a body becomes hotter and hotter, the color of light (determined by the frequency of light waves) changes. The image at the top of this post shows the spectrum of thermal radiation given off by the body heat of a cat named fluffy — it’s mostly infrared light, which cannot be seen by the human eye but can be captured with an infrared camera. If you’ve ever seen iron or steel glow red once it gets heated up to much hotter temperatures, then you’ve seen how hotter temperatures shift the thermal radiation spectrum into the range that’s visible to our eyes.

Kirchhoff’s challenge was to find the precise mathematical function for the spectrum of light given off by each temperature, which he proved would have to take the same form for any material object. But since most objects not only emit thermal radiation but also reflect ambient light from the room, the only perfectly clean way to measure this was with objects that don’t reflect any light — black bodies. Willy Wien and his colleagues at PTR set out to do this in 1895 with an oven with a small hole poked in it — the hole served as a perfect black body in the sense that since it wasn’t a surface at all, it couldn’t reflect any light, it only emitted or absorbed light into the oven. It’s true that part of the motivation/funding for PTR having them research this was to establish a new industry standard for thermal radiation which might help the city government of Berlin decide whether gas or electric lighting was more efficient to use in their streetlights. This is likely the origin of the myth, as it’s the closest connection any of their work had to lightbulbs.

In 1895 Wien came up with an empirical formula based on the data, and by early 1900 Planck had been able to reproduce Wien’s formula from a more theoretical equation specifying how the entropy of the walls of the oven depended on their energy. Planck’s equation for the entropy of the walls was motivated by his own views on how entropy should work, which Lord Rayleigh soon pointed out was not consistent with Maxwell & Boltzmann’s Law of Equipartition (that energy should get divided up equally between all modes of oscillation or vibration in a system). Planck didn’t believe in the Law of Equipartition because it was motivated by the statistical view of entropy (which was in turn motivated by a belief in atoms and molecules — neither of which Planck was willing to accept yet).

Just as Planck shared his derivation of Wien’s formula from entropy, which he had worked on for many years, the oven experiments at PTR started to show that there was something wrong with Wien’s formula for lower frequencies of light — it no longer matched the data. Lord Rayleigh suggested this might be because the Equipartition Theorem should be used to understand the lower frequencies. It was known that there were some issues with applying the Equipartition Theorem to higher frequencies (later, this began to be referred to as “the ultraviolet catastrophe”). In Oct 1900, new data confirmed Rayleigh’s suggestion so Planck began to take Equipartition seriously.

He modified his previous entropy formula, pasting it together with another formula based on the Law of Equipartition for low energies, and then calculated what effect this would have on the overall spectrum. Lo and behold, it fit the data perfectly. Planck had found Kirchhoff’s universal radiation law, which is now known as Planck’s Law.

But Planck still wasn’t happy with it, as all he had done was paste together two different entropy formulas (the fancy math term for this is “interpolation”). He still needed to find some kind of physical motivation for his new expression for the entropy. Looking at the entropy formula he came up with, he realized it looked almost exactly like a formula he had seen in one of Boltzmann’s papers from 1877.

In defending the statistical interpretation of the Second Law (ironically, against people like Planck who didn’t believe in atoms), Boltzmann had used a “toy model” of the world where energy was discrete instead of continuous. He never intended this as an actual hypothesis about how the world works, he just knew that probability was very confusing and difficult to deal with when infinities are involved, so pretending energy came in tiny little chunks instead of being infinitely divisible greatly simplified his argument. It allowed him to compute probabilities just based on combinatorics — counting up the total number of possible states of a system. With this method, Boltzmann showed that the entropy was proportional to the logarithm of the number of accessible states of a system.

Ludwig Boltzmann’s tombstone

The equation S=k log W is engraved on Boltzmann’s tombstone, but ironically it was Planck who first wrote down this equation. Boltzmann had only argued that entropy (S) was proportional to the log of the number of states (W); the proportionality constant k was introduced by Planck later with another constant he is more famous for, h. Both h and k were introduced (and estimated) by Planck in late 1900 in order to find a physical interpretation for his entropy formula. But h became known as Planck’s constant and k became known as Boltzmann’s constant. Which sort of made sense since Boltzmann was most of the way there with k, but hadn’t come close to finding h.

Planck noticed that if he made the same assumption Boltzmann had made — that energy was discrete rather than continuous (in the information age, we’d probably say “digital” rather than “analog”), then the new entropy formula just arose naturally from first principles. In fact, it looked almost identical to a formula Boltzmann had already written — the only additional assumption Planck needed to make was that the energy of a single mode of oscillation (which he called a “resonator”) was proportional to its frequency of oscillation:

E = hf

This was a completely unprecedented leap of faith, but it was the only way he could get his entropy formula to match Boltzmann’s. Prior to that, energy had always been considered to be independent of frequency — frequency had to do with how fast something was oscillating (or the color of light or the pitch of a note of sound) while energy had to do with how intense it was (volume, brightness, etc.) There was no reason at all in classical physics why one might think the 2 are connected. Planck was the first to discover that they were.

Few physicists at the time paid much attention to Planck’s bizarre hypothesis, and the ones who did mostly criticized it for not being compatible with the known laws of physics. But it did catch the eye of a little known 26 year old patent clerk in Switzerland named Albert Einstein. Einstein found Planck’s idea very intriguing. Planck had intended his energy quantization to apply only to the oscillations within the walls of black-body ovens (or similar heated matter), but Einstein boldly extended this idea in 1905 to apply to light itself. Einstein referred to the smallest energy oscillations which a light wave could make as energy quanta. Later they became known as photons. deBroglie then realized that perhaps matter particles were also the same kind of energy quanta and had a wave interpretation. And with that, quantum mechanics was off and running. The rest is history, but it was all originally thanks to entropy! (Not lightbulbs — sorry Gizmodo.)