# I wasn't confused by Thermodynamics

*Motivation: Practicing writing in a classical style *

For years, I was confused by thermodynamics. Why isn't a sufficiently small change in heat a differential? What separates heat from internal energy? What even is entropy, physically? Like, where did it come from? And so on. This was in spite of the fact that I was good at statistical mechanics. In fact, I loved it. A beautiful subject. But I think learning it alongside thermodynamics confused me.

Then I stumbled on Yuxi-on-the-wired's [beautiful essay on the conceptual foundations of thermodynamics](https://yuxi.ml/essays/posts/equilibrium-thermoeconomics/), and his references to some truly wonderful books: "Thermodynamic Weirdness" by D. Lemons and "Thermodynamics" by Fermi. One book is a conceptual history and the other a lucid textbook. They both treat thermodynamics as a first-class theory: no mention of probabilities, Boltzmann distributions or the like are required.

Instead, Thermodynamics is the study of heat and how it can be transformed into work or vice versa. I.e. things you can actually feel. Its ontology is our own. You can just play with the referents of the theory yourself. Which certainly helps to bind the formalism to your intuitive world model.

To see that, consider history, where thermodynamics was understood before statistical mechanics.

It wasn't until about 1700s that we began investigating heat seriously. Before then, we did not have a reliable way to even measure temperature, let alone heat. After the invention of air thermometers, consistent measurements of temperature and the flow of heat became possible. 

At that point, the common theory of heat was that it was a conserved fluid clinging to bodies that repelled itself. This theory worked well for systems far from a phase transition that can be described by thermal energy alone.

The idea that heat was a form of motion was flirted with, since motion against friction was correlated with increases in heat. But that was not enough because it was hard to interpret useful equations in terms of motion. And they were useful.

Thermodynamics was developed alongside heat-engines, used to drain mines, drive steam locomotives and turn cotton mills. No one knew why steam engines suffered if the pistons were cooled alongside the steam. They just knew it worked. Until Carnot.

His core insight was that a heat engine can only produce work in the presence of heat flowing from a hot body to a cold body. This was the second law of thermodynamics. Together with his idealized heat engine, this was enough to understand what made some steam engines better than others: if you do not extract work from the movement of heat, then it is wasted, and you cannot undo the movement without equivalent work.

For his engine, the Carnot engine, was reversible. No heat engine could be more efficient, for then you could run it, then the Carnot engine in reverse and get work out with no net movement of heat. Absurd. This anticipated the conservation of energy.

In 1842, Mayer published his discovery of energy: a conserved property of heat and motion. An outsider, his writings were illegible to many in the field. Joule then independently discovered this idea, that mechanical work is converted into heat at a fixed rate and vice versa. His work was rigorous, and more importantly legible, in a way that Mayer[^qqkaj3cecvm]'s was not. But their contemporaries, like Kelvin, were unconvinced. Conservation of energy conflicted with conservation of caloric, which Carnot based his theory about.

Or so it seemed. In 1850, Clausius, a pupil of Kelvin, showed that Carnot's derivation made no use of the fact that caloric is conserved. Conservation of energy could be used instead. And so, after gaining its second law, thermodynamics gained its first.

Nowhere did they use notions of probability, or even entropy. Clausius did not develop that idea until 1854. And it wasn't until Maxwell and Boltzmann appeared that its connections to statistics and mechanics were understood. And not until the Ehrenfests[^m5ygtf255ur]' and their dog-flea model were those connections accepted. Because the connections are odd. How could something so deterministic as the flow of heat, so tangible as pressure, be rooted in uncertainty?

That is what confused me for so long. Not thermodynamics. And perhaps that's true for you, too. 

[^qqkaj3cecvm]: He jumped off a three story building after people failed to understand the significance of his work. He failed, was commited to a mental asylum for years, left and lost his zeal for science.  

[^m5ygtf255ur]: The Ehrenfests had a tragic life. Paul fell into depression, killed his son and then killed himself. His wife Tatyana survived him.