## Gibbs’ Elementary Principles in Statistical Mechanics

I had another discovery from the collection of books at archive.org, now that I thought to look for it: Josiah Willard Gibbs’s Elementary Principles in Statistical Mechanics, originally published in 1902 and reprinted 1960 by Dover, which gives you a taste of Gibbs’s writings by its extended title, Developed With Especial Reference To The Rational Foundation of Thermodynamics. Gibbs was an astounding figure even in a field that seems to draw out astounding figures, and he’s a good candidate for the title of “greatest scientist to come from the United States”.

He lived in walking distance of Yale (where his father and then he taught) nearly his whole life, working nearly isolated but with an astounding talent for organizing the many complex and confused ideas in the study of thermodynamics into a neat, logical science. Some great scientists have the knack for finding important work to do; some great scientists have the knack for finding ways to express work so the masses can understand it. Gibbs … well, perhaps it’s a bit much to say the masses understand it, but the language of modern thermodynamics and of quantum mechanics is very much the language he spoke a century-plus ago.

My understanding is he published almost all his work in the journal Transactions of the Connecticut Philosophical Society, in a show of hometown pride which probably left the editors baffled but, I suppose, happy to print something this fellow was very sure about.

To give some idea why they might have found him baffling, though, consider the first paragraph of Chapter 1, which is accurate and certainly economical:

We shall use Hamilton’s form of the equations of motion for a system of n degrees of freedom, writing $q_1, \cdots q_n$ for the (generalized) coördinates, $\dot{q}_1, \cdots \dot{q}_n$ for the (generalized) velocities, and

$F_1 q_1 + F_2 q_2 + \cdots + F_n q_n$ [1]

for the moment of the forces. We shall call the quantities $F_1, \cdots F_n$ the (generalized) forces, and the quantities $p_1 \cdots p_n$, defined by the equations

$p_1 = \frac{d\epsilon_p}{d\dot{q}_1}, p_2 = \frac{d\epsilon_p}{d\dot{q}_2}, etc.,$ [2]

where $\epsilon_p$ denotes the kinetic energy of the system, the (generalized) momenta. The kinetic energy is here regarded as a function of the velocities and coördinates. We shall usually regard it as a function of the momenta and coördinates, and on this account we denote it by $\epsilon_p$. This will not prevent us from occasionally using formulas like [2], where it is sufficiently evident the kinetic energy is regarded as function of the $\dot{q}$‘s and $q$‘s. But in expressions like $d\epsilon_p/dq_1$, where the denominator does not determine the question, the kinetic energy is always to be treated in the differentiation as function of the p’s and q’s.

(There’s also a footnote I skipped because I don’t know an elegant way to include it in WordPress.) Your friend the physics major did not understand that on first read any more than you did, although she probably got it after going back and reading it a touch more slowly. And his writing is just like that: 240 pages and I’m not sure I could say any of them could be appreciably tightened.

Also, I note I finally reached 9,000 page views! Thank you; I couldn’t have done it without at least twenty of you, since I’m pretty sure I’ve obsessively clicked on my own pages at minimum 8,979 times.