I'd second the recommendation of An Introduction to Thermal Physics by Daniel V. Schroeder if anyone is looking for an undergrad level text. It is on my shortlist of "best undergrad textbooks read". It is also physically quite a small book; the quantum physicists must have some insight into density.
The way large numbers of particles come together and start to behave like a continuum, is quite well represented by looking at simulations of a bunch of spheres with forces applied.
For example the phenomenon of surface tension, which is usually taught as "surface energy" in a continuum way, but you can also view it "bottom-up" by looking at attractive forces between the individual particles. (Video by Eugene Khutoryansky, who has a large number of these sorts of animations):
Hi, this is the author. It's been really fun to see these articles getting a little more attention. I wanted to repeat something I said when my website was posted here a couple of weeks ago, which is that I'm a private math tutor. I've been filling up a lot more than usual lately thanks to HN, but I do still have a little space! More info here: https://nicf.net/tutoring/
I want to like these resources, but they seem to suffer from the common approach to teaching mathematics.
Namely, a complete disregard for a numerical approach, and sprinkling latex symbols throughout large blocks of text without a clear through line.
Reading education material like this feels like reading code from an llm. I am spending more time cross referencing and checking the statements than doing the thing I came to the material to work on.
> If we interpret the path integral in terms of the Wiener measure, then it does in fact solve our Wick-rotated Schrödinger equation.
I guess I’ll takes your word for it.
If you understand the material as well as your word count suggests than why omit numerical elucidation?
If you are unable to perform numerical analysis of these formulae than why should I trust your symbol mangling of them?
I do appreciate the feedback. (And I should mention that there is a supplement to the thermodynamics article that this post is about which goes through some numerical experiments; you can see it here: https://nicf.net/articles/toy-thermodynamic/.) I do think that this may just be a situation where the thing I made isn't the thing you wanted, which is fine, but I can say a little bit about why I made the decisions I made.
The audience that I had in mind for these pieces is mostly people who've had some amount of training in pure math; it came about because I felt (and still feel) like a lot of the existing resources for this stuff aren't presented in a way that lines up well with the way that people with that training often like to absorb information.
Speaking for myself, I don't think I would enjoy reading an article about the Feynman path integral that's full of tables and graphs with results of numerical experiments. My reaction to a piece like that would be "Okay, that's nice, but can you please just say what mathematical objects we're even talking about here first?" It sounds like you have the opposite set of expectations, which is totally fine, and that probably just means these aren't the articles for you, but there (I hope!) are also other people whose expectations are more in alignment with the presentation I went for. I don't think either of us is "wrong"; our preferences are just different.
This is such a beautiful reply. I am bookmarking it. There is value in distilling out the key abstractions that wants to be heard from the sea of data.
Standard results omitting tedious derivations is a standard practice in all educational material of upper level physics. "Derivation left to reader", etc. I personally greatly enjoy the fact that i get to decide how much detail i want. I can get the gist, a moderate exploration, or a deep dive if i have the time. Keep in mind, this person did not have to make this resource, and if you don't like it you don't have to read it. I greatly appreciate a breadth of resources when learning a topic and this is an amazing addition to my learning resources. If you absolutelyneed every single detail of every single assertion then look it up, but i can never expect a human doing this in their free time as a free gift to write dozens of textbooks about every facet of physics. Keeping every single detail in would make these prohibitively time consuming to create in a way that i don't think a free resource could support
As fellow mathematician tutoring my son I found all of the articles incredibly useful. Thank you for writing them and making them freely accessible. Awesome work.
I have to take this opportunity to recite the opening passage of the introduction to thermodynamics and statistical mechanics in David L. Goodstein’s «States of Matter» from 1985:
“Ludwig Boltzmann, who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics.”
What an attention grabber! If only more textbooks introduced their topics so as to capture the imagination and set the reader on a path of possible danger and intrigue.
I really enjoyed my chemical kinetics, statistical mechanics and thermodynamics classes in undergrad chem. Learning that temperature as measured was an emergent property, and that entropy increases with higher temperature due to the larger number of microstates / ensembles, was pretty fucking kewl! :) ;p haha
The parallels between statistical mechanics and machine learning are eerily profound. Optimization theory describes systems and their evolution toward lowest-energy states, the landscape that optimization traverses and the dynamics by which it does so are described by statistical mechanical principles.
I had an experience in an interview for a Sr MLE role describing some previous work. Was unable to get through to the interviewer that the answer to the question of "which loss function did you use" can't always be found in a Medium listicle.
Hint: loss functions are negative log-likelihoods.
Off-topic, but this whole website is a gold mine. I just finished reading "Coordinate-Free Linear Algebra" and "Hamiltonian and Lagrangian Mechanics", making tensor products, Lie derivatives and Poisson brackets finally click. Thermodynamics is next on my list.
If the author ever lurks here, thank you so much for putting these online!
I do! Thanks for the kind words. I'm happy to hear someone enjoyed the linear algebra article --- after I finished that one and heard from some people who tried to read it I ended up with the impression that I'd pitched it too hard, so it's nice to know someone got something out of it.
