I came to your article in a terrifically circuitous fashion (via a serendipitous Google Image search), but am so glad that I did. Now it appears I'll end up spending the rest of my day reading all your links and references. I work in areas of information theory and molecular biology and have been filling in my knowledge of mathematics over time. Having just finished up six months of work in Lie groups and algebras, it would appear that category theory is the next sandbox in which I should play. Thanks especially for the textbook references as well as the journal articles!
#categorytheory #mathematics #abstractalgebra #topology
First, let's unpack your question a bit. What we're really doing is taking mathematical form, function, and theory and apply it in some sort of unifying manner to unify the areas of information theory and statistical thermodynamics.
You can also ask your question in the reverse direction to apply the mathematics of information to thermodynamics as well as there should be a mathematical duality of sorts, if in fact the two are directly related.
First, let's unpack your question a bit. What we're really doing is taking mathematical form, function, and theory and apply it in some sort of unifying manner to unify the areas of information theory and statistical mechanics and thermodynamics.
You can also ask your question in the reverse direction to apply the mathematics of information to thermodynamics as well as there should be a mathematical duality of sorts, if in fact the two are directly related.
The original link between the two areas stems from a paper by E.T. Jaynes entitled "Information Theory and Statistical Mechanics" (downloadable from http:/
Since then, in my opinion, the best source tying the fields together mathematically is A Farewell To Entropy: Statistical Thermodynamics Based on Information by Arieh Ben-Naim [http:/
As for other connections, I've not seen anything as direct, primarily as there aren't any other major concepts with the exact same name. However, because of the underlying mathematics, I strongly suspect that there are many similar 'concepts'. A few in particular, I have a feeling will come out of the more rigorous mathematical frameworks based on Lieb and Yngvason’s concptualization of thermodynamics (see Resource Convertibility (Part 1)[https:/
Another fruitful area of conceptual similarities is slowly grown out of quantum information theory and the statistical mechanics of black holes, though there is some heavily rarified air surrounding these given the required background needed to operate with them both simultaneously.
I'll turn this question around to suggest that instead of taking notes from your math/physics textbooks, that you're FAR better off PUTTING notes INTO them! Those margins are meant for writing down the parts of problems and examples that the author implicitly leaves out.
One typically wouldn't take notes from a Spanish, French, or Latin textbook would they? Like most languages, mathematics should be read and written to practice it (and maybe even spoken).
Knowing math or physics is best demonstrated by actually doing problems - and the majority of the time, this is what is going to be on the test too, so just pick up a pencil or pen and start working out the answers.
These subjects aren't like history, philosophy, or psychology with multiple choice or essay type questions that might benefit from note-taking, so just jump right in. Give the book a short read and start plugging away at problems.
If you have problems getting started, take a look at some of the examples provided by the author (or in other books), cover up the answer, and try to recreate the solution.
Drafting off of the Quora question "Why aren't math textbooks more straightforward?" (https:/
The Math Major Who Never Reads Math http:/
Certainly an interesting perspective on a very pervasive problem. I saw the provocative question "Why Aren’t Math Textbooks More Straightforward?" on Quora just the other day. I take much the same view as you (though I religiously read not only my assigned math texts for classes, but also read their competitors in search of increased clarity and beauty.) I spent a few minutes writing up a few thoughts on what is going on in math textbooks taking a broader view of the forest instead of complaining about the trees.
Perhaps some of the tentative will gain a better appreciation for mathematics: http:/
The repetition that a good mathematician (or really any good thinker) reads books with a pencil in hand needs to be drilled down to younger learners.
A parting math textbook joke:
Q. Why did Nicolas Bourbaki quit writing mathematics textbooks?
A. "He" realized that Serge Lang was just one person.