## September 30, 2024

### The Space of Physical Frameworks (Part 5)

#### Posted by John Baez

In Part 4, I presented a nifty result supporting my claim that classical statistical mechanics reduces to thermodynamics when Boltzmann’s constant $k$ approaches zero. I used a lot of physics jargon to explain why I care about this result. I also used some math jargon to carry out my argument.

This may have been off-putting. But to understand the result, you only need to know calculus! So this time I’ll state it without all the surrounding rhetoric, and then illustrate it with an example.

At the end, I’ll talk about the physical meaning of it all.

## September 27, 2024

### Axiomatic Set Theory 2: The Axioms, Part One

#### Posted by Tom Leinster

*Previously: Part 1. Next: Part 3*

We’ve just finished the second week of my undergraduate Axiomatic Set Theory course, in which we’re doing Lawvere’s Elementary Theory of the Category of Sets but without mentioning categories.

This week, we covered the first six of the ten axioms: notes here.

## September 25, 2024

### ACT 2025 and the Adjoint School

#### Posted by John Baez

Here’s some basic information about the next big annual applied category theory conference — Applied Category Theory 2025 — and the school that goes along with that: the Adjoint School.

James Fairbanks will hold ACT2025 and the Adjoint School at the University of Florida, in Gainesville, on these dates:

- Adjoint School: May 26–30, 2025.
- ACT 2025: June 2–6, 2025.

More information will eventually appear on a website somewhere, and I’ll try to remember to let you know about it!

## September 22, 2024

### Axiomatic Set Theory 1: Introduction

#### Posted by Tom Leinster

*Next: Part 2*

I’m teaching Edinburgh’s undergraduate Axiomatic Set Theory course, and the axioms we’re using are Lawvere’s Elementary Theory of the Category of Sets — with the twist that everything’s going to be done directly in terms of sets and functions, without invoking categories. That is, I’ll neither assume nor teach the general notion of category.

I thought I’d share my notes so far.

## September 15, 2024

### The Space of Physical Frameworks (Part 4)

#### Posted by John Baez

In Part 1, I explained my hopes that classical statistical mechanics reduces to thermodynamics in the limit where Boltzmann’s constant $k$ approaches zero. In Part 2, I explained exactly what I mean by ‘thermodynamics’. I also showed how, in this framework, a quantity called ‘negative free entropy’ arises as the Legendre transform of entropy.

In Part 3, I showed how a Legendre transform can arise as a limit of something like a Laplace transform.

Today I’ll put all the puzzle pieces together. I’ll explain exactly what I mean by ‘classical statistical mechanics’, and how negative free entropy is defined in *this* framework. Its definition involves a Laplace transform. Finally, using the result from Part 3, I’ll show that as $k \to 0$, negative free entropy in classical statistical mechanics approaches the negative free entropy we’ve already seen in thermodynamics!

## September 9, 2024

### The Space of Physical Frameworks (Part 3)

#### Posted by John Baez

In Part 1, I explained how statistical mechanics is connected to a rig whose operations depend on a real parameter $\beta$ and approach the ‘tropical rig’, with operations $\min$ and $+$, as $\beta \to +\infty$. I explained my hope that if we take equations from classical statistical mechanics, expressed in terms of this $\beta$-dependent rig, and let $\beta \to +\infty$, we get equations in thermodynamics. That’s what I’m slowly trying to show.

As a warmup, last time I explained a bit of thermodynamics. We saw that some crucial formulas involve Legendre transforms, where you take a function $f \colon \mathbb{R} \to [-\infty,\infty]$ and define a new one $\tilde{f} \colon \mathbb{R} \to [-\infty,\infty]$ by

$\tilde{f}(s) = \inf_{x \in \mathbb{R}} (s x - f(x))$

I’d like the Legendre transform to be something like a limit of the Laplace transform, where you take a function $f$ and define a new one $\hat{f}$ by

$\hat{f}(s) = \int_{-\infty}^\infty e^{-s x} f(x) \, d x$

Why do I care? As we’ll see later, classical statistical mechanics features a crucial formula that involves a Laplace transform. So it would be great if we could find some parameter $\beta$ in that formula, take the limit $\beta \to +\infty$, and get a corresponding equation in thermodynamics that involves a Legendre transform!

As a warmup, let’s look at the purely mathematical question of how to get the Legendre transform as a limit of the Laplace transform — or more precisely, something *like* the Laplace transform. Once we understand that, we can tackle the physics in a later post.

## September 7, 2024

### The Space of Physical Frameworks (Part 2)

#### Posted by John Baez

I’m trying to work out how classical statistical mechanics can reduce to thermodynamics in a certain limit. I sketched out the game plan in Part 1 but there are a lot of details to hammer out. While I’m doing this, let me stall for time by explaining more precisely what I mean by ‘thermodynamics’. Thermodynamics is a big subject, but I mean something more precise and limited in scope.