John C. Baez's Blog, page 8

The Space of Physical Frameworks (Part 5)

In Part 4, I presented a nifty result supporting my claim that classical statistical mechanics reduces to thermodynamics when Boltzmann’s constant 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 ...

Ceres

In 1596, Kepler claimed that the planetary orbits would only follow “God’s design” if there were two more planets: one between Mars and Jupiter and one between Mercury and Venus. Later folks came up with the Titius–Bode law. This says that for each n there should be a planet whose distance from the Sun is

0.4 + 0.3 × 2ⁿ

times the distance between the Earth and Sun.

• For n = 0 we get Venus.
• For n = 1 we get Earth.
• For n = 2 we get Mars.
• For n = 3 we get… NOTHING???
• For n = 4 we get J...

ACT 2025 and the Adjoint School

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!

The Gravo-Thermal Catastrophe

In 1962, V. A. Antonov did some remarkable simulations showing that even in Newtonian mechanics, gravitating systems can violate the usual rules of thermodynamics. Instead of reaching equilibrium they can get hotter and hotter!

Suppose you put a lot of stars in a large sphere, and suppose (unrealistically) that they bounce elastically off the walls of this sphere. In fact suppose they’re point masses, so they never collide, and interact only gravitationally. Also suppose they’re ‘gravitatio...

The Space of Physical Frameworks (Part 4)

In Part 1, I explained my hopes that classical statistical mechanics reduces to thermodynamics in the limit where Boltzmann’s constant 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’l...

The Space of Physical Frameworks (Part 3)

In Part 1, I explained how statistical mechanics is connected to a rig whose operations depend on a real parameter and approach the ‘tropical rig’, with operations and as I explained my hope that if we take equations from classical statistical mechanics, expressed in terms of this -dependent rig, and let 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 ...

The Space of Physical Frameworks (Part 2)

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.

Thermostatic systems

A lot of what we call ‘thermodynamics’, or more precisely ‘classical thermodynamics’...

The Space of Physical Frameworks (Part 1)

Besides learning about individual physical theories, students learn different frameworks in which physical theories are formulated. I’m talking about things like this:

• classical statics
• classical mechanics
• quantum mechanics
• thermodynamics
• classical statistical mechanics
• quantum statistical mechanics

A physical framework often depends on some physical constants that we can imagine varying, and in some limit one framework may reduce to another. This suggests that we should study a ...

Stirling’s Formula from Statistical Mechanics

Physicists like to study all sorts of simplified situations, but here’s one I haven’t seen them discuss. I call it an ‘energy particle’. It’s an imaginary thing with no qualities except energy, which can be any number

I hate it when on Star Trek someone says “I’m detecting an energy field” — as if energy could exist without any specific form. That makes no sense! Yet here I am, talking about energy particles.

Earlier on the n-Café, I once outlined a simple proof of Stirling’s formula usin...

Bernoulli Numbers and the Harmonic Oscillator

I keep wanting to understand Bernoulli numbers more deeply, and people keep telling me stuff that’s fancy when I want to understand things simply. But let me try again.

The Bernoulli numbers can be defined like this:

and if you grind them out, you get

and so on. The pattern is quite strange.

Bernoulli numbers are connected to hundreds of interesting things. For example if you want to figure out a sum like

you can use Bernoulli numbers—indeed Jakob Bernoulli boasted

It took me less ...