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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 kk 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→0k \to 0, negative free entropy in classical statistical mechanics approaches the negative free entropy we’ve already seen in thermodynamics!

Continue reading “The Space of Physical Frameworks (Part 4)” …
Posted at 8:00 PM UTC | Permalink | Followups (2)

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 β\beta and approach the ‘tropical rig’, with operations min\min and ++, as β→+∞\beta \to +\infty. I explained my hope that if we take equations from classical statistical mechanics, expressed in terms of this β\beta-dependent rig, and let β→+∞\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:ℝ→[−∞,∞]f \colon \mathbb{R} \to [-\infty,\infty] and define a new one f˜:ℝ→[−∞,∞]\tilde{f} \colon \mathbb{R} \to [-\infty,\infty] by

f˜(s)=inf x∈ℝ(sx−f(x)) \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 ff and define a new one f^\hat{f} by

f^(s)=∫ −∞ ∞e −sxf(x)dx \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 β\beta in that formula, take the limit β→+∞\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.

Continue reading “The Space of Physical Frameworks (Part 3)” …
Posted at 3:20 PM UTC | Permalink | Followups (8)

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.

Continue reading “The Space of Physical Frameworks (Part 2)” …
Posted at 12:00 PM UTC | Permalink | Followups (7)