### 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.

### From Legendre to Laplace: the rough idea

In Part 1 we saw that for each $\beta \gt 0$ we can make $(0,\infty]$ into a rig with addition

$x \oplus_\beta y = -\frac{1}{\beta} \ln(e^{-\beta x} + e^{-\beta y})$

As $\beta \to +\infty$, this operation approaches

$x \oplus_\infty y = x \min y$

and we get a version of the tropical rig. (There’s a more popular version using $\max$ instead of $\min$, but I decided to use $\min$.)

Suppose we want to define a notion of integration with $\oplus_\beta$ replacing ordinary addition. We could call it ‘$\beta$-integration’ and denote it by $\int_\beta$. Then it’s natural to try this:

$\int_\beta f(x) \, d x = - \frac{1}{\beta} \ln \int e^{-\beta f(x)} d x$

If the function $f$ is nice enough, we could hope that

$\lim_{\beta \to +\infty} \int_\beta f(x) \, d x \; = \; \inf_x f(x)$

Then, we could hope to express the Legendre transform

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

as the $\beta \to + \infty$ limit of some transform involving $\beta$-integration.

Indeed, in Section 6 here, Litvinov claims that the Legendre transform is the $\beta = \infty$ analogue of the Laplace transform:

- Grigory L. Litvinov, Tropical mathematics, idempotent analysis, classical mechanics and geometry.

But he doesn’t state any result saying that it’s a *limit* of the Laplace transform, or something like that.. Touchette states a result along these lines here:

- Hugo Touchette, Legendre–Fenchel transforms in a nutshell.

He even applies it to classical statistical mechanics! But he’s operating as a physicist, not a mathematician, so he doesn’t state a precise theorem. I’d like to take a crack at that, just to be sure I’m not fooling myself.

### The Legendre transform as a limit

Touchette’s formula gives an Legendre transform involving a sup rather than an inf. I slightly prefer a version with an inf. We’ll get the Legendre transform as a limit of something that is not exactly a Laplace transform, but close enough for our physics application:

**Almost Proved Theorem.** Suppose that $f \colon [0,\infty) \to \mathbb{R}$ is a concave function with continuous second derivative. Suppose that for some $s \gt 0$ the function $s x - f(x)$ has a unique minimum at $x_0$, and $f''(x_0) \lt 0$. Then as $\beta \to +\infty$ we have

$\displaystyle{ -\frac{1}{\beta} \ln \int_0^\infty e^{-\beta (s x - f(x))} \, d x \quad \longrightarrow \quad \inf_x \left( s x - f(x)\right) }$

**Almost Proof.** Laplace’s method should give the asymptotic formula

$\displaystyle{ \int_0^\infty e^{-\beta (s x - f(x))} \, d x \quad \sim \quad \sqrt{\frac{2\pi}{-\beta f''(x_0)}} \; e^{-\beta(s x_0 - f(x_0)} }$

Taking the logarithm of both sides and dividing by $-\beta$ we get

$\displaystyle{\lim_{\beta \to +\infty} -\frac{1}{\beta} \ln \int_0^\infty e^{-\beta(s x - f(x))} \, d x \; = \; s x_0 - f(x_0) }$

since

$\displaystyle{ \lim_{\beta \to +\infty} \frac{1}{\beta} \; \ln \sqrt{\frac{2\pi}{-\beta f''(x_0)}} = 0 }$

Since $s x - f(x)$ has a minimum at $x_0$ we get the desired result:

$\displaystyle{ -\frac{1}{\beta} \ln \int_0^\infty e^{-\beta (s x - f(x))} \, d x \quad \longrightarrow \quad \inf_x \left( s x - f(x)\right) }$

The only tricky part is that Laplace’s method as proved here requires finite limits of integration, while we are integrating from $0$ all the way up to $\infty$. However, the function $s x - f(x)$ is concave, with a minimum at $x_0$, and it has positive second derivative there since $f''(x_0) \lt 0$. Thus, it grows at least linearly for large $x$, so as $\beta \to +\infty$ the integral

$\displaystyle \int_0^\infty e^{-\beta(s x - f(x))} \, d x$

can be arbitrarily well approximated by an integral over some finite range $[0,a]$. ∎

Someone must have studied the hell out of this issue somewhere — do you know where?

Now, let’s look at the key quantity in the above result:

$\displaystyle{ - \frac{1}{\beta} \ln \int_0^\infty e^{-\beta (s x - f(x))} \, d x }$

We’ve got a $-1/\beta$ and then the logarithm of an integral that is not exactly a Laplace transform… but it’s close! In fact it’s almost the Laplace transform of the function

$g_\beta(x) = e^{\beta f(x)}$

since

$\displaystyle{ \int_0^\infty e^{-\beta (s x - f(x))} \, d x = \int_0^\infty e^{-\beta s x} \, g_\beta(x) \, d x }$

The right hand side here would be the Laplace transform of $g_\beta$ if it weren’t for that $\beta$ in the exponential.

So, it seems to be an exaggeration to say the Legendre transform is a limit of the Laplace transform. It seems to be the limit of something that’s *related* to a Laplace transform, but more complicated in a number of curious ways.

This has made my life difficult (and exciting) for the last few weeks. Right now I believe that all the curious complications do exactly what we want them to in our physics applications. But I should hold off on declaring this until I write up all the details: I keep making computational mistakes and fixing them, with a roller-coaster of emotions as things switch between working and not working.

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

David Corfield may be happy to hear that I’m finally answering his question from April 2014:

Igor Khavkine’s reply to that question was spot on, since Laplace’s method is a cousin to the saddle point or steepest descent method. But there seem to be some subtleties… which I’m tackling now.