“Entropy is geometry. And geometry is entropy.”

Curt explains a new finding from physicist Gabriele Carcassi, simply.

“Entropy is geometry. And geometry is entropy.”

This is a new finding by Gabriele Carcassi, and I’ll explain the reasoning below, along with the math. Don’t worry, I’ll hold your hand (metaphorically, of course, unless you’re into that).

Firstly, note that entropy is NOT a measure of disorder… It’s a way you count states. From this, we’ll see that geometry is entropy, and entropy is geometry.

Let’s start with classical mechanics. Why do we use phase space (position q and momentum p)?

To Gabriele, it’s because phase space counts configurations. The key is something called the symplectic form, ω.

This is what I described in a previous post below…

For a single degree of freedom, it’s simple: ω = dq∧dp. In some ways, symplectic forms are just fancy area calculators in an abstract space. Why area, though? Because area in phase space corresponds to the number of possible states.

To be specific, if you have a region U in phase space, the integral μ(U) = ∫_U ω gives you the “volume” (actually, the count) of states within U.

The Role of Symplectic Forms

So, the symplectic form ω is the foundation. It lets you count configurations. But where does entropy come in? First, let’s put a probability distribution ρ onto our phase space. Probability is always there (at least epistemically), so we’re justified in its usage.

The Gibbs-Shannon entropy, S = -∫ ρ log ρ ω, measures the “spread” or “variability” of your probability distribution ρ across phase space. The reason we need the geometry to define entropy is because we need the ω.

Now, consider a uniform distribution ρ_U over a region U. We know that this is what maximizes entropy, and it’s for this special case that the entropy simplifies to S[ρ_U] = log(μ(U)). The entropy is the logarithm of the number of configurations, and we see that Boltzmann’s famous S = k log W is a special case.

Boltzmann is what you get with a uniform distribution. Gibbs-Shannon is the more general form.

Undergirding Classical With Entropy (and Vice Versa)

So far, so trivial (this is covered in stat mech courses), but the part that makes this work is that this relationship is invertible.

If you know the entropy of all possible distributions, you can recover the measure μ(U) (and therefore the symplectic form ω).

The magic formula given by Gabriele Carcassi is below… That “sup” isn’t me asking you about your day. It’s something that finds the distribution with maximum entropy within U (which is always the uniform one).

\(\mu(U)=e^{\sup S(\{\rho \mid \rho(x)=0, x \notin U\})}\)

So, knowing all the entropies allows you to reconstruct the geometry of phase space. Now you can start doing statistical mechanics and recover the standard mechanics in the limit case where the ensembles are more and more precise.

Basically, standard mechanics and statistical mechanics are the same thing.

You can’t define one without implicitly defining the other.

What I found more interesting is that there’s an answer to the question of “why are position (q) and momentum (p) so special?” It’s because they’re required for defining an entropy that’s invariant across observers.

You can derive the conjugate variables (i.e., position and momentum) from the requirement that entropy be observer-independent.

Geometry and Entropy in Quantum Theory

What about quantum mechanics? The “geometry” here is defined by the inner product, ⟨ψ | φ⟩. This measures the “overlap” between quantum states |ψ⟩ and |φ⟩.

PS: I quibbled with Gabriele on WhatsApp about this because “geometry” usually means “metric,” and thus the Born rule is like the geometry on quantum space, but the symplectic form is the dynamics. This is a pettifogging aside.

We can make this elementary by taking a spin up / down system. The Born rule, which gives the probability of measuring |φ⟩ after preparing |ψ⟩, is |⟨ψ | φ⟩|² = cos²(θ/2). This θ/2 is half the angle between the states on the Bloch sphere. Interestingly, the Born rule in quantum mechanics is just another way of saying “geometry = entropy.” It too is secretly counting states! The indeterminacy of quantum mechanics gives you the entropy of an equal mixture of two pure states.

See Carcassi’s paper for the full derivation.

The important part is that the function which gets you from geometry to entropy is invertible. It’s one-to-one. That means you can describe something in geometric terms or in entropic terms.

The TLDR is that both classically and quantum mechanically, the geometry of the state space is the very structure that defines and determines entropy.

It’s fundamental. Geometry is entropy, and entropy is geometry.

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