Last time I ended with a formula for the ‘Gibbs distribution’: the probability distribution that maximizes entropy subject to constraints on the expected values of some observables.
This formula is well-known, but I’d like to derive it here. My argument won’t be up to the highest standards of rigor: I’ll do a bunch of computations, and it would take more work to state conditions under which these computations are justified. But even a nonrigorous approach is worthwhile, since the computations will give us more than the mere formula for the Gibbs distribution.
I’ll start by reminding you of what I claimed last time. I’ll state it in a way that removes all unnecessary distractions, so go back to Part 20 if you want more explanation.
The Gibbs distribution
Take a measure space with measure Suppose there is a probability distribution on that maximizes the entropy
subject to the requirement that some integrable functions on have expected values equal to some chosen list of numbers
(Unlike last time, now I’m writing and with superscripts rather than subscripts, because I’ll be using the Einstein summation convention: I’ll sum over any repeated index that appears once as a a superscript and once as a subscript.)
Furthermore, suppose depends smoothly on I’ll call it to indicate its dependence on Then, I claim is the so-called Gibbs distribution
is the entropy of
Let’s show this is true!
Finding the Gibbs distribution
So, we are trying to find a probability distribution that maximizes entropy subject to these constraints:
We can solve this problem using Lagrange multipliers. We need one Lagrange multiplier, say for each of the above constraints. But it’s easiest if we start by letting range over all of that is, the space of all integrable functions on Then, because we want to be a probability distribution, we need to impose one extra constraint
To do this we need an extra Lagrange multiplier, say
So, that’s what we’ll do! We’ll look for critical points of this function on
Here I’m using some tricks to keep things short. First, I’m dropping the dummy variable x which appeared in all of the integrals we had: I’m leaving it implicit. Second, all my integrals are over so I won’t say that. And third, I’m using the Einstein summation convention, so there’s a sum over i implicit here.
Okay, now let’s do the variational derivative required to find a critical point of this function. When I was a math major taking physics classes, the way physicists did variational derivatives seemed like black magic to me. Then I spent months reading how mathematicians rigorously justified these techniques. I don’t feel like a massive digression into this right now, so I’ll just do the calculations—and if they seem like black magic, I’m sorry!
We need to find obeying
or in other words
First we need to simplify this expression. The only part that takes any work, if you know how to do variational derivatives, is the first term. Since the derivative of is we have
The second and third terms are easy, so we get
Thus, we need to solve this equation:
That’s easy to do:
Good! It’s starting to look like the Gibbs distribution!
We now need to choose the Lagrange multipliers and to make the constraints hold. To satisfy this constraint
we must choose so that
or in other words
Plugging this into our earlier formula
we get this:
Great! Even more like the Gibbs distribution!
By the way, you must have noticed the “1” that showed up here:
It buzzed around like an annoying fly in the otherwise beautiful calculation, but eventually went away. This is the same irksome “1” that showed up in Part 19. Someday I’d like to say a bit more about it.
Now, where were we? We were trying to show that
minimizes entropy subject to our constraints. So far we’ve shown
is a critical point. It’s clear that
so really is a probability distribution. We should show it actually maximizes entropy subject to our constraints, but I will skip that. Given that, will be our claimed Gibbs distribution if we can show
This is interesting! It’s saying our Lagrange multipliers actually equal the so-called conjugate variables given by
where is the entropy of
There are two ways to show this: the easy way and the hard way. The easy way is to reflect on the meaning of Lagrange multipliers, and I’ll sketch that way first. The hard way is to use brute force: just compute and show it equals This is a good test of our computational muscle—but more importantly, it will help us discover some interesting facts about the Gibbs distribution.
The easy way
Consider a simple Lagrange multiplier problem where you’re trying to find a critical point of a smooth function
subject to the constraint
for some smooth function
and constant c. (The function f here has nothing to do with the f in the previous sections.) To answer this we introduce a Lagrange multiplier and seek points where
This works because the above equation says
Geometrically this means we’re at a point where the gradient of points at right angles to the level surface of
Thus, to first order we can’t change by moving along the level surface of
But also, if we start at a point where
and we begin moving in any direction, the function will change at a rate equal to times the rate of change of . That’s just what the equation says! And this fact gives a conceptual meaning to the Lagrange multiplier
Our situation is more complicated, since our functions are defined on the infinite-dimensional space and we have an n-tuple of constraints with an n-tuple of Lagrange multipliers. But the same principle holds.
So, when we are at a solution of our constrained entropy-maximization problem, and we start moving the point by changing the value of the ith constraint, namely the rate at which the entropy changes will be times the rate of change of So, we have
But this is just what we needed to show!
The hard way
Here’s another way to show
We start by solving our constrained entropy-maximization problem using Lagrange multipliers. As already shown, we get
Then we’ll compute the entropy
Then we’ll differentiate this with respect to and show we get
Let’s try it! The calculation is a bit heavy, so let’s write for the so-called partition function
and the entropy is
This is the sum of two terms. The first term
is times the expected value of with respect to the probability distribution all summed over But the expected value of is so we get
The second term is easier:
since integrates to 1 and the partition function doesn’t depend on
Putting together these two terms we get an interesting formula for the entropy:
This formula is one reason this brute-force approach is actually worthwhile! I’ll say more about it later.
But for now, let’s use this formula to show what we’re trying to show, namely
where we played a little Kronecker delta game with the second term.
Now we just need to compute the third term:
Ah, you don’t know how good it feels, after years of category theory, to be doing calculations like this again!
Now we can finish the job we started:
We’ve learned the formula for the probability distribution that maximizes entropy subject to some constraints on the expected values of observables. But more importantly, we’ve seen that the anonymous Lagrange multipliers that show up in this problem are actually the partial derivatives of entropy! They equal
Thus, they are rich in meaning. From what we’ve seen earlier, they are ‘surprisals’. They are analogous to momentum in classical mechanics and have the meaning of intensive variables in thermodynamics:
|| Classical Mechanics
|| Probability Theory
|| Shannon entropy
Furthermore, by showing the hard way we discovered an interesting fact. There’s a relation between the entropy and the logarithm of the partition function:
(We proved this formula with replacing but now we know those are equal.)
This formula suggests that the logarithm of the partition function is important—and it is! It’s closely related to the concept of free energy—even though ‘energy’, free or otherwise, doesn’t show up at the level of generality we’re working at now.
This formula should also remind you of the tautological 1-form on the cotangent bundle namely
It should remind you even more of the contact 1-form on the contact manifold namely
Here is a coordinate on the contact manifold that’s a kind of abstract stand-in for our entropy function
So, it’s clear there’s a lot more to say: we’re seeing hints of things here and there, but not yet the full picture.
For all my old posts on information geometry, go here:
• Information geometry.