Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

June 18, 2022

Compositional Thermostatics (Part 4)

Posted by John Baez

guest post by Owen Lynch

This is the fourth and final part of a blog series on this paper:

• John Baez, Owen Lynch and Joe Moeller, Compositional thermostatics.

In Part 1, we went over our definition of thermostatic system: it’s a convex space XX of states and a concave function S:X[,]S \colon X \to [-\infty, \infty] saying the entropy of each state. We also gave examples of thermostatic systems.

In Part 2, we talked about what it means to compose thermostatic systems. It amounts to constrained maximization of the total entropy.

In Part 3 we laid down a categorical framework for composing systems when there are choices that have to be made for how the systems are composed. This framework has been around for a long time: operads and operad algebras.

In this post we will bring together all of these parts in a big synthesis to create an operad of all the ways of composing thermostatic systems, along with an operad algebra of thermostatic systems!

Recall that in order to compose thermostatic systems (X 1,S 1),,(X n,S n), (X_1, S_1), \ldots, (X_n, S_n), we need to use a ‘parameterized constraint’, a convex subset

RX 1××X n×Y R \subseteq X_1 \times \cdots \times X_n \times Y

where Y Y is some other convex set. We end up with a thermostatic system on Y Y, with S:Y[,] S \colon Y \to [-\infty,\infty] defined by

S(y)=sup (x 1,,x n,y)RS 1(x 1)++S n(x n) S(y) = \sup_{(x_1,\ldots,x_n,y) \in R} S_1(x_1) + \cdots + S_n(x_n)

In order to model this using operads and operad algebras, we will make an operad 𝒞ℛ \mathcal{CR} which has convex sets as its types, and convex relations as its morphisms. Then we will make an operad algebra that assigns to any convex set X X the set of concave functions

S:X[,] S \colon X \to [-\infty,\infty]

This operad algebra will describe how, given a relation RX 1××X n×Y R \subseteq X_1 \times \cdots \times X_n \times Y, we can ‘push forward’ entropy functions on X 1,,X n X_1,\ldots,X_n to form an entropy function on Y Y.

The operad 𝒞ℛ \mathcal{CR} is built using a construction from Part 3 that takes a symmetric monoidal category and produces an operad. The symmetric monoidal category that we start with is ConvRel, \mathsf{ConvRel}, which has convex sets as its objects and convex relations as its morphisms. This symmetric monoidal category has Conv \mathsf{Conv} (the category of convex sets and convex-linear functions) as a subcategory with all the same objects, and ConvRel \mathsf{ConvRel} inherits a symmetric monoidal structure from the bigger category Conv. \mathsf{Conv}.

Following the construction from Part 3, we see that we get an operad

𝒞ℛ=Op(ConvRel) \mathcal{CR} = \mathrm{Op}(\mathsf{ConvRel})

exactly as described before: namely it has convex sets as types, and

𝒞ℛ(X 1,,X n;Y)=ConvRel(X 1××X n,Y) \mathcal{CR}(X_1,\ldots,X_n;Y) = \mathsf{ConvRel}(X_1 \times \cdots \times X_n, Y)

Next we want to make an operad algebra on 𝒞ℛ \mathcal{CR}. To do this we use a lax symmetric monoidal functor Ent \mathrm{Ent} from ConvRel \mathsf{ConvRel} to Set, \mathsf{Set}, defined as follows. On objects, Ent \mathrm{Ent} sends any convex set X X to the set of entropy functions on it:

Ent(X)={S:X[,]Sis concave} \mathrm{Ent}(X) = \{ S \colon X \to [-\infty,\infty] \mid S \: \text{is concave} \}

On morphisms, Ent \mathrm{Ent} sends any convex relation to to the map that “pushes forward” an entropy function along that relation:

Ent(RX×Y)=(ysup (x,y)RS(x)) \mathrm{Ent}(R \subseteq X \times Y) = (y \mapsto \sup_{(x,y) \in R} S(x))

And finally, the all-important laxator ϵ \epsilon produces an entropy function on X 1×X 2 X_1 \times X_2 by summing an entropy function on X 1 X_1 and an entropy function on X 2 X_2:

ϵ X 1,X 2=((S 1,S 2)S 1+S 2) \epsilon_{X_1,X_2} = ((S_1,S_2) \mapsto S_1 + S_2)

The proof that all this indeed defines a lax symmetric monoidal functor can be found in our paper. The main point is that once we have proven this really is a lax symmetric monoidal functor, we can invoke the machinery of lax symmetric monoidal functors and operad algebras to prove that we get an operad algebra! This is very convenient, because proving that we have an operad algebra directly would be somewhat tedious.

We have now reached the technical high point of the paper, which is showing that this operad algebra exists and thus formalizing what it means to compose thermostatic systems. All that remains to do now is to show off a bunch of examples of composition, so that you can see how all this categorical machinery works in practice. In our paper we give many examples, but here let’s consider just one.

Consider the following setup with two ideal gases connected by a movable divider.

The state space of each individual ideal gas is >0 3 \mathbb{R}^3_{> 0}, with coordinates (U,V,N) (U,V,N) representing energy, volume, and number of particles respectively. Let (U 1,V 1,N 1) (U_1, V_1, N_1) be the coordinates for the left-hand gas, and (U 2,V 2,N 2) (U_2, V_2, N_2) be the coordinates for the right-hand gas. Then as the two gases move to thermodynamic equilibrium, the conserved quantities are U 1+U 2 U_1 + U_2, V 1+V 2 V_1 + V_2, N 1 N_1 and N 2 N_2. We picture this with the following diagram.

Ports on the inner circles represent variables for the ideal gases, and ports on the outer circle represent variables for the composed system. Wires represent relations between those variables. Thus, the entire diagram represents an operation in 𝒞ℛ \mathcal{CR}, given by

U 1+U 2=U e U_1 + U_2 = U^e V 1+V 2=V e V_1 + V_2 = V^e N 1=N 1 e N_1 = N_1^e N 2=N 2 e N_2 = N_2^e

We can then use the operad algebra to take entropy functions S 1,S 2: >0 3[,] S_1,S_2 \colon \mathbb{R}^3_{> 0} \to [-\infty, \infty] on the two inner systems (the two ideal gases), and get an entropy function S e: >0 4[,] S^e \colon \mathbb{R}^4_{> 0} \to [-\infty,\infty] on the outer system.

As a consequence of this entropy maximization procedure, the inner state (U 1,V 1,N 1),(U 2,V 2,N 2) (U_1,V_1,N_1), (U_2,V_2,N_2) are such that the temperature and pressure equilibriate between the two ideal gases. This is because constrained maximization with the constraint U 1+U 2=U e U_1 + U_2 = U^e leads to the following equations at a maximizer:

1T 1=S 1U 1=S 2U 2=1T 2 \displaystyle{ \frac{1}{T_1} = \frac{\partial S_1}{\partial U_1} = \frac{\partial S_2}{\partial U_2} = \frac{1}{T_2} }

(where T 1 T_1 and T 2 T_2 are the respective temperatures), and

p 1T 1=S 1V 1=S 2V 2=p 2T 2 \displaystyle{ \frac{p_1}{T_1} = \frac{\partial S_1}{\partial V_1} = \frac{\partial S_2}{\partial V_2} = \frac{p_2}{T_2} }

(where p 1 p_1 and p 2 p_2 are the respective pressures).

Thus we arrive at the expected conclusion, which is that temperature and pressure equalize when we maximize entropy under constraints on the total energy and volume.

And that concludes this series of blog posts! For more examples of thermostatic composition, I invite you to read our paper, which has some “thermostatic systems” that one does not normally see thought of in this way, such as heat baths and probabilistic systems! And if you find this stuff interesting, don’t hesitate to reach out to me! Just drop a comment here or email me at the address in the paper.

Posted at June 18, 2022 11:28 PM UTC

TrackBack URL for this Entry:

2 Comments & 0 Trackbacks

Ising spins please!

Please somebody compose Ising spins!

I would love to see the real space/block spin/Kadanoff renormalization stuff (more ambitiously and/or in parallel, even conformal field theory) come out of “doing this backwards.”

Since it’s spin models all the way down, you could even inform Yang-Mills with such an effort.

Posted by: Steve Huntsman on June 20, 2022 2:27 PM | Permalink | Reply to this

Re: Ising spins please!

Yes, Ising models would fit quite nicely in our framework. In Example 18 we treat the example of lattice systems in the infinite-volume limit, but the finite-volume case is easier to handle and better for composing.

Posted by: John Baez on June 21, 2022 12:36 AM | Permalink | Reply to this

Post a New Comment