John Baez, **Owen Lynch**, and Joe Moeller

Applied Category Theory 2022

Compositional Thermostatics represents the beginning of an attempt to integrate *applied category theory* and *thermodynamics*.

But mostly, this is a result of trying to learn thermodynamics, and then trying to tidy it all up into nice categorical presentations.

- Thermostatic systems
- Composition of thermostatic systems
- Examples
- Conclusion

A **thermostatic system** consists of a convex space \(X\) called the **state space** and a concave function \(S:X \to \bar{\mathbb{R}}\) called the **entropy function**.

A **convex space** is a set \(X\) along with an operation \(c_{\lambda} \colon X \times X \to X\) for every \(\lambda\in[0,1]\) such that

- \(c_{\lambda}(x,x) = x\)
- \(c_{1}(x,y) = x\)
- \(c_{\lambda}(x,y) = c_{1-\lambda}(y,x)\)
- \(c_{\lambda}(c_{\gamma}(x,y),z) = c_{\lambda \gamma}(x,c_{\eta}(y,z))\) if \(\lambda(1-\gamma)=(1-\lambda\gamma)\eta\).

A subset \(X \subset \mathbb{R}^{n}\) for some \(n\) such that for \(x,y \in X\) and \(\lambda \in [0,1]\)

\[\lambda x + (1-\lambda)y \in X\]

is a convex space.

The extended reals \(\bar{\mathbb{R}}=[-\infty,+\infty]\) is a convex space where for \(\lambda \in (0,1)\)

\[ c_\lambda(x,y) = \lambda x + (1-\lambda) y \quad \text{for $x,y \in \mathbb{R}$} \]

\[ c_\lambda(x,+\infty) = +\infty \quad \text{for $x \in (-\infty,+\infty]$} \]

\[ c_\lambda(x,-\infty) = -\infty \quad \text{for $x \in [-\infty,+\infty]$} \]

A **concave function** \(S \colon X \to \bar{\mathbb{R}}\) is a function such that for all \(x,y \in X\),

\[ S(c_\lambda(x,y)) \geq c_\lambda(S(x),S(y)) \]

A **constant heat-capacity system** has state space \(X=\mathbb{R}_{>0}\) and the entropy function

\[ S(U) = C \log U\]

Why does this make sense?

\[ \frac{1}{T} = \frac{\partial S}{\partial U} \]

\[ CT = U \]

A **heat bath** at constant temperature \(T\) has state space \(X=\mathbb{R}\) and entropy function

\[ S(\Delta U) = \frac{\Delta U}{T} \]

A **probabilistic system** has state space \(X=\mathcal{P}(\Omega)\) for some finite set \(\Omega\), and

\[S(p) = -\sum_{\omega \in \Omega} p(\omega) \log p(\omega)\]

(Shannon entropy)

A **quantum system** has state space \(X\) equal to the collection of *mixed states* of a quantum system, and \(S\) given by *von Neumann* entropy.

There are generalizations of probability theory used in quantum foundations that also fit into this framework.

State space \(X_{1} = \mathbb{R}_{>0}\), \(S_{1}(U_{1}) = C_{1} \log U_{1}\)

State space \(X_{2} = \mathbb{R}_{>0}\), \(S_{2}(U_{2}) = C_{2} \log U_{2}\)

Total energy conserved, i.e.

\[U = U_1 + U_2\]

Found by maximizing \(S_1(U_1) + S_2(U_2)\) with respect to constraint \(U_1 + U_2 = U\).

Found by maximizing \(S_1(U_1) + S_2(U_2)\) with respect to constraint \(U_1 + U_2 = U\).

From calculus, at the maximum we can derive

\[ \frac{\partial S_{1}}{\partial U_{1}} = \frac{\partial S_{2}}{\partial U_{2}} \]

that is, \[\frac{1}{T_1} = \frac{1}{T_2}\]

Found by maximizing \(S_1(U_1) + S_2(U_2)\) with respect to constraint \(U_1 + U_2 = U\).

Also,

\[ S(U) = \sup_{U_1+U_2 = U} S_1(U_1) + S_2(U_2) = (C_{1} + C_{2}) \log U + K \]

Pie+icecream can be viewed as a single thermostatic system with variable \(U \in \mathbb{R}_{>0}\)

The **independent composition** of thermostatic systems \((X_{1},S_{1}),\ldots,(X_{n},S_{n})\) is

\[(X_{1} \times \cdots \times X_{n}, S_{1} + \cdots + S_{n})\]

The **coarse-graining** of a thermostatic system \((X, S)\) via a *convex relation* \(R \subset X \times Y\) is the thermostatic system \((Y, R_{\ast} S)\) where

\[R_{\ast} S(y) = \sup_{(x,y) \in R} S(x)\]

Composition of thermostatic systems is independent composition followed by coarse-graining.

A **convex relation** \(R\) between convex spaces \(X\) and \(Y\) is a convex subspace of \(X \times Y\) and is written \(R \colon X \not \to Y\).

The composite of \(R_{1} \colon X \not\to Y\) and \(R_{2} \colon Y \not\to Z\) defined by

\[ R_{2} \circ R_{1} = \{ (x,z) \in X \times Z \mid \exists y \in Y, (x,y) \in R_{1}, (y,z) \in R_{2} \}\]

is a convex subspace of \(X \times Z\).

There is a category \(\mathsf{ConvRel}\) that has as objects convex spaces, and as morphisms convex relations.

There is a functor \(\mathrm{Ent} \colon \mathsf{ConvRel} \to \mathsf{Set}\) defined in the following way.

\[ \mathrm{Ent}(X) = \{ S \colon X \to \bar{\mathbb{R}} \mid \text{$S$ is concave} \}\]

\[ \mathrm{Ent}(R \colon X \not\to Y) = R_{\ast}\]

where

\[ R_{\ast}S(y) = \sup_{(x,y) \in R} S(x) \]

This formalizes “coarse-graining” entropy functions.

For each \(n\), there is a natural transformation

\[ \mu \colon \mathrm{Ent}(-) \times \cdots \times \mathrm{Ent}(-) \Rightarrow \mathrm{Ent}(- \times \cdots \times -)\]

that for convex spaces \(X_1 \times \cdots \times X_n\) looks like the following

\[\mu_{X_{1},\ldots,X_{n}} \colon \mathrm{Ent}(X_1) \times \cdots \times \mathrm{Ent}(X_n) \to \mathrm{Ent}(X_1 \times \cdots \times X_n)\]

and is defined by

\[(S_{1},\ldots,S_{n}) \mapsto S_{1} + \cdots + S_{n}\]

This formalizes “independent composition”

(J.C. Baez, O.L., J. Moeller, 2021). \(\mathrm{Ent}\) along with \(\mu\) as defined before forms a lax symmetric monoidal functor from \(\mathsf{ConvRel}\) to \(\mathsf{Set}\).

Given thermostatic systems \((X_{1},S_{1}), \ldots, (X_{n},S_{n})\), a convex space \(Y\), and a relation \(R \colon (X_{1} \times \cdots \times X_{n}) \not\to Y\), the central dogma builds a thermostatic system on \(Y\) via

\[ \mathrm{Ent}(X_{1}) \times \cdots \times \mathrm{Ent}(X_{n}) \xrightarrow{\mu} \mathrm{Ent}(X_1 \times \cdots \times X_n) \xrightarrow{R_{\ast}} \mathrm{Ent}(Y) \]

\[ (S_1, \ldots, S_n) \mapsto S_1 + \cdots + S_n \mapsto R_\ast(S_1 + \cdots + S_n) \]

This composition can be formalized using operads.

An **operad** \(\mathcal{O}\) consists of

- A collection \(\mathcal{O}_{0}\) of
**types** - For all types \(X_1,\ldots,X_n,Y\), a collection \(\mathcal{O}(X_{1},\ldots,X_{n};Y)\) of
**operations**

along with a way of composing the operations.

There is an operad \(\mathcal{CR}\) with types convex spaces and where

\[ \mathcal{CR}(X_1,\ldots,X_n;Y) = \mathsf{ConvRel}(X_1 \times \cdots \times X_n, Y) \]

An operad algebra \(F\) for an operad \(\mathcal{O}\) associates a set of things \(F(X)\) to each type \(X \in \mathcal{O}_{0}\), and a function

\[F(f) \colon F(X_{1}) \times \cdots \times F(X_{n}) \to F(Y)\]

to every operation \(f \in \mathcal{O}(X_{1},\ldots,X_{n};Y)\), in a way compatible with composition in the operad.

(J.C. Baez, O.L., J. Moeller, 2021). There is an operad algebra \(E\) on \(\mathcal{CR}\), where \(E(X) = \mathrm{Ent}(X)\) is the set of entropy functions on \(X\), and given a relation \(R \in \mathcal{CR}(X_{1},\ldots,X_{n};Y)\), \(E(R)\) is the central dogma formula of

\[ \mathrm{Ent}(X_{1}) \times \cdots \times \mathrm{Ent}(X_{n}) \xrightarrow{\mu} \mathrm{Ent}(X_1 \times \cdots \times X_n) \xrightarrow{R_{\ast}} \mathrm{Ent}(Y) \]

\[ (S_1, \ldots, S_n) \mapsto S_1 + \cdots + S_n \mapsto R_\ast(S_1 + \cdots + S_n) \]

that is,

\[ E(R)(S_{1},\ldots,S_{n}) = R_\ast(S_1 + \cdots + S_n) \]

Start with \(X_{1} = X_{2} = Y = \mathbb{R}_{>}\), and \(R \colon X_{1} \times X_{2} \not\to Y\) given by

\[ R = \{ (U_{1},U_{2},U) \mid U_{1} + U_{2} = U \} \]

Take entropy functions \(S_{i}(U_{i}) = C_{i} \log U_{i}\)

Use the operad algebra to apply \(R\) to \(S_{1}\) and \(S_{2}\)

End up with an entropy function on \(Y\)

This is just the central dogma, but enshrined in math

- What happens when we attach a probabilistic system to a heat bath?

Probabilistic system, \(X_{1}=\mathcal{P}(\Omega)\)

\[ S(p) = - \sum_{\omega \in \Omega} p(\omega) \log p(\omega)\]

Heat bath \(X_{2}=\mathbb{R}\)

\[S_2(\Delta U) = \frac{1}{T} \Delta U\]

Let \(H \colon \Omega \to \mathbb{R}\) be an observable for the probabilistic system

Then let the following relation hold for \(p \in \mathcal{P}(\Omega) = X_1\) and \(\Delta U \in \mathbb{R} = X_2\)

\[ \mathbb{E}_p H + \Delta U = 0\]

The point of maximum entropy here is the

**canonical distribution**\[ p(\omega) = \frac{e^{-H(\omega)/T}}{Z_{1/T}}\]

where \(Z_{1/T}\) is the

**partition function**, a normalizing factor.

- Operads and operad algebras are a toolbox for complex open systems
- We can get pretty far by just thinking about constrained entropy maximization
- Modeling composition of systems with lax symmetric monoidal functors is productive