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.
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
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
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
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.