# Compositional Thermostatics

Applied Category Theory 2022

## Statement of Purpose

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.

## Overview

1. Thermostatic systems
2. Composition of thermostatic systems
3. Examples
4. Conclusion

## Thermostatic systems

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 $\lambda(1-\gamma)=(1-\lambda\gamma)\eta$.

## Example convex spaces

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.

Convex subset

## Example convex spaces

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

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

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

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

## Concave functions

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

$S(c_\lambda(x,y)) \geq c_\lambda(S(x),S(y))$

## Examples of thermostatic systems

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?

$\frac{1}{T} = \frac{\partial S}{\partial U}$

$CT = U$

## Examples of thermostatic systems

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

$S(\Delta U) = \frac{\Delta U}{T}$

## Examples of thermostatic systems

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.

## Example composition

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 $U_1 + U_2 = U$.

## Example composition

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, $\frac{1}{T_1} = \frac{1}{T_2}$

## Example composition

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 Central Dogma

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.

## Formalization of the central dogma

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.

## Convex relations visualized

Pie and icecream relation

## Formalization of the central dogma

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.

## Formalization of the central dogma

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”

## Putting it together

(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,

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

## Examples: Pie and icecream again

Pie and icecream again

## Examples: Pie and icecream again

• 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

## Examples: Statistical mechanics

• 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 $X_{2}=\mathbb{R}$

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

## Statistical mechanics

• 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.

## Summary and Conclusions

• 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