# Axiomatic Information Thermodynamics

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## Abstract

**:**

## 1. Introduction

## 2. Eidostates

**Axiom**

**1.**

- (a)
- Every $A\in \mathcal{E}$ is a finite nonempty set with a finite prime Cartesian factorization.
- (b)
- $A+B\in \mathcal{E}$ if and only if $A,B\in \mathcal{E}$.
- (c)
- Every nonempty subset of an eidostate is also an eidostate.

## 3. Processes

**Axiom**

**2.**

- (a)
- If $A\sim B$, then $A\to B$.
- (b)
- If $A\to B$ and $B\to C$, then $A\to C$.
- (c)
- If $A\to B$, then $A+C\to B+C$.
- (d)
- If $A+s\to B+s$, then $A\to B$.

- natural if $A\to B$;
- antinatural if $B\to A$;
- possible if $A\to B$ or $B\to A$ (which may be written $A\rightleftharpoons B$);
- impossible if it is not possible;
- reversible if $A\leftrightarrow B$; and
- irreversible if it is possible but not reversible.

**Axiom**

**3.**

**Axiom**

**4.**

- (a)
- Suppose $A,{A}^{\prime}\in \mathcal{E}$ and $b\in \mathcal{S}$. If $A\to b$ and ${A}^{\prime}\subseteq A$ then ${A}^{\prime}\to b$.
- (b)
- Suppose A and B are uniform eidostates that are each disjoint unions of eidostates: $A={A}_{1}\cup {A}_{2}$ and $B={B}_{1}\cup {B}_{2}$. If ${A}_{1}\to {B}_{1}$ and ${A}_{2}\to {B}_{2}$ then $A\to B$.

## 4. Process Algebra and Irreversibility

- If $\mathrm{\Gamma},\Delta \in \widehat{\mathcal{P}}$, then $\mathbb{I}(\mathrm{\Gamma}+\Delta )=\mathbb{I}(\mathrm{\Gamma})+\mathbb{I}(\Delta )$.
- The value of $\mathbb{I}$ determines the type of the processes in $\widehat{\mathcal{P}}$:
- $\mathbb{I}(\mathrm{\Gamma})>0$ whenever $\mathrm{\Gamma}$ is natural irreversible.
- $\mathbb{I}(\mathrm{\Gamma})=0$ whenever $\mathrm{\Gamma}$ is reversible.
- $\mathbb{I}(\mathrm{\Gamma})<0$ whenever $\mathrm{\Gamma}$ is antinatural irreversible.

**Theorem**

**1.**

**Proof.**

**Theorem**

**2**(“Hahn–Banach theorem” for Abelian groups)

**.**

## 5. Information and Entropy

**Axiom**

**5.**

**Theorem**

**3.**

**Proof.**

- $\mathbb{S}(A+B)=\mathbb{S}(A)+\mathbb{S}(B)$.
- If $\u2329A,B\u232a$ is natural irreversible, then $\mathbb{S}(A)<\mathbb{S}(B)$.
- If $\u2329A,B\u232a$ is reversible, then $\mathbb{S}(A)=\mathbb{S}(B)$.

## 6. Demons

**Axiom**

**6.**

- (a)
- There exists $I\in \mathcal{I}$ such that $b\to a+I$.
- (b)
- For any $I\in \mathcal{I}$, either $a\to b+I$ or $b+I\to a$.

- There exists an information state $J\in \mathcal{I}$ such that either $A\to B+J$ or $B\to A+J$.
- There exists an information process $\u2329I,J\u232a\in {\mathcal{P}}_{I}$ such that $\u2329A,B\u232a+\u2329I,J\u232a$ is possible; that is, either $A+I\to B+J$ or $B+J\to A+I$.

**Theorem**

**4.**

**Proof.**

**Axiom**

**7.**

## 7. Irreversibility for Singleton Processes

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

## 8. Components of Content and Entropy

- We can construct components of content, which are the abstract versions of conserved quantities. A component of content Q is an additive function on $\mathcal{S}$ such that, if the singleton process $\u2329a,b\u232a$ is possible, then $Q(a)=Q(b)$. (In conventional thermodynamics, components of content include energy, particle number, etc.)
- We can find a sufficient set of components of content. The singleton process $\u2329a,b\u232a$ is possible if and only if $Q(a)=Q(b)$ for all Q in the sufficient set.
- We can use $\mathbb{I}$ to define a quasi-entropy $\mathbb{S}$ on $\mathcal{S}$ as follows: $\mathbb{S}(a)=\mathbb{I}(\u2329a,2a\u232a)$. This is an additive function on states in $\mathcal{S}$ such that $\mathbb{I}(\u2329a,b\u232a)=\mathbb{S}(b)-\mathbb{S}(a)$.

**Axiom**

**8.**

- (a)
- If $l,m\in \mathcal{M}$, then $l+m\in \mathcal{M}$.
- (b)
- For $l,m\in \mathcal{M}$, if $l\to m$ then $m\to l$.

**Theorem**

**8.**

- (a)
- For any $a,b\in \mathcal{S}$, $\mathbb{S}(a+b)=\mathbb{S}(a)+\mathbb{S}(b)$.
- (b)
- For any $a,b\in \mathcal{S}$ and component of content Q, $Q(a+b)=Q(a)+Q(b)$.
- (c)
- For any $a,b\in \mathcal{S}$, $a\to b$ if and only if $\mathbb{S}(a)\le \mathbb{S}(b)$ and $Q(a)=Q(b)$ for every component of content Q.
- (d)
- $\mathbb{S}(m)=0$ for all $m\in \mathcal{M}$.

## 9. State Equivalence

**Axiom**

**9.**

**Theorem**

**9**(Uniform eidostate thermodynamics)

**.**

- (a)
- For any $E,F\in \mathcal{U}$, $\mathbb{S}(E+F)=\mathbb{S}(E)+\mathbb{S}(F)$.
- (b)
- For any $E,F\in \mathcal{U}$ and component of content Q, $Q(E+F)=Q(E)+Q(F)$.
- (c)
- For any $E,F\in \mathcal{U}$, $E\to F$ if and only if $\mathbb{S}(E)\le \mathbb{S}(F)$ and $Q(E)=Q(F)$ for every component of content Q.
- (d)
- $\mathbb{S}(m)=0$ for all $m\in \mathcal{M}$.

## 10. Entropy for Uniform Eidostates

**Theorem**

**10.**

**Proof.**

- There are disjoint information states ${J}_{k}$ containing ${m}_{k}$ record states.
- There are disjoint information states ${J}_{k}^{*}$ containing ${m}_{k}+1$ record states.
- $J={J}_{1}\cup {J}_{2}$ and ${J}^{*}={J}_{1}^{*}\cup {J}_{2}^{*}$ have ${m}_{1}+{m}_{2}$ and ${m}_{1}+{m}_{2}+2$ record states, respectively.
- We have$${S}_{0}+\mathrm{log}{m}_{k}\le \mathbb{S}({E}_{k})<{S}_{0}+\mathrm{log}({m}_{k}+1).$$

**Theorem**

**11.**

## 11. Probability

**Theorem**

**12.**

**Proof.**

## 12. A Model for the Axioms

- A set $\mathcal{S}$ of states and a collection $\mathcal{E}$ of finite nonempty subsets of $\mathcal{S}$ to be designated as eidostates.
- A rule for interpreting the combination of states (+) in $\mathcal{S}$.
- A relation → on $\mathcal{E}$.
- A designated set $\mathcal{M}\subseteq \mathcal{S}$ of mechanical states (which might be empty).
- Proofs of Axioms 1–9 within the model, including the general properties of →, the existence of record states and information states, etc.

- Either ${N}_{A}$ and ${N}_{B}$ both do not exist, or ${N}_{A}\sim {N}_{B}$.
- Either ${U}_{A}$ and ${U}_{B}$ both do not exist, or only one exists and its Q-value is 0, or both exist and $Q({U}_{A})=Q({U}_{B})$.
- Either ${U}_{A}$ and ${U}_{B}$ both do not exist, or only ${U}_{A}$ exists and $\mathbb{S}({U}_{A})=0$, or only ${U}_{B}$ exists and $\mathbb{S}({U}_{B})\ge 0$, or both exist and $\mathbb{S}({U}_{A})\le \mathbb{S}({U}_{B})$.

**Axiom****1**- The basic properties of eidostates follow by construction.
**Axiom****2**- Part (a) holds because $A\sim B$ implies that A and B can have the same NU-decomposition. Part (b) holds because similarity, equality (for Q) and inequality (for $\mathbb{S}$) are all transitive. Parts (c) and (d) make use of the general facts that ${N}_{A+B}\sim {N}_{A}+{N}_{B}$ and ${U}_{A+B}\sim {U}_{A}+{U}_{B}$.
**Axiom****3**- If ${N}_{A}\nsim {N}_{B}$, then $A\nrightarrow B$. If ${N}_{A}\sim {N}_{B}$, then it must be true that ${U}_{B}\u228a{U}_{A}$, and so $\mathbb{S}({U}_{A})>\mathbb{S}({U}_{B})$. The $\mathbb{S}$-criterion fails, so $A\nrightarrow B$ in this case as well.
**Axiom****4**- For Part (a), we note that A must be uniform, and so ${A}^{\prime}\subseteq A$ is also uniform. The statement follows from the $\mathbb{S}$-criterion. Part (b) also follows from the $\mathbb{S}$-criterion.
**Axiom****5**- The atomic state r with $Q(r)=0$ and $\mathbb{S}(r)=0$ is a record state, as is $r+r$, etc. We can take our bit state to be ${I}_{\mathrm{b}}=\{r,r+r\}$. Since every information state $I\in \mathcal{I}$ is uniform with $Q(I)=0$, every information process (including ${\Theta}_{\mathrm{b}}=\u2329r,{I}_{\mathrm{b}}\u232a$) is possible.
**Axiom****6**- Since all of the states of the form $a+I$ are uniform, the statements in this axiom follow from the $\mathbb{S}$-criterion.
**Axiom****7**- Suppose $nA\to nB+J$. Since J is uniform, it must be that $n{N}_{A}\sim n{N}_{B}$, from which it follows that ${N}_{A}\sim {N}_{B}$. The $\mathbb{S}$-criterion for ${U}_{A}$ and ${U}_{B}$ follows from a typical stability argument—that is, if $nx\le ny+z$ for arbitrarily large values of n, then it must be true $x\le y$.
**Axiom****8**- The set $\mathcal{M}$ of mechanical states may be defined to include all states that can be constructed from the zero-entropy atomic state ${s}_{0}$ (${s}_{0}+{s}_{0}$, ${s}_{0}+({s}_{0}+{s}_{0})$, etc.). The required properties of $\mathcal{M}$ follow.
**Axiom****9**- The uniform eidostate E has $Q(E)=q\ge 0$ and $\mathbb{S}(E)=\sigma \ge 0$. Choose an integer $n>\sigma $. Now, let $e=q{s}_{0}$ (or $e=r$ if $q=0$), $x=n{s}_{0}$, and $y=n{s}_{\lambda}$ where $\lambda =\sigma /n$. We find that $Q(e)=q$, $Q(x)=Q(y)=n$, $\mathbb{S}(e)=\mathbb{S}(x)=0$ and $\mathbb{S}(y)=n(\sigma /n)=\sigma $. It follows that $x\to y$ and $E+x\leftrightarrow e+y$.

## 13. A Simple Quantum Model

**Axiom****1**- This follows from our construction of the eidostates $\mathcal{E}$.
**Axiom****2**- All of these basic properties of the → relation follow from the subspace dimension criterion.
**Axiom****3**- If $B\u228aA$, then ${d}_{A}>{d}_{B}$, and so $A\nrightarrow B$.
**Axiom****4**- Again, both parts of this axiom follow from the subspace dimension criterion. If eidostate A is a disjoint union of eidostates ${A}_{1}$ and ${A}_{2}$, then ${d}_{A}={d}_{{A}_{1}}+{d}_{{A}_{2}}$.
**Axiom****5**- Any state r with ${d}_{r}=1$ functions as a record state. We have assumed that such a state exists. We can take ${I}_{\mathrm{b}}=\{r,r+r\}$. The bit process ${\Theta}_{\mathrm{b}}=\u2329r,{I}_{\mathrm{b}}\u232a$ is natural ($r\to {I}_{\mathrm{b}}$) by the subspace dimension criterion. Notice that, for any information state I, ${d}_{I}=\#(I)$.
**Axiom****6**- For any $b\in \mathcal{S}$ and $I\in \mathcal{I}$, we have ${d}_{b+I}={d}_{b}\xb7\#(I)$. For Part (a), we can always find a large enough information state so that ${d}_{b}\le {d}_{a}\xb7\#(I)$. For Part (b), either ${d}_{a}\le {d}_{b+I}$ or ${d}_{b+I}\le {d}_{a}$.
**Axiom****7**- If ${({d}_{A})}^{n}\le {({d}_{B})}^{n}\xb7\#(J)$ for arbitrarily large values of n, then ${d}_{A}\le {d}_{B}$.
**Axiom****8**- It is consistent to take $\mathcal{M}=\varnothing $.
**Axiom****9**- All of our eidostates are uniform. For any eidostate E, we can choose e so that ${d}_{e}={d}_{E}$. (Recall that we have assumed states with every positive subspace dimension.) If we chose $x=y$ to be any state, then $E+x\leftrightarrow e+y$.

## 14. Remarks

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

**eidostate**is a set whose elements are called

**states**. The collection of eidostates is $\mathcal{E}$ and the collection of states is $\mathcal{S}$. An element $a\in \mathcal{S}$ may be identified with the singleton eidostate $\left\{a\right\}\in \mathcal{E}$.

**similar**(written $A\sim B$) if they are made up of the same Cartesian factors, perhaps combined in a different way.

**uniform**if, for all $a,b\in A$, either $a\to b$ or $b\to a$. A formal

**process**is a pair of eidostates $\u2329A,B\u232a$. We say that a process $\u2329A,B\u232a$ is

**possible**if either $A\to B$ or $B\to A$.

**record state**r is a state for which there exists another state a such that $a\to a+r$ and $a+r\to a$ (denoted $a\leftrightarrow a+r$). An

**information state**is an eidostate containing only record states, and the set of information states is called $\mathcal{I}$. A

**bit state**${I}_{\mathrm{b}}$ is an information state containing exactly two distinct record states. A

**bit process**is a formal process $\u2329r,{I}_{\mathrm{b}}\u232a$, where r is a record state and ${I}_{\mathrm{b}}$ is a bit state.

**Axiom****1**- (Eidostates.) $\mathcal{E}$ is a collection of sets called eidostates such that:
**(a)**- Every $A\in \mathcal{E}$ is a finite nonempty set with a finite prime Cartesian factorization.
**(b)**- $A+B\in \mathcal{E}$ if and only if $A,B\in \mathcal{E}$.
**(c)**- Every nonempty subset of an eidostate is also an eidostate.

**Axiom****2**- (Processes.) Let eidostates $A,B,C\in \mathcal{E}$, and $s\in \mathcal{S}$.
**(a)**- If $A\sim B$, then $A\to B$.
**(b)**- If $A\to B$ and $B\to C$, then $A\to C$.
**(c)**- If $A\to B$, then $A+C\to B+C$.
**(d)**- If $A+s\to B+s$, then $A\to B$.

**Axiom****3**- If $A,B\in \mathcal{E}$ and B is a proper subset of A, then $A\nrightarrow B$.
**Axiom****4**- (Conditional processes.)
**(a)**- Suppose $A,{A}^{\prime}\in \mathcal{E}$ and $b\in \mathcal{S}$. If $A\to b$ and ${A}^{\prime}\subseteq A$ then ${A}^{\prime}\to b$.
**(b)**- Suppose A and B are uniform eidostates that are each disjoint unions of eidostates: $A={A}_{1}\cup {A}_{2}$ and $B={B}_{1}\cup {B}_{2}$. If ${A}_{1}\to {B}_{1}$ and ${A}_{2}\to {B}_{2}$ then $A\to B$.

**Axiom****5**- (Information.) There exist a bit state and a possible bit process.
**Axiom****6**- (Demons.) Suppose $a,b\in \mathcal{S}$ and $J\in \mathcal{I}$ such that $a\to b+J$.
**(a)**- There exists $I\in \mathcal{I}$ such that $b\to a+I$.
**(b)**- For any $I\in \mathcal{I}$, either $a\to b+I$ or $b+I\to a$.

**Axiom****7**- (Stability.) Suppose $A,B\in \mathcal{E}$ and $J\in \mathcal{I}$. If $nA\to nB+J$ for arbitrarily large values of n, then $A\to B$.
**Axiom****8**- (Mechanical states.) There exists a subset $\mathcal{M}\subseteq \mathcal{S}$ of mechanical states such that:
**(a)**- If $l,m\in \mathcal{M}$, then $l+m\in \mathcal{M}$.
**(b)**- For $l,m\in \mathcal{M}$, if $l\to m$ then $m\to l$.

**Axiom****9**- (State equivalence.) If E is a uniform eidostate then there exist states $e,x,y\in \mathcal{S}$ such that $x\to y$ and $E+x\leftrightarrow e+y$.

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**Figure 1.**A one-particle gas may freely expand to occupy a larger volume, but the reverse process would violate the Second Law.

**Figure 2.**A Maxwell’s demon device acquires one bit of information about the location of a gas particle, allowing it to contract the gas to a smaller volume.

**Figure 3.**Maxwell’s demon interacting with a one-particle gas, illustrating a bit process $\u2329r,\{{r}_{0},{r}_{1}\}\u232a$.

**Figure 4.**A state equivalence thought-experiment involving states of an arbitrary thermodynamic system and a one-particle gas.

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Hulse, A.; Schumacher, B.; Westmoreland, M.D. Axiomatic Information Thermodynamics. *Entropy* **2018**, *20*, 237.
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Hulse A, Schumacher B, Westmoreland MD. Axiomatic Information Thermodynamics. *Entropy*. 2018; 20(4):237.
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Hulse, Austin, Benjamin Schumacher, and Michael D. Westmoreland. 2018. "Axiomatic Information Thermodynamics" *Entropy* 20, no. 4: 237.
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