# Landauer’s Principle as a Special Case of Galois Connection

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Related Work

#### 2.1. Entropy and Ordering in Thermodynamics

**Definition**

**1.**

- Monotonicity: For two comparable states X and Y$$X\preccurlyeq Y\phantom{\rule{1.em}{0ex}}\iff \phantom{\rule{1.em}{0ex}}S\left(X\right)\le S\left(Y\right);$$
- Additivity: For two states X and Y, the entropy of compound state $(X,Y)$ is$$S(X,Y)=S\left(X\right)+S\left(Y\right);$$
- Extensivity: For $\lambda >0$ and a state X$$S\left(\lambda X\right)=\lambda S\left(X\right).$$

- Reversible adiabatic process:$$X\sim Y\phantom{\rule{1.em}{0ex}}\iff \phantom{\rule{1.em}{0ex}}S\left(X\right)=S\left(Y\right);$$
- Irreversible adiabatic process:$$X\prec \prec Y\phantom{\rule{1.em}{0ex}}\iff \phantom{\rule{1.em}{0ex}}S\left(X\right)<S\left(Y\right).$$

#### 2.2. Galois Connection

- monotone if for any $x,y\in C$, if $x\preccurlyeq y$, then $Fx\u2291Fy$;
- order-embedding if for all $x,y\in C$, $x\preccurlyeq y\iff Fx\u2291Fy$; and
- order-isomorphism iff F is surjective order-embedding.

**Definition**

**2**

**.**Suppose that $\mathcal{C}=(C,\preccurlyeq )$ and $\mathcal{D}=(D,\u2291)$ are two posets, and let $F:\mathcal{C}\to \mathcal{D}$ and $G:\mathcal{D}\to \mathcal{C}$ be a pair of functors such that for all $c\in C$, $d\in D$,

**Theorem**

**1**

- 1.
- F and G are both monotone;
- 2.
- for all $c\in C$, $d\in D$, $c\preccurlyeq GFc$ and $FGd\u2291d$; and
- 3.
- $FGF=F$ and $GFG=G$.

## 3. Main Results

- state-space (G-Set) + entropy → total ordering;
- total ordering → poset (G-poset) structure; and
- two posets → Galios (Landauer’s) connection between them.

**Step 1.**The main point is to introduce state-space set $\Gamma $. In the case of thermodynamic systems, the scaling of the system is modeled as an action of the multiplicative group $({\mathbb{R}}^{+},\xb7,1)$ on the set $\Gamma $, which preserves ordering (as described in [3,4] and Section 2.1). Such kind of element $\{\Gamma ,({\mathbb{R}}^{+},\xb7,1)\}$ is the object of a G-Set category [24], namely ${\mathbb{R}}^{+}$-Set category. However, in non-thermodynamic systems (e.g., information theory), there is usually no group action and we have the following options: narrowly describe the state-space to the category set, select the category of G-Set with the trivial group $(1,\xb7,1)$, or model state-space on the ${\mathbb{R}}^{+}$-Set category with trivial group action. We select the last possibility since it gives a more uniform approach.

**Definition**

**3.**

**Definition**

**4.**

**Step 2.**The existence of total ordering ≼ on $\Gamma $ allows us to define a poset structure. However, for accounts of additional group structure of scaling, the more general approach would be to use G-Pos category [25], i.e., posets with a group action. The group action is needed only when the scaling is present in the system (in particular thermodynamics). In other systems without scaling, the group action is trivial. Therefore, we omit the group action/scaling part when it is not important in the context and provide modifications in the presence of G-Pos structure of group action/scaling later.

**Definition**

**5.**

**Step 3.**The third key ingredient to formulate the Landauer’s connection is the Galois connection from category theory. The short overview of its definition and basic properties are collected in Section 2.2 for the reader’s convenience.

**Definition**

**6.**

- a set part of the functor: $F:{\Gamma}_{1}\to {\Gamma}_{2}$; and
- a function ϕ that is a surjective group endomorphism of $({\mathbb{R}}^{+},\xb7,1)$,

**Corollary**

**1.**

**Proof.**

**Corollary**

**2.**

- 1.
- For ${c}_{1},{c}_{2}\in C$, if ${c}_{1}\preccurlyeq {c}_{2}$ then ${S}_{2}\left(F{c}_{1}\right)\le {S}_{2}\left(F{c}_{2}\right)$; analogously for G functor. In other words, F and G are monotone functors.
- 2.
- For all $c\in C$, $d\in D$, ${S}_{1}\left(c\right)\le {S}_{1}\left(GFc\right)$ and ${S}_{2}\left(FGd\right)\le {S}_{2}\left(d\right)$.
- 3.
- For all $c\in C$, $d\in D$, ${S}_{2}\left(FGFc\right)={S}_{2}\left(Fc\right)$, and ${S}_{1}\left(GFGd\right)={S}_{2}\left(Gd\right)$.

**Theorem**

**2.**

**Theorem**

**3.**

**Construction**

**1.**

**Corollary**

**3.**

- 1.
- if states $p,{p}^{\prime}$ are ordered as follows: $p\preccurlyeq {p}^{\prime}$, then $p\preccurlyeq {p}^{\prime}\preccurlyeq GF\left({p}^{\prime}\right)$ and $p\preccurlyeq GF\left(p\right)\preccurlyeq GF\left({p}^{\prime}\right)$; and
- 2.
- if states $p,{p}^{\prime}$ are ordered as follows: $q\u2291{q}^{\prime}$, then $FG\left(q\right)\u2291q\u2291{q}^{\prime}$ and $FG\left(q\right)\u2291FG\left({q}^{\prime}\right)\u2291{q}^{\prime}$.

**Lemma**

**1.**

- 1.
- $GF\left(p\right)=p,\phantom{\rule{1.em}{0ex}}\forall p\in {\Gamma}_{1}$.
- 2.
- F is surjective,
- 3.
- G is injective.

**Corollary**

**4.**

**Definition**

**7.**

**Theorem**

**4.**

**Proof.**

## 4. Examples

#### 4.1. Toy Example

**Case 1.**Consider $F:{\Gamma}_{1}\to {\Gamma}_{2}$ defined as $F\left(z\right)=\lceil \frac{z}{3}\rceil $ and $G:{\Gamma}_{2}\to {\Gamma}_{1}$ given by $G\left(z\right)=3z$. We have $F\u22a3G$ since it fulfills Equation (7), i.e.

- Consider now the following map $f:{\Gamma}_{1}\to {\Gamma}_{1}$ given by a simple shift $f\left(z\right)=z+0.2$. Take $x=1\in {\Gamma}_{1}$ for which $S\left(x\right)=1$. Then, $\overline{x}=f\left(x\right)=1.2$ and $S\left(f\right(x\left)\right)=1.2$ and therefore process $x\to \overline{x}$ is irreversible (entropy increases). We have $y=F\left(x\right)=1$ with $S\left(y\right)=1$, and $\overline{y}=F\left(\overline{x}\right)=Ff\left(\overline{x}\right)=1$ with $S\left(\overline{y}\right)=1$, and therefore, the irreversible process in ${\Gamma}_{1}$ is mapped by F to reversible process on the level of ${\Gamma}_{2}$.
- Take the same map $f\left(x\right)=x+0.2$ with initial point $x=2.9$. It gives $\overline{x}=f\left(x\right)=3.1$ and therefore $S\left(x\right)=2.9$ and $S\left(\overline{x}\right)=3.1$—irreversible process in ${\Gamma}_{1}$. Using functor F, we get $y=F\left(x\right)=1$ and $\overline{y}=F\left(\overline{y}\right)=2$. Therefore, irreversible process in ${\Gamma}_{1}$ is mapped to irreversible process in ${\Gamma}_{2}$.
- If we take $f\left(x\right)=x$, then reversible (trivial) process in ${\Gamma}_{1}$ is mapped to reversible process in ${\Gamma}_{2}$
- No irreversible process in ${\Gamma}_{2}$ can be realized by a reversible process in ${\Gamma}_{1}$.

**Case 2.**We now take $F:{\Gamma}_{2}\to {\Gamma}_{1}$ defined as $F\left(x\right)=3x$ and $G:{\Gamma}_{1}\to {\Gamma}_{2}$ given by $G\left(x\right)=\lfloor \frac{x}{3}\rfloor $. This also defines the Galois connection $F\u22a3G$ as it is easily checked. We have the following examples of processes:

- The process in ${\Gamma}_{1}$, e.g., the shift $f\left(z\right)=z+3$ that irreversibly maps $x=6$ to $\overline{x}=9$ on the level of ${\Gamma}_{2}$ gives the map from $y=G\left(x\right)=2$ into $\overline{y}=G\left(\overline{x}\right)=3$ which is also irreversible.
- For the irreversible shift $f\left(z\right)=z+0.1$ on ${\Gamma}_{1}$ that maps $x=2$ to $\overline{x}=2.1$, we have reversible (identity) process in ${\Gamma}_{2}$ that maps $y=G\left(x\right)=0$ to $\overline{x}=G\left(\overline{y}\right)=0$ that is obviously reversible.
- Identity (reversible) process in ${\Gamma}_{1}$ is trivially mapped into reversible process in ${\Gamma}_{2}$.
- There is no mapping of an irreversible process on ${\Gamma}_{1}$ to a reversible process in ${\Gamma}_{2}$.

**Case 3.**For an example of Case 3 of Theorem 4, consider identity mapping $F=I{d}_{{\Gamma}_{1}}=G$ between two copies of ${\Gamma}_{1}$.

#### 4.2. Landauer’s Functors and Maxwell’s demon

- In this state, there is no information on localization of the particle, and therefore information entropy ${S}_{i}=2$ as the state is the mixture of two states 01 and 10. This state is associated to 00 bit description (reset) and transferred to the memory.
- Particle was localized (for example) in the left chamber (10) so the information entropy is now ${S}_{i}=1$. This state is transferred into memory, where reversible operation (e.g., $NOT\otimes Id$) is performed that change 00 into 10.
- Partition starts to move freely. There is adiabatic decompression of a single particle gas. State of the memory is the same as in the previous step.
- Partition is pushed maximally to the right. The work done by the particle is $W={k}_{B}T{\int}_{V/2}^{V}\frac{dV}{V}={k}_{B}Tln2$, where V is the volume of the box, ${k}_{B}$ is the Boltzmann constant, and T is the temperature. Since no heat flow was present, thermodynamic entropy is still constant and the internal energy of the gas decreased. New cycle will start.
- This transition is the restart of the cycle. The partition is placed in the middle of the box, and therefore information on localization of the particle is lost. Information entropy is now ${S}_{i}=2$. State become 00 and it is correlated with the state of the memory—irreversible operation (e.g., $f\left(x\right)=00\phantom{\rule{1.em}{0ex}}AND\phantom{\rule{1.em}{0ex}}x$) is performed on the memory, which results in expelling, via Landauer’s principle, at least ${k}_{B}Tln2$ of heat form its physical part D. Cycle repeats.

#### 4.3. DNA Computation

#### 4.4. Is 42 the Meaning of Life?

## 5. Discussion

## 6. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The Landauer’s connection between box with ideal gas E, memory M of the Maxwell’s demon and its physical realization D in the Maxwell’s demon experiment.

**Figure 2.**Maxwell’s demon experiment with a single particle and movable partition. On the left, there is a box with movable partition and, on the right, a corresponding memory state.

**Figure 3.**Schematic structure of computation in DNA. On the level of computation realm (information theory), there is information encoded in DNA strand. It is Galois connected with biochemical system which realizes computations by means of chemical reactions. This system contains an Environment (Env.) and embeds inside the System (Sys.) with DNA, chemical elements and enzymes, where actual computation takes place. The Environment interacts with the System for conducting specific chemical reactions that realizes logical operations. The System and Environment overall fulfill the second law of thermodynamics and therefore the total entropy can remain constant or increase, i.e., $\Delta {S}_{Env}+\Delta {S}_{Sys.}=\Delta {S}_{T}\ge 0$. Defining the enthalpy change (dispersed heat of the System) as $\Delta H=-T\Delta {S}_{Sys.}$ and the Gibbs free energy change as $\Delta G=-T\Delta {S}_{T}$ on gets the famous equation $\Delta G=\Delta H-T\Delta {S}_{Sys}$. All reactions in the system are spontaneous if $\Delta {S}_{T}>0$, that is $\Delta G<0$. This is the principle of interaction between the System and the Environment.

**Table 1.**Table 1 from [11] defining connection between memory operations and their realizations in thermodynamic system. For definition of (ir)reversibility, see below.

Possibilities | Thermodynamically Reversible | Thermodynamically Irreversible |
---|---|---|

Logically reversible | YES | YES |

Logically irreversible | NO | YES |

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Kycia, R.A.
Landauer’s Principle as a Special Case of Galois Connection. *Entropy* **2018**, *20*, 971.
https://doi.org/10.3390/e20120971

**AMA Style**

Kycia RA.
Landauer’s Principle as a Special Case of Galois Connection. *Entropy*. 2018; 20(12):971.
https://doi.org/10.3390/e20120971

**Chicago/Turabian Style**

Kycia, Radosław A.
2018. "Landauer’s Principle as a Special Case of Galois Connection" *Entropy* 20, no. 12: 971.
https://doi.org/10.3390/e20120971