# Thermodynamics of an Empty Box

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## Abstract

**:**

## 1. Introduction

#### 1.1. Scope

#### 1.2. Outline

## 2. Thermodynamics

#### 2.1. Mathematical Thermodynamics

#### Thermodynamic Potentials

#### 2.2. Scalar Potentials, Gradients, and Forces

#### 2.3. The Internal Energy of an Empty Box

#### Entropy—More than Statistics

## 3. Chopping the Box—Quantization of Space

- bulk voxels,
- face voxels,
- edge voxels,
- vertex voxels.

## 4. Applications of the Quantized Box Model

#### 4.1. Thermodynamics of Geometric Objects

#### 4.2. Dimensionless Entities

#### 4.3. Squeezing the Box

#### Special Relativity

#### 4.4. Evolution of the Box

#### 4.5. Translating the Box

#### 4.5.1. Newton’s Laws

- So far, we cannot say anything about how the quantity $\overrightarrow{p}$, defined by Equation (115), is related to the velocity $\overrightarrow{v}$. Of course, Equation (117) is only equivalent to the statement usually thought of as Newton’s first law if $\overrightarrow{p}$ is proportional to the velocity.
- Energy conservation in a closed system should also hold for the more general case $\overrightarrow{F}\ne \overrightarrow{0}$ (whereas Equation (114) suggests a direct dependence of $\dot{E}$ on $\overrightarrow{F}$, which motivated the first implication in Equation (117)). A position-dependent potential does not lead to a violation of energy conservation.

#### 4.5.2. The Unruh Effect

#### 4.5.3. Position-Dependent Volume

#### 4.5.4. Uncertainty Relation

## 5. Beyond the Empty Box

- all three laws of Newton (classical mechanics),
- the harmonic oscillator equation,
- the Unruh equation,
- an uncertainty relation.

#### 5.1. Oriented Surfaces

#### 5.2. Shearing and Twisting the Box

#### 5.3. Filling the Box

## 6. Summary and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Legendre Transformations

## Appendix B. Relation between Dual-State Entropy and Boltzmann Entropy

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**Figure 1.**Visualization of the different voxel types: face voxels of the different faces (blue, green, red), edge voxels (light gray, yellow), and vertex voxels (orange).

**Figure 2.**Visualization of a transition between a cube ($cos\left(\theta \right)=1$) and a plane ($cos\left(\theta \right)=0)$. The volume is kept the same in all cases.

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**MDPI and ACS Style**

Schmitz, G.J.; te Vrugt, M.; Haug-Warberg, T.; Ellingsen, L.; Needham, P.; Wittkowski, R.
Thermodynamics of an Empty Box. *Entropy* **2023**, *25*, 315.
https://doi.org/10.3390/e25020315

**AMA Style**

Schmitz GJ, te Vrugt M, Haug-Warberg T, Ellingsen L, Needham P, Wittkowski R.
Thermodynamics of an Empty Box. *Entropy*. 2023; 25(2):315.
https://doi.org/10.3390/e25020315

**Chicago/Turabian Style**

Schmitz, Georg J., Michael te Vrugt, Tore Haug-Warberg, Lodin Ellingsen, Paul Needham, and Raphael Wittkowski.
2023. "Thermodynamics of an Empty Box" *Entropy* 25, no. 2: 315.
https://doi.org/10.3390/e25020315