# Thermodynamics of an Empty Box

^{1}

^{2}

^{3}

^{4}

^{*}

## 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

## References

- te Vrugt, M.; Needham, P.; Schmitz, G.J. Is thermodynamics fundamental? arXiv
**2022**, arXiv:2204.04352. [Google Scholar] [CrossRef] - Jaynes, E.T. Probability Theory: The Logic of Science, 1st ed.; Cambridge University Press: Cambridge, UK, 2003. [Google Scholar]
- Doran, C.; Lasenby, A. Geometric Algebra for Physicists, 1st ed.; Cambridge University Press: Cambridge, UK, 2003. [Google Scholar]
- Bekenstein, J. Black holes and the second law. Lett. Nuovo C.
**1972**, 4, 737–740. [Google Scholar] [CrossRef] - Schmitz, G.J. Entropy of Geometric Objects. Entropy
**2018**, 20, 453. [Google Scholar] [CrossRef] [PubMed] - Planck, M. Eight Lectures on Theoretical Physics Delivered at Columbia University in 1909; Library of Alexandria, Columbia University Press: New York, NY, USA, 1915. [Google Scholar]
- Turner, M.S. Why Is the Temperature of the Universe 2.726 Kelvin? Science
**1993**, 262, 861–867. [Google Scholar] [CrossRef] - Valery, A.R.; Dmitry, S.G. Introduction to the Theory of the Early Universe: Hot Big Bang Theory; World Scientific Publishing Company: Singapore, 2011. [Google Scholar]
- Lemaître, G. Un univers homogène de masse constante et de rayon croissant rendant compte de la vitesse radiale des nébuleuses extra-galactiques. Ann. Soc. Sci. Brux.
**1927**, 47, 49–59. [Google Scholar] - Friedman, A. Über die Krümmung des Raumes. Z. Physik.
**1922**, 10, 377–386. [Google Scholar] [CrossRef] - Verlinde, E. On the Origin of Gravity and the Laws of Newton. J. High Energy Phys.
**2011**, 2011, 29. [Google Scholar] [CrossRef] - Kepler, J. Epitome Astronomiae Copernicanae; Johann Planck: Linz, Austria, 1618. [Google Scholar]
- Schwarzschild, K. Über das Gravitationsfeld einer Kugel aus inkompressibler Flüssigkeit nach der Einsteinschen Theorie. Sitzungsber. Preuss. Akad. Wiss. Berlin
**1916**, 189–196. [Google Scholar] - Einstein, A. Über die Spezielle und die Allgemeine Relativitätstheorie, 24th ed.; Springer: Berlin, Germany, 2009. [Google Scholar]
- t’Hooft, G. Dimensional Reduction in Quantum Gravity. arXiv
**2009**, arXiv:gr-qc/9310026. [Google Scholar] [CrossRef] - Bransden, B.; Joachain, C. Quantum Mechanics, 2nd ed.; Prentice Hall: Harlow, UK, 2000. [Google Scholar]
- Planck, M. Ueber das Gesetz der Energieverteilung im Normalspectrum. Ann. Phys.
**1901**, 309, 553–563. [Google Scholar] [CrossRef] - Binasch, G.; Grünberg, P.; Saurenbach, F.; Zinn, W. Enhanced magnetoresistance in layered magnetic structures with antiferromagnetic interlayer exchange. Phys. Rev. B
**1989**, 39, 4828–4830. [Google Scholar] [CrossRef] - von Weizsäcker, C.F. Zur Theorie der Kernmassen. Z. Phys.
**1935**, 96, 431–458. [Google Scholar] [CrossRef] - Bekenstein, J.D. Bekenstein-Hawking entropy. Scholarpedia
**2008**, 3, 7375. [Google Scholar] [CrossRef] - Schmitz, G.J. Thermodynamics of Diffuse Interfaces. In Interface and Transport Dynamics; Emmerich, H., Nestler, B., Schreckenberg, M., Eds.; Springer: Berlin, Germany, 2003; pp. 47–64. [Google Scholar]
- Crispino, L.C.B.; Higuchi, A.; Matsas, G.E.A. The Unruh effect and its applications. Rev. Mod. Phys.
**2008**, 80, 787–838. [Google Scholar] [CrossRef] - Barceló, C.; Liberati, S.; Visser, M. Analogue gravity. Living Rev. Relativ.
**2011**, 14, 3. [Google Scholar] [CrossRef] [PubMed] - te Vrugt, M.; Frohoff-Hülsmann, T.; Heifetz, E.; Thiele, U.; Wittkowski, R. From a microscopic inertial active matter model to the Schrödinger equation. Nat. Commun. 2023; in press. [Google Scholar] [CrossRef]
- Gibbs, J.W. On the Equilibrium of Heterogeneous Substances: First Part. Trans. Conn. Acad. Arts Sci.
**1878**, 3, 108–248. [Google Scholar] - Gibbs, J.W. On the Equilibrium of Heterogeneous Substances: Second Part. Trans. Conn. Acad. Arts Sci.
**1878**, 3, 343–524. [Google Scholar] - Gibbs, J.W. Elementary Principles in Statistical Mechanics; John Wilson and Son: Cambridge, MA, USA, 1902. [Google Scholar]
- Guggenheim, E. Modern Thermodynamics by the Methods of Willard Gibbs. J. Phys. Chem.
**1934**, 38, 713. [Google Scholar] [CrossRef] - Gibbs, J. On the Fundamental Formula of Statistical Mechanics, with Applications to Astronomy and Thermodynamics (Abstract). Proc. Amer. Assoc. Adv. Sci.
**1884**, XXXIII, 57–58. [Google Scholar] - Lukas, H.; Fries, S.G.; Sundman, B. Computational Thermodynamics: The Calphad Method, 1st ed.; Cambridge University Press: Cambridge, UK, 2007. [Google Scholar]
- Andersson, J.O.; Helander, T.; Höglund, L.; Shi, P.; Sundman, B. Thermo-Calc & DICTRA, computational tools for materials science. Calphad
**2002**, 26, 273–312. [Google Scholar] [CrossRef] - Bale, C.; Bélisle, E.; Chartrand, P.; Decterov, S.; Eriksson, G.; Gheribi, A.; Hack, K.; Jung, I.H.; Kang, Y.B.; Melançon, J.; et al. FactSage thermochemical software and databases, 2010–2016. Calphad
**2016**, 54, 35–53. [Google Scholar] [CrossRef] - Chen, S.L.; Daniel, S.; Zhang, F.; Chang, Y.; Yan, X.Y.; Xie, F.Y.; Schmid-Fetzer, R.; Oates, W. The PANDAT software package and its applications. Calphad
**2002**, 26, 175–188. [Google Scholar] [CrossRef] - Saunders, N.; Guo, U.; Li, X.; Miodownik, A.; Schillé, J.P. Using JMatPro to model materials properties and behavior. JOM
**2003**, 55, 60–65. [Google Scholar] [CrossRef] - Wallace, D. Thermodynamics as control theory. Entropy
**2014**, 16, 699–725. [Google Scholar] [CrossRef] [Green Version] - Myrvold, W.C. The Science of ΘΔ
^{cs}. Found. Phys.**2020**, 50, 1219–1251. [Google Scholar] [CrossRef] - Myrvold, W.C. Beyond Chance and Credence: A Theory of Hybrid Probabilities, 1st ed.; Oxford University Press: Oxford, UK, 2021. [Google Scholar]
- Callen, H.B. Thermodynamics and an Introduction to Thermostatistics, 2nd ed.; Wiley: New York, NY, USA, 1991. [Google Scholar]
- Elder, K.; Provatas, N. Phase-Field Methods in Materials Science and Engineering; Wiley-VCH: Weinheim, Germany, 2010. [Google Scholar]
- MICRESS
^{®}—The MICRostructure Evolution Simulation Software. Available online: https://micress.rwth-aachen.de/ (accessed on 12 June 2022). - Schmitz, G.J.; Böttger, B.; Eiken, J.; Apel, M.; Viardin, A.; Carré, A.; Laschet, G. Phase-field based simulation of microstructure evolution in technical alloy grades. Int. J. Adv. Eng. Sci. Appl. Math.
**2010**, 2, 126–139. [Google Scholar] [CrossRef] - Okano, A.; Matsumoto, T.; Kato, T. Gaussian Curvature Entropy for Curved Surface Shape Generation. Entropy
**2020**, 22, 353. [Google Scholar] [CrossRef] - Schmitz, G.J. A phase-field perspective on mereotopology. AppliedMath
**2022**, 2, 54–103. [Google Scholar] [CrossRef] - Gibbs, J.W. A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces. Trans. Conn. Acad. Arts Sci.
**1873**, 2, 309–342. [Google Scholar] - Gibbs, J.W. Graphical Methods in the Thermodynamics of Fluids. Trans. Conn. Acad. Arts Sci.
**1873**, 2, 309–342. [Google Scholar] - Ram, B. Engineering Mathematics, 1st ed.; Pearson: New Dehli, India, 2009. [Google Scholar]
- Noether, E. Invariante Variationsprobleme. Nachr. Ges. Wiss. Gott. Math.-Phys. Kl.
**1918**, 235–257. [Google Scholar] - Hermann, S.; Schmidt, M. Noether’s theorem in statistical mechanics. Commun. Phys.
**2021**, 4, 176. [Google Scholar] [CrossRef] - Mori, H. Transport, Collective motion, and Brownian motion. Prog. Theor. Phys.
**1965**, 33, 423–455. [Google Scholar] [CrossRef] - Zwanzig, R. Ensemble Method in the Theory of Irreversibility. J. Chem. Phys.
**1960**, 33, 1338–1341. [Google Scholar] [CrossRef] - te Vrugt, M.; Wittkowski, R. Mori-Zwanzig projection operator formalism for far-from-equilibrium systems with time-dependent Hamiltonians. Phys. Rev. E
**2019**, 99, 062118. [Google Scholar] [CrossRef] - te Vrugt, M.; Wittkowski, R. Projection operators in statistical mechanics: A pedagogical approach. Eur. J. Phys.
**2020**, 41, 045101. [Google Scholar] [CrossRef] - Camargo, D.; de la Torre, J.A.; Duque-Zumajo, D.; Español, P.; Delgado-Buscalioni, R.; Chejne, F. Nanoscale hydrodynamics near solids. J. Chem. Phys.
**2018**, 148. [Google Scholar] [CrossRef] [Green Version] - Treumann, R.A.; Baumjohann, W. A note on the entropy force in kinetic theory and black holes. Entropy
**2019**, 21, 716. [Google Scholar] [CrossRef] [PubMed] - Shannon, C.E. A mathematical theory of communication. Bell Syst. Tech. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef] - Frigg, R. A field guide to recent work on the foundations of statistical mechanics. In The Ashgate Companion to Contemporary Philosophy of Physics; Rickles, D., Ed.; Ashgate: London, UK, 2008; pp. 99–196. [Google Scholar]
- Bronstein, I.N.; Semendyayev, K.A.; Musiol, G.; Muehlig, H. Handbook of Mathematics, 5th ed.; Springer: Berlin, Germany, 2007. [Google Scholar]
- Hahn, T.; Wigger, D.; Kuhn, T. Entropy Dynamics of Phonon Quantum States Generated by Optical Excitation of a Two-Level System. Entropy
**2020**, 22, 286. [Google Scholar] [CrossRef] - Schmitz, G.J. Quantitative mereology: An essay to align physics laws with a philosophical concept. Phys. Essays
**2020**, 33, 479–488. [Google Scholar] [CrossRef] - Gerla, G.; Mirandam, A. Mathematical Features of Whitehead’s Point-free Geometry. Handb. Whiteheadian Process Thought
**2008**, II, 119–129. [Google Scholar] - Roeper, P. Region-Based Topology. J. Philos. Log.
**1997**, 26, 251–309. [Google Scholar] [CrossRef] - Johnstone, P.T. The point of pointless topologies. Bull. Am. Math. Soc.
**1983**, 8, 41–53. [Google Scholar] - Cullity, B.D.; Stock, S.R. Elements of X-ray Diffraction, 3rd ed.; Pearson Education Limited: Harlow, UK, 2014. [Google Scholar]
- Siqveland, L.M.; Skjaeveland, S.M. Derivations of the Young-Laplace equation. Capillarity
**2021**, 4, 23–30. [Google Scholar] [CrossRef] - Greaves, G.; Greer, A.; Lakes, R.; Rouxel, T. Poisson’s ratio and modern materials. Nat. Mater.
**2011**, 10, 823–837. [Google Scholar] [CrossRef] - Tiesinga, E.; Mohr, P.J.; Newell, D.B.; Taylor, B.N. CODATA Recommended Values of the Fundamental Physical Constants: 2018. J. Phys. Chem. Ref. Data
**2021**, 50, 033105. [Google Scholar] [CrossRef] - Gibson, J.G. Alpha and electroweak coupling. arXiv
**2001**, arXiv:quant-ph/0112033. [Google Scholar] [CrossRef] - Akarsu, Ö.; Kılınç, C.B. Bianchi type III models with anisotropic dark energy. Gen. Relativ. Gravit.
**2010**, 42, 763–775. [Google Scholar] [CrossRef] - te Vrugt, M.; Hossenfelder, S.; Wittkowski, R. Mori-Zwanzig formalism for general relativity: A new approach to the averaging problem. Phys. Rev. Lett.
**2021**, 127, 231101. [Google Scholar] [CrossRef] [PubMed] - Le Delliou, M.; Deliyergiyev, M.; del Popolo, A. An Anisotropic Model for the Universe. Symmetry
**2020**, 12, 1741. [Google Scholar] [CrossRef] - Larena, J.; Alimi, J.M.; Buchert, T.; Kunz, M.; Corasaniti, P.S. Testing backreaction effects with observations. Phys. Rev. D
**2009**, 79, 083011. [Google Scholar] [CrossRef] - Buchert, T.; Carfora, M.; Ellis, G.F.R.; Kolb, E.W.; MacCallum, M.A.H.; Ostrowski, J.J.; Räsänen, S.; Roukema, B.F.; Andersson, L.; Coley, A.A.; et al. Is there proof that backreaction of inhomogeneities is irrelevant in cosmology? Class. Quantum Grav.
**2015**, 32, 215021. [Google Scholar] [CrossRef] - Brown, H.R. Physical Relativity: Space-Time Structure from a Dynamical Perspective; Clarendon Press: Oxford, UK, 2005. [Google Scholar]
- Ghedini, E.; Friis, J.; Goldbeck, G.; Hashibon, A.; Schmitz, G.J.; Moruzzi, S.; Varzi, A.C. The Elementary Multiperspective Material Ontology. 2022; Unpublished Work. [Google Scholar]
- Ferretti, E. The Cell Method: An Enriched Description of Physics Starting from the Algebraic Formulation. Comput. Mater. Contin.
**2013**, 36, 49–71. [Google Scholar] [CrossRef] - Atkins, P.; Friedman, R. Molecular Quantum Mechanics, 5th ed.; Oxford University Press: Oxford, UK, 2011. [Google Scholar]
- Liu, Z.K. Theory of cross phenomena and their coefficients beyond Onsager theorem. Mater. Res. Lett.
**2022**, 10, 393–439. [Google Scholar] [CrossRef]

**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.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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