# A Combined Entropy/Phase-Field Approach to Gravity

## Abstract

**:**

## 1. Introduction

## 2. Entropy

#### 2.1. Scalar Entropy

#### 2.2. Entropy Formulations Comprising Gradient Terms

^{*)}in this equation and also in Figure 1c denotes a difference and not the Laplacian operator. Generalization to three dimensions then yields $\overrightarrow{\mathit{\Nabla}\Phi}\text{}*\overrightarrow{a},$ where $\overrightarrow{a}$ represents a vector characterizing the metric imposed by the underlying crystal structure of the solid phase. The inclusion of such gradient type contributions due to diffuse interfaces—revealing a characteristic length scale defined by the vector $\overrightarrow{a}$—in the classical free energy/entropy formulation of thermodynamics is extremely interesting for the description of evolving structures, and introduces a length scale into thermodynamics.

## 3. Phase-Field Models

## 4. Lagrange Formalism

## 5. Derivation of the Gravitational Law

^{2}function of the angle between the vectors $\overrightarrow{\mathrm{n}}\text{}\mathrm{and}\text{}\overrightarrow{\mathsf{\nabla}}{\Phi}_{i}$. When applying the Lagrange formalism, this cos

^{2}function will eventually lead to a non-linear generalization of the Newton–Poisson equation as, for instance, that used in the modified Newtonian dynamic (MOND) approaches [19,20,21,22]. Further aspects of this generalization are detailed and discussed in Section 6. In the following, the function is first considered as a constant making the product $\overrightarrow{\mathrm{n}}{\mathrm{cos}}^{2}(\overrightarrow{\mathrm{n}},\overrightarrow{\mathit{\Nabla}}{\Phi}_{i})=1$. The overall entropy term only comprises terms related to $\overrightarrow{\mathit{\Nabla}}{\Phi}_{i}(\overrightarrow{r},t)$ and no terms related to ${\Phi}_{i}(\overrightarrow{r},t)\text{}or\text{}{\dot{\Phi}}_{i}(\overrightarrow{r},t)$. Thus, only the gradient related terms of the functional derivative become active in the Lagrange formalism:

- the Poisson equation of gravity (Newton’s law);
- a term related to curvature of space (which can probably be related to Einstein’s general theory of relativity);
- a term introducing the mass density of vacuum (which seems related to the cosmological constant); and

## 6. Modified Newtonian Dynamics

^{2}(φ) ~ 0, the classical Newton-Poisson equation is recovered, while for angles of φ approaching $\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$, the MOND terms generate additional contributions. In both cases, the curvature related term persists. For comparison, the MOND Eulerian [20] reads:

_{0}is a new fundamental constant (a

_{0}~10

^{−8}cm s

^{−2}) which marks the transition between the Newtonian and deep-MOND regimes. Agreement with Newtonian mechanics requires μ(x) → 1 for x >> 1, and consistency with astronomical observations requires μ(x) → x for x << 1. Beyond these limits, the interpolating function is not specified by the theory, although it is possible to weakly constrain it empirically……” [20]. Examples for the MOND interpolation function are the “standard” [22] and the “simple” [24] interpolation functions:

_{0}reads:

^{2}). The tangent of the angle $\varphi ,$ between the radial direction and the diagonal of the parallelepiped, can then be approximated as a function of R by:

## 7. Summary and Future Perspectives

- the Poisson equation of gravity (Newton’s law);
- a term related to the curvature of space (which probably can be related to Einstein’s general theory of relativity);
- a term introducing the mass density of vacuum (which seems related to the cosmological constant);
- terms related to a nonlinear generalization of the Newton–Poisson equation as used in modified Newtonian dynamic (MOND) approaches.

^{2}) introduced by the scalar products, (iii) a careful comparison with other theories, and (iv) a comparison with experimental observations.

## Acknowledgments

^{®}group at ACCESS e.V. for numerous discussions and constructive criticism. Special thanks are due to Bernd Böttger, Gottfried Laschet, and Markus Apel for commenting on the final manuscript and to Ulrike Hecht for drawing my attention to the MOND at just the right time.

## Conflicts of Interest

## References

- Verlinde, E.P. On the Origin of Gravity and the Laws of Newton. J. High Energy Phys.
**2011**, 29. [Google Scholar] [CrossRef] - Verlinde, E.P. Emergent Gravity and the Dark Universe. arXiv
**2016**. [Google Scholar] - Schmitz, G.J. Thermodynamics of diffuse interfaces. In Interface and Transport Dynamics; Emmerich, H., Nestler, B., Schreckenberg, M., Eds.; Springer Lecture Notes in Computational Science and Engineering; Springer: Berlin/Heidelberg, Germany, 2003; pp. 47–64. [Google Scholar]
- Woodruff, D. The Solid Liquid Interface; Cambridge University Press: Cambridge, UK, 1973. [Google Scholar]
- Flory, P.J. Principles of Polymer Chemistry; Cornell University Press: New York, NY, USA, 1953. [Google Scholar]
- Jackson, K.A. Liquid Metals and Solidification; ASM: Cleveland, OH, USA, 1958. [Google Scholar]
- Jaynes, E.T. Probability Theory: The Logic of Science, Chapter 11. Available online: http://omega.albany.edu:8008/JaynesBook.html) (accessed on 30 March 2017).
- Schmitz, G.J.; Böttger, B.; Apel, M.; Eiken, J.; Laschet, G.; Altenfeld, R.; Berger, R.; Boussinot, G.; Viardin, A. Towards a metadata scheme for the description of materials—The description of microstructures. Sci. Technol. Adv. Mater.
**2016**, 17, 410–430. [Google Scholar] [CrossRef] [PubMed] - Temkin, D.E. Crystallization Processes; Sirota, N.N., Gorskii, F.K., Varikash, V.M., Eds.; English Translation; Consultants Bureau: New York, NY, USA, 1966. [Google Scholar]
- Langer, J. Manuscript on a concept of phase field models. 1978; unpublished work. [Google Scholar]
- Fix, G. Phase field models for free boundary problems. In Free Boundary Problems Vol. II; Fasano, A., Primicerio, M., Eds.; Piman: Boston, MA, USA, 1983. [Google Scholar]
- Caginalp, G.; Fife, P.C. Phase-field methods for interfacial boundaries. Phys. Rev. B
**1986**, 33, 7792–7794. [Google Scholar] [CrossRef] - Kobayashi, R. Modeling and numerical simulations of dendritic crystal growth. Physica D
**1993**, 63, 410–423. [Google Scholar] [CrossRef] - Steinbach, I.; Pezzolla, F.; Nestler, B.; Seeßelberg, M.; Prieler, R.; Schmitz, G.J.; Rezende, J.L.L. A phase field concept for multiphase systems. Physica D
**1996**, 94, 135–147. [Google Scholar] [CrossRef] - Elder, K.R.; Katakowski, M.; Haataja, M.; Grant, M. Modeling Elasticity in Crystal Growth. Phys. Rev. Lett.
**2002**, 88, 245701. [Google Scholar] [CrossRef] [PubMed] - 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, 126–129. [Google Scholar] [CrossRef] - Provatas, N.; Elder, K. Phase-Field Methods in Materials Science and Engineering; Wiley VCH: Weinheim, Germany, 2010. [Google Scholar]
- Steinbach, I. Phase-field models in Materials Science–Topical Review: Modelling Simul. Mater. Sci. Eng.
**2009**, 17, 073001. [Google Scholar] - Bekenstein, J.; Milgrom, M. Does the missing mass problem signal the breakdown of Newtonian gravity? Astrophys. J.
**1984**, 286, 7–14. [Google Scholar] [CrossRef] - Modified Newtonian Dynamics. Available online: https://en.wikipedia.org/wiki/Modified_Newtonian_dynamics (accessed on 30 March 2017).
- Milgrom, M. The MOND. Available online: https://arxiv.org/pdf/1404.7661.pdf (accessed on 30 March 2017).
- Milgrom, M. A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis. Astrophys. J.
**1983**, 270, 365–370. [Google Scholar] [CrossRef] - Die kosmologische Konstante. Available online: https://de.wikipedia.org/wiki/Kosmologische_Konstante (accessed on 30 March 2017).
- Famaey, B.; Binney, J. Modified Newtonian Dynamics in the Milky Way. Mon. Not. R. Astron. Soc.
**2005**, 363, 603–608. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Kossel’s model of a growing crystal (figure adapted from [4]). Atoms attach to the interface in layers. The model assumes that the atoms may only adhere to the top of already solid atoms (‘solid on solid’). A smooth transition with a finite interface thickness can be identified when averaging the fraction of solid atoms parallel to the interface (see Figure 1d). (

**b**) The Jackson model [6] assumes an interface which is restricted to a single layer separating the two bulk regions. The entropic term in the Jackson model reads: ΦlnΦ + (1−Φ)ln(1−Φ). (

**c**) The Temkin model [9] assumes multiple steps separating the two bulk regions. The entropy of an intermediate layer n in this case is also defined by its adjacent layer n − 1. The entropic term in the Temkin model reduces to the Jackson model in the case of a single interface layer and reads: ${\displaystyle \sum}_{\mathrm{n}\text{}=\text{}-\infty}^{\infty}}({\Phi}_{\mathrm{n}-1}-{\Phi}_{\mathrm{n}})\mathrm{ln}({\Phi}_{\mathrm{n}-1}-{\Phi}_{\mathrm{n}})={\displaystyle {\displaystyle \sum}_{\mathrm{n}\text{}=\text{}-\infty}^{\infty}}({\mathsf{\Delta}}^{*}{\Phi}_{\mathrm{n}})\mathrm{ln}({\mathsf{\Delta}}^{*}{\Phi}_{\mathrm{n}})$ (* see footnote of Equation (10)). (

**d**) The Schmitz model [3] is an extension of the Temkin model for small step widths and approximates the discrete formulation of Temkin by a continuous gradient and by turning the sum of the individual contributions of the discrete layers into an integral (see text): $\iiint}\overrightarrow{\mathrm{a}}\overrightarrow{\mathsf{\nabla}}\Phi \mathrm{ln}(\overrightarrow{\mathrm{a}}\overrightarrow{\mathsf{\nabla}}\Phi )$.

**Figure 2.**Scalar triple product: ${\overrightarrow{e}}_{z}({\overrightarrow{e}}_{x}\times {\overrightarrow{e}}_{y})=\mathrm{v}$ The volume being spanned by these vectors is v. The diagonal vector of this coordinate system reads ${\overrightarrow{e}}_{x}+{\overrightarrow{e}}_{y}+{\overrightarrow{e}}_{z}=\overrightarrow{\mathrm{n}}$ with its norm being ${({\overrightarrow{e}}_{x}^{2}+{\overrightarrow{e}}_{y}^{2}+{\overrightarrow{e}}_{z}^{2})}^{0.5}=|\overrightarrow{\mathrm{n}}|$ in the case of an orthonormal system. This vector is also the normal vector to the tangential plane to that volume.

**Figure 3.**Basic setting for the description of a complex shaped object by an order parameter. The order parameter field $\mathsf{\Phi}\text{}(\overrightarrow{\mathrm{r}},\mathrm{t})$ takes the value 1 wherever and whenever the object is present. A diffuse continuous interface marks the transition from the object to the “non-object”, as shown here for a solid object in a liquid.

**Figure 4.**Some important mathematical relations related to the phase-field description of a massive sphere.

**Figure 5.**The proposed explanation for a change in the direction of the normal vector of space leading to an angle φ which increases with increasing radius R (see text).

**Table 1.**Basic scheme to derive equations of motion using a Lagrange formalism acting on different entropy formulations being constrained by different conservation laws.

Lagrange Derivative | Entropy Term | Conserved Quantity | Lagrange Multipliers | |
---|---|---|---|---|

established procedures | $\frac{\partial}{\partial {\Phi}_{i}}$ | $s({\Phi}_{i})$ | energy | scalar (λ) |

this work | $\frac{\partial}{\partial {\Phi}_{i}}$ | $s({\Phi}_{i},\overrightarrow{\mathit{\Nabla}}{\Phi}_{i})$ | mass | scalar (g) |

$\overrightarrow{\mathit{\Nabla}}\frac{\partial}{\partial \overrightarrow{\mathit{\Nabla}}{\Phi}_{i}}$ | $s({\Phi}_{i},\overrightarrow{\mathit{\Nabla}}{\Phi}_{i})$ | |||

Future topics | $\frac{\partial}{\partial t}\frac{\partial}{\partial {\dot{\Phi}}_{i}}$ | $s({\Phi}_{i},\overrightarrow{\mathit{\Nabla}}{\Phi}_{i},{\dot{\Phi}}_{i})$ | mass, energy, charge, momentum, spin | scalars: (λ,g,ε,…) vectors: $\overrightarrow{c}$ and $\overrightarrow{h}$ |

tensor type formulations | …. | energy-momentum tensor, charge, spin | … to be continued.… |

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Schmitz, G.J.
A Combined Entropy/Phase-Field Approach to Gravity. *Entropy* **2017**, *19*, 151.
https://doi.org/10.3390/e19040151

**AMA Style**

Schmitz GJ.
A Combined Entropy/Phase-Field Approach to Gravity. *Entropy*. 2017; 19(4):151.
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**Chicago/Turabian Style**

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2017. "A Combined Entropy/Phase-Field Approach to Gravity" *Entropy* 19, no. 4: 151.
https://doi.org/10.3390/e19040151