# Historical and Physical Account on Entropy and Perspectives on the Second Law of Thermodynamics for Astrophysical and Cosmological Systems

## Abstract

**:**

**PACS Codes:**01.70.+w; 05.20.−y; 05.70.−a; 95.30.Tg

## 1. Introduction

## 2. Key Aspects of the Historical Development of the Concepts of Entropy and the Second Law

_{1}…p

_{n}and the corresponding velocities…”. If the system is a gas or a solid, it is not stated. The energy of the system is given by E = L + V

_{a}+ V

_{i}, where L is the kinetic energy, V

_{i}is the internal potential and V

_{a}is a potential representing external influences on the system. As Einstein poses [13] (p. 418):

“Two kinds of external forces shall act upon the masses of the system. One kind of force shall be derivable from potential V_{a}and shall represent external conditions (gravity, effect of rigid walls without thermal effects, etc.); their potential may contain time explicitly, but its derivative with respect to time should be very small. The other forces shall not be derivable from a potential and shall vary rapidly. They have to be conceived as the forces that produce the influx of heat. If such forces do not act, but V_{a}depends explicitly on time, then we are dealing with an adiabatic process.”

_{a}, at the end of the article, Einstein adds “No assumptions had to be made about of the forces that correspond to potential V

_{a}, not even that such forces occur in nature” [13] (p. 433).

_{B}is the Boltzmann constant and T is the temperature of the gas. Note how, in the line of reasoning about ϕ, the potential U is disregarded and the parameter V is introduced with the plain argument that the allowed room for each particle of the gas is the volume of the container. At this point, the author of the book inserts a footnote stating that this could be done by making a potential energy of a particle go to infinity once it tries to penetrate the wall of the container [14] (p. 64). Thus, containers are often poorly defined in statistical mechanics and this is crucial for the scientific realism involved in this subject and in this manuscript. We have to consider that the concept of wall is critical to the definition of pressure, a fundamental thermodynamic (and highly phenomenological) parameter [15] (Vol. I, p.1–3). The physical parameter pressure is defined as the ratio of force to the area over which that force is distributed. In the corpuscular constructive model, the force is attributed to the collisions of the gas particles on the walls. That definition implies the existence of a surface. Historically this surface is often considered to be the wall of the gas container [9] (p. 12).

## 3. Entropy and the Second Law in Astrophysics and Cosmology

_{bh}, he states similarly “But we emphasize that one should not regard T

_{bh}as the temperature of the black hole; such an identification can easily lead to all sorts of paradoxes, and is thus not useful.” Despite the shortcomings, Bekenstein concludes that “In fact, one can say that the black-hole state is the maximum entropy state of a given amount of matter”.

_{hc}) involves the relation between heat (Q) and temperature (T) of a body:

“… formulas are meaningless unless they bear on non-mathematical experiences. In other words, we can use formulas only after we have made sense of the world to the point of asking questions about it and have established the bearing of the formulas on the experience that they are to explain. Mathematical reasoning about experience must include, beside the antecedent non-mathematical finding and shaping of experience, the equally non-mathematical relating of mathematics to such experience and the eventual, also non-mathematical, understanding of experience elucidated by mathematical theory.”

- SCENARIO 1: Consider a system consisted of an artificial satellite orbiting the Earth. Equation (13) holds valid for this case. Imagine that a small perturbation slightly deflects the satellite’s trajectory, starting a falling process. While the satellite is falling, its potential energy is decreasing and the satellite is gaining kinetic energy.

_{bh}in his 1973 manuscript. Clausius proposed Equation (1) to define entropy, in an attempt to understand the workings of steam engines. Later on, Bekenstein and Hawking took the reverse route, departing from a concept of entropy derived from information theory and adapted to the realm of black holes, and then used Equation (17) to assign a temperature to the black hole. It is noteworthy that after Hawking published his manuscript about what is now called “Hawking radiation”, the expression for T

_{bh}has been considered (even by Bekenstein) the actual temperature of the black hole, despite Bekenstein’s disclaimer in his prior to 1975 manuscripts.

## 4. Perspectives on the Second Law of Thermodynamics

- The low entropy of our Universe: As peculiar as it may seem, a negative entropy variation for an isolated system by the action of gravity can be considered as a reasonable explanation for our fairly organized Universe.
- Energy sources: In thermodynamics, increasing entropy is commonly linked to energy degradation (the energy can no longer be converted into work). On the other hand, we can think about useful energy being generated in a decreasing entropy process. According to our model, the solar system is the result of a gravitational gas cloud collapse, i.e., a negative entropy variation process. Note that practically all sources of energy we have come essentially from the Sun: solar, fossil fuels, biofuels, biomass, hydroelectric, wind, and wave. Regarding nuclear energy, we have to remember that 99% of the so called baryonic matter of our Universe is hydrogen and helium. All heavier elements were formed inside stars or other gravity dependent cosmic phenomena. Note that tidal and geothermal energy are also connected to gravity. It is worth noticing that this entropic view on energy sources may open new routes for scientific endeavors in progress, such as batteries and hydrogen fusion.
- The reverse elsewhere argument: Consider the following process: a system consisted of two bodies with dissimilar temperatures are put in contact in order to reach thermal equilibrium. It can be proved that, after thermal equilibrium, the entropy of the system is higher than the previous configuration when the two bodies had different temperatures. This specific analysis is correct. It is also accepted that, the higher the temperature of the system, the higher its entropy. This leads to a particularly delicate discussion regarding things getting colder spontaneously in nature (like winter). In fact, these processes are usually despised along with an “elsewhere argument”. This meets with the day-by-day notion that, if cold and coherent structures emerge, like a refrigerator’s interior or a city, then consequently heat and disorder have been generated in the surroundings (such as power plants for instance). In the example of thermal equilibrium between two objects with distinct temperatures, we should verify how one object became warmer than the other in the first place. If we track down the energy source, we should find a negative entropy variation process (see item ii) from where useful energy has been created. Let us reverse the usual argument: we can say that a local increase in entropy is possible when it is coupled with a decrease in entropy elsewhere. That would be the reverse elsewhere argument. Therefore, practically all of our daily experience with automobiles, cities, flora, fauna, rivers, winds, hot sources, cold sinks, “room temperature” i.e., our familiar high entropy environment is possible because there was once a huge gas cloud that collapsed. We have also to consider, in this argument, a negative entropy variation related to nuclear processes.
- Our special location: Thermodynamically speaking, one of the special features of our planet is something that most people take for granted and is known as “room temperature”. In the scope of statistical mechanics, this explains why the canonic ensemble (that requires a heat bath or heat reservoir) has been more commonplace than the microcanonical ensemble. This fact reinforces our argument that models that have been proven successful to our day-by-day phenomena are likely to be fruitless when applied to astrophysical systems. The second law is very useful to understand our quotidian processes, however once we leave our particular environment we should look for a more comprehensive model. Our “room temperature” means we live under an energy balance, in the sense that, in first approximation, the same amount of energy that we receive from the Sun are irradiated back to space. In this sense, the unbalance between the Sun and our planet is entropic, as we receive from the Sun a low entropy form of energy (mainly visible light) irradiating back a high entropy form of energy. For every 1 high energy photon we receive, the Earth radiates about 20 low energy photons. This entropic dissymmetry reinforces item ii.
- Understanding life: This issue is connected to item iv. When scientists study life under a cosmological perspective, such as in the field of astrobiology, it is always mentioned that life can occur only under very special conditions. The second law and our rather high entropy environment appear to be connected to the necessary conditions for life. It is interesting to note that Schrödinger introduced the concept of “negative entropy” while developing his model for life [46]. It is connected to the amount of entropy that a living organism has to export to the environment in order to keep its own entropy low. The term has been shortened to “negentropy” and has close relation with free enthalpy applied to molecular biology.
- Entropy equilibrium and the arrow of time: In our view, the rather low entropy of our Universe is due to a balance between processes that decrease entropy (such as gravitational gas cloud collapses) and processes that increase entropy (such as thermal processes). In this scope, the statistical mechanical quandary regarding the connection between process reversibility and the arrow of time loses much of its weight.

## 5. Statistical Mechanical Evaluation of a Gravitational Gas Cloud Collapse

_{ij}is the distance between the i

_{th}and j

_{th}particles. Given this general formalization, several different attractive and repulsive potentials can be considered. For simplicity, we will consider a monoatomic gas. Assuming no internal energy, the total energy of the gas can be expressed by:

_{ρ}from the origin. This particular case represents a state in which the total kinetic energy of the system is attributed to a single component of a single particle’s momentum. Thus, we can think of the radius p

_{ρ}as a characteristic momentum of the system.

_{ρ}was defined as a characteristic moment of the system, a characteristic position q

_{ρ}can also be defined for the system. Assuming that the total potential energy of the system can be attributed to a position q

_{ρ}of a single component of a single particle related to the potential of Equation (22), the expression for q

_{ρ}is:

_{ρ}:

_{0}, and n = −1. Hence:

## 6. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

- Renn, J. Einstein’s Controversy with Drude and the Origin of Statistical Mechanics: A New Glimpse from the “Love Letters”. Arch. Hist. Exact Sci
**1997**, 51, 315–354. [Google Scholar] - Bourget, D.; Chalmers, D.J. What Do Philosophers Believe? Available online: http://philpapers.org/archive/BOUWDP.pdf (accessed on 30 July 2014).
- Norden, R.V. Persistence pays off for Crystal Chemist. Nature
**2011**, 478, 165–166. [Google Scholar] - Carroll, S.M. Is our Universe natural? Nature
**2006**, 440, 1132–1136. [Google Scholar] - Benguigui, L. The different paths to entropy. Eur. J. Phys
**2013**, 34, 303–321. [Google Scholar] - Atkins, P.W. The Second Law; Scientific American Books: New York, NY, USA, 1984. [Google Scholar]
- Uffink, J. Bluff Your Way in the Second Law of Thermodynamics. Stud. Hist. Phil. Mod. Phys
**2001**, 32, 305–394. [Google Scholar] - Carnot, N.L.S. Reflections on the Motive Power of Heat, Accompanied by Kelvin W. T., “An Account of Carnot’s Theory”, 2nd ed.; Thurston, R.H., Ed.; John Willey & Sons: New York, NY, USA, 1897. [Google Scholar]
- Atkins, P.W. The Four Laws that Drive the Universe; Oxford University Press: New York, NY, USA, 2007. [Google Scholar]
- Clausius, R.J.E. The Mechanical Theory of Heat—With its Applications to the Steam-Engine and the Physical Properties of Bodies; Archer Hirst, T., Ed.; Taylor and Francis: London, UK, 1870. [Google Scholar]
- Bordoni, S. Routes towards an Abstract Thermodynamics in the Late Nineteenth Century. Eur. Phys. J. H
**2013**, 38, 617–660. [Google Scholar] - Mehra, J. Einstein and the Foundation of Statistical Mechanics. Physica
**1975**, 79A, 447–477. [Google Scholar] - Einstein, A. Kinetic theory of Thermal Equilibrium and of the Second Law of Thermodynamics. Ann. Phys
**1902**, 9, 417–433. [Google Scholar] - Reif, F. Fundamentals of Statistical and Thermal Physics; McGraw-Hill Book Company: Boston, UK, 1965. [Google Scholar]
- Feynman, R.; Leighton, R.; Sands, M. The Feynman Lectures on Physics; Pearson Addison Wesley: San Francisco, CA, USA, 2006. [Google Scholar]
- Einstein, A. A theory of the Foundations of Thermodynamics. Ann. Phys
**1903**, 11, 170–187. [Google Scholar] - Winsberg, E. Bumps on the Road to Here (from Eternity). Entropy
**2012**, 14, 390–406. [Google Scholar] - Bekenstein, J.D. Black Holes and Entropy. Phys. Rev. D
**1973**, 7, 2333–2346. [Google Scholar] - Bekenstein, J.D. Generalized second law of thermodynamics in black-hole physics. Phys. Rev. D
**1974**, 9, 3292–3300. [Google Scholar] - Hawking, S.W. Black holes and thermodynamics. Phys. Rev. D
**1976**, 13, 191–197. [Google Scholar] - Hawking, S.W. Black hole explosions? Nature
**1974**, 248, 30–31. [Google Scholar] - Ashcroft, N.W.; Mermin, N.D. Física do Estado Sólido; Cengage Learning: São Paulo, Brasil, 2011. (In Portuguese) [Google Scholar]
- Lynden-Bell, D. Negative Specific Heat in Astronomy, Physics and Chemistry. Physica A
**1999**, 263, 293–304. [Google Scholar] - Silva, R.; Alcaniz, J.S. Non-extensive statistics and the stellar polytrope index. Physica A
**2004**, 341, 208–214. [Google Scholar] - Vega, H.J.; Sánchez, N. Statistical Mechanics of the Self-Gravitating Gas: I. Thermodynamic Limit and Phase Diagrams. Nucl. Phys. B
**2002**, (02), 00025–1. [Google Scholar] [CrossRef] - Ruppeiner, G. Equations of State of Large Gravitating Gas Clouds. Astrophys. J
**1996**, 464, 547–555. [Google Scholar] - Chavanis, P-H. Phase transitions in self-gravitating systems: Self-gravitating fermions and hard-sphere models. Phys. Rev. E
**2002**, 65, 1–23. [Google Scholar] - Hooft, G,‘t. Dimensional Reduction in Quantum Gravity.
**2009**. arXiv:gr-qc/9310026v2. [Google Scholar] - Bousso, R. The Holographic Principle. Rev. Mod. Phys
**2002**, 74, 825. - Padmanabhan, T. Gravity and the Thermodynamics of Horizons. Phys. Rept
**2005**, 406, 49–125. [Google Scholar] - Padmanabhan, T. Thermodynamical Aspects of Gravity: New insights. Rep. Prog. Phys
**2010**, 73, 046901. [Google Scholar] - Jamil, M.; Saridakis, E.N.; Setare, M.R. The generalized second law of thermodynamics in Horava-Lifshitz cosmology. JCAP
**2010**, 11, 032. [Google Scholar] - Verlinde, E. On the Origin of Gravity and the Laws of Newton. JHEP
**2011**, 4, 029. [Google Scholar] - Cai, R.-G.; Cao, L.-M.; Ohta, N. Friedmann Equations from Entropic Force. Phys. Rev. D
**2010**. [Google Scholar] [CrossRef] - Eling, C.; Guedens, R.; Jacobson, T. Non-equilibrium Thermodynamics of Spacetime. Phys. Rev. Lett
**2006**, 96, 121301. [Google Scholar] - Akbar, M.; Cai, R.-G. Thermodynamic Behavior of Friedmann Equation at Apparent Horizon of FRW Universe. Phys. Rev. D
**2007**, 75, 084003. [Google Scholar] - Jamil, M.; Saridakis, E.N.; Setare, M.R. Thermodynamics of Dark Energy Interacting with Dark Matter and Radiation. Phys. Rev. D
**2010**, 81, 023007. [Google Scholar] - Nojiri, S.; Odintsov, S.D. Unified cosmic history in modified gravity: From F(R) theory to Lorentz non-invariant models. Phys. Rep
**2011**, 505, 59–144. [Google Scholar] - Bamba, K.; Capozziello, S.; Nojiri, S.; Odintsov, S.D. Dark energy cosmology: The equivalent description via different theoretical models and cosmography tests. Astrophys. Space Sci
**2012**, 342, 155–228. [Google Scholar] - Schrödinger, E.R.J.A. Statistical Thermodynamics; Dover: New York, NY, USA, 1989. [Google Scholar]
- Bozorth, R.M. Ferromagnetism; IEEE Press: New York, NY, USA, 1993. [Google Scholar]
- Polanyi, M. Tacit Knowing: Its Bearing on Some Problems of Philosophy. Rev. Mod. Phys
**1962**, 34, 601–616. [Google Scholar] - Baez, J. Available online: http://math.ucr.edu/home/baez/entropy.html (accessed on 14 July 2014).
- Shapiro, H.N.; Moran, M.J. Fundamentals of Engineering Thermodynamics, 7 ed.; John Wiley & Sons: New York, NY, USA, 2011; chapter 5.1—Introducing the second law—System C. [Google Scholar]
- Sewell, G.L. Entropy and Entropy Generation—Fundamentals and Applications; Shiner, J.S., Ed.; Kluwer Academic Publishers: New York, NY, USA, 1996; Volume Chapter 2. [Google Scholar]
- Schrödinger, E. O que é vida: O Aspecto Físico da Célula Vida. Mente e Matéria. Fragmentos Autobiográficos; UNESP/Cambridge: São Paulo, Brazil, 1997. (In Portuguese) [Google Scholar]

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Schoenmaker, J.
Historical and Physical Account on Entropy and Perspectives on the Second Law of Thermodynamics for Astrophysical and Cosmological Systems. *Entropy* **2014**, *16*, 4420-4442.
https://doi.org/10.3390/e16084420

**AMA Style**

Schoenmaker J.
Historical and Physical Account on Entropy and Perspectives on the Second Law of Thermodynamics for Astrophysical and Cosmological Systems. *Entropy*. 2014; 16(8):4420-4442.
https://doi.org/10.3390/e16084420

**Chicago/Turabian Style**

Schoenmaker, Jeroen.
2014. "Historical and Physical Account on Entropy and Perspectives on the Second Law of Thermodynamics for Astrophysical and Cosmological Systems" *Entropy* 16, no. 8: 4420-4442.
https://doi.org/10.3390/e16084420