The purpose of this research is to characterize shapes in thermodynamic terms, namely, in terms of total energy, dissipative energy, entropy, and temperature. As case studies, polygons and some well-known curves were taken, and they were characterized using physical terms. The relation between entropy and curvature was elucidated, and the black hole surface gravity and temperature were criticized and reinterpreted from this point of view. Particular energy attributions were evaluated by comparing the position of any edge of a polygon (i.e. its angle with the horizontal axis) with a broken crystal surface. Energies of all edges were added up at all positions between 0˚？- 360˚. In regular polygons, the total energy decreases with the increase of the number of edges. Entropy increases in the reverse order, and the increase of the number of edges increases entropy. It implies that the circle has the lowest energy but the highest surface entropy. In curves (circle, sine-curve, spiral, and exponential curve), the total energy, dissipative energy, and entropy all depend on amplitude and also on specific variables. Black hole entropy expressed in terms of the surface area is a configurational entropy and not thermal entropy; therefore, it does not involve a varying temperature term. The surface gravity of a black hole is connected to acceleration and thus to curvature. To relate it with the temperature needs to be reinterpreted, because, surface gravity behaves like an attractive force not exactly like temperature. Hawking radiation is still possible, but the black hole does not get warmer as it evaporates. Material loss from the black hole gets faster as its radius decreases due to the curvature effect, i.e. by a mechanism similar to the Gibbs-Thomson effect.
References
[1]
Weinhold, F. (1976) Thermodynamics and Geometry. Physics Today, 29, 23-30 https://physicstoday.scitation.org/doi/10.1063/1.3023366 https://doi.org/10.1063/1.3023366
[2]
Ruppeiner, G. (1979) Thermodynamics: A Riemannian Geometric Model. Physical Review A, 20, 1608-1613. https://doi.org/10.1103/PhysRevA.20.1608 https://journals.aps.org/pra/abstract/10.1103/PhysRevA.20.1608
[3]
Gilmore, R. (1984) Length and Curvature in the Geometry of Thermodynamics. Physical Review A, 30, 1994-1997. https://doi.org/10.1103/PhysRevA.30.1994 https://journals.aps.org/pra/abstract/10.1103/PhysRevA.30.1994
[4]
Ruppeiner, G. (2013) Unitary Thermodynamics from Thermodynamic Geometry. https://arxiv.org/pdf/1310.2566.pdf
[5]
Andresen, B. (2015) Metrics and Energy Landscapes in Irreversible Thermodynamics. Entropy, 17, 6304-6317. https://doi.org/10.3390/e17096304 https://www.mdpi.com/1099-4300/17/9/6304
[6]
Lee, J., McGough, L. and Safdi, B.R. (2014) Rényi Entropy and Geometry. https://arxiv.org/pdf/1407.8171.pdf
[7]
Mandelbrot, B. (1982) The Fractal Geometry of Nature. WH Freeman & Co, San Francisco. https://onlinelibrary.wiley.com/doi/abs/10.1002/esp.3290080415
[8]
Anders, G., Klotsa, D., Ahmed, N.K., Engel, M. and Glotzer, S. (2014) Understanding Shape Entropy through Local Dense Packing. Proceedings of the National Academy of Sciences of the United States of America, 111, E4812-E4821. https://doi.org/10.1073/pnas.1418159111 https://www.pnas.org/content/111/45/E4812
Parker, M.C. and Jeynes, C. (2020) Fullerene Stability by Geometrical Thermodynamics. ChemistrySelect, 5, 5-14. https://doi.org/10.1002/slct.201903633 https://chemistry-europe.onlinelibrary.wiley.com/doi/10.1002/slct.201903633
[11]
Oddo, L.A. (1992) Global Shape Entropy: A Mathematically Tractable Approach to Building Extraction in Aerial Imagery. The 20th AIPR Workshop: Computer Vision Applications: Meeting the Challenges, McLean, 17-18 October 1991, 91-101. https://spie.org/Publications/Proceedings/Paper/10.1117/12.58059?SSO=1
[12]
Narayanan, K.R. and Srinivasa, A.R. (2012) Shannon-Entropy-Based Nonequilibrium “Entropic” Temperature of a General Distribution. Physical Review E, 85, Article ID: 031151. https://doi.org/10.1103/PhysRevE.85.031151 https://journals.aps.org/pre/abstract/10.1103/PhysRevE.85.031151
[13]
Sahyun, M.R.V. (2018) Aesthetics and Entropy III. Aesthetic Measures. Preprints, Article ID: 2018010098. https://www.preprints.org/manuscript/201801.0098/v1
[14]
Arnheim, R. (1971) Entropy and Art. University of California Press, Berkeley. https://www.aakkozzll.com/pdf/arnheim.pdf
[15]
Casti, J. and Karlqvist, A. (2003) Art and Complexity. JAI Press, Greenwich. https://www.elsevier.com/books/art-and-complexity/casti/978-0-444-50944-4
[16]
Burns, K. (2015) Entropy and Optimality in Abstract Art: An Empirical Test of Visual Aesthetics. Journal of Mathematics and the Arts, 9, 77-90. https://www.tandfonline.com/doi/full/10.1080/17513472.2015.1096738 https://doi.org/10.1080/17513472.2015.1096738
[17]
Birkhoff, G.D. (1933) Aesthetic Measure. Harvard University Press, Cambridge.
[18]
Kotecky, R., Shlosman, S. and Dobrushin, R. (1992) Wulff Construction: A Global Shape from Local Interaction. Vol. 104, American Mathematical Society, Providence. https://bookstore.ams.org/mmono-104
[19]
Wouts, M. (2009) Surface Tension in the Dilute Ising Model: The Wulff Construction. Communications in Mathematical Physics, 289, 157-204. https://link.springer.com/article/10.1007/s00220-009-0782-8 https://doi.org/10.1007/s00220-009-0782-8
[20]
Li, H., Zhao, M. and Jiang, Q. (2009) Cohesive Energy of Clusters Referenced by Wulff Construction. The Journal of Physical Chemistry C, 113, 7594-7597 https://pubs.acs.org/doi/abs/10.1021/jp902319z https://doi.org/10.1021/jp902319z
[21]
Mitra, M.K., Menon, G.I. and Rajesh, R.J. (2008) Asymptotic Behavior of Inflated Lattice Polygons. Journal of Statistical Physics, 131, 393-404. https://link.springer.com/article/10.1007/s10955-008-9512-4 https://doi.org/10.1007/s10955-008-9512-4
[22]
Winklmann, S. (2006) A Note on the Stability of the Wulff Shape. Archiv der Mathematik, 87, 272-279. https://link.springer.com/article/10.1007/s00013-006-1685-y https://doi.org/10.1007/s00013-006-1685-y
[23]
Sekerka, R.F. (2005) Analytical Criteria for Missing Orientations on Three-Dimensional Equilibrium Shapes. Crystal Research and Technology, 40, 291-306. https://onlinelibrary.wiley.com/doi/abs/10.1002/crat.200410342 https://doi.org/10.1002/crat.200410342
[24]
Miracle-Sole, S. (1995) Surface Tension, Step Free Energy, and Facets in the Equilibrium Crystal. Journal of Statistical Physics, 79, 183-214. https://link.springer.com/article/10.1007/BF02179386 https://doi.org/10.1007/BF02179386
[25]
Leibler, S., Singh, R.R.P. and Fisher, M.E. (1987) Thermodynamic Behavior of Two-Dimensional Vesicles. Physical Review Letters, 59, 1989-1992. https://pubmed.ncbi.nlm.nih.gov/10035389/ https://doi.org/10.1103/physrevlett.59.1989
[26]
Rajesh, R. and Dhar, D. (2005) Convex Lattice Polygons of Fixed Area with Perimeter-Dependent Weights. Physical Review E, 71, Article ID: 016130. https://journals.aps.org/pre/abstract/10.1103/PhysRevE.71.016130 https://doi.org/10.1103/PhysRevE.71.016130
[27]
Porter, D.A. and Easterling, K.E. (1992) Phase Transformations in Metals and Alloys. Chapman & Hall, London; New York. https://www.worldcat.org/title/phase-transformations-in-metals-and-alloys/oclc/623259187
[28]
Van der Eeerden, J.P. (1989) Morphology and Surface Structure of Simple Crystals. In: Sunagawa, I., Ed., Morphology and Growth Unit of Crystals, Terra Scientific Publishing Co., Tokyo, 37-47. https://www.terrapub.co.jp/e-library/mguc/index.html
[29]
Sunagawa, I. (1999) Growth and Morphology of Crystals. Forma, 14, 147-166. https://www.academia.edu/6291100/Growth_and_Morphology_of_Crystals
[30]
Nehrke, G., Reichart, G.J., Van Cappellen, P., Meile, C. and Bijma, J. (2007) Dependence of Calcite Growth Rate and Sr Partitioning on Solution Stoichiometry: Non-Kossel Crystal Growth. Geochimica et Cosmochimica Acta, 71, 2240-2249. https://core.ac.uk/download/pdf/11759888.pdf https://doi.org/10.1016/j.gca.2007.02.002
[31]
Gündüz, G. and Gündüz, U. (2005) The Mathematical Analysis of the Structure of Some Songs. Physica A, 357, 565-592. https://doi.org/10.1016/j.physa.2005.03.042 https://www.sciencedirect.com/science/article/abs/pii/S0378437105003547
[32]
Arnheim, R. (1971) Entropy and Art, an Essay on Disorder and Order. University of California Press, Berkeley. https://www.aakkozzll.com/pdf/arnheim.pdf
[33]
Sigaki, H.Y.D., Perc, M. and Ribeiro, H.V. (2018) History of Art Paintings through the Lens of Entropy and Complexity. Proceedings of the National Academy of Sciences of the United States of America, 115, E8585-E8594. https://www.pnas.org/content/115/37/E8585/tab-article-info https://doi.org/10.1073/pnas.1800083115
[34]
Hyde, S., Andersson, S., Larsson, K., Blum, Z., Landh, T., Lidin, S. and Ninham, B.W. (1997) The Language of Shape. Elsevier Science, Amsterdam. https://www.elsevier.com/books/the-language-of-shape/hyde/978-0-444-81538-5
[35]
Gündüz, G. and Gündüz, Y. (2016) A Thermodynamical View on Asset Pricing. International Review of Financial Analysis, 47, 310-327. https://www.sciencedirect.com/science/article/abs/pii/S1057521916000144 https://doi.org/10.1016/j.irfa.2016.01.013
[36]
Gündüz, G. and Gündüz, A. (2017) Viscoelasticity and Pattern Formations in Stock Market Indices. The European Physical Journal B, 90, Article No. 104 https://link.springer.com/article/10.1140/epjb/e2017-70711-x https://doi.org/10.1140/epjb/e2017-70711-x
[37]
Gündüz, G. (2018) Pattern Formation in Time Series Systems Due to Viscoelastic Behavior: Case Studies in Uniform Distribution, Normal Distribution, Stock Market Index, and Music. International Journal of Modern Physics C, 29, Article ID: 1850085. https://doi.org/10.1142/S0129183118500857 https://www.worldscientific.com/doi/10.1142/S0129183118500857
[38]
Gündüz, G. and Gündüz, Y. (2010) Viscoelastic Behavior of Stock Indices. Physica A, 389, 5776-5784. https://doi.org/10.1016/j.physa.2010.09.010 https://www.sciencedirect.com/science/article/abs/pii/S0378437110007909
[39]
Christodoulou, D. (1971) Reversible and Irreversible Transformations in Black-Hole Physics. Physical Review Letters, 25, 1596-1597. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.25.1596 https://doi.org/10.1103/PhysRevLett.25.1596
[40]
Bekenstein, J.D. (1973) Black Holes and Entropy. Physical Review D, 7, 2333-2346 https://journals.aps.org/prd/abstract/10.1103/PhysRevD.7.2333 https://doi.org/10.1103/PhysRevD.7.2333
[41]
Bardeen, J.M., Carter, B. and Hawking, S. (1973) The Four Laws of Black Hole Mechanics. Communications in Mathematical Physics, 31, 161-170.
[42]
Hawking, S.W. (1975) Particle Creation by Black Boles. Communications in Mathematical Physics, 43, 199-220. https://doi.org/10.1007/BF02345020 https://link.springer.com/article/10.1007/BF02345020
Smarr, L. (1973) Mass Formula for Kerr Black Holes. Physical Review Letters, 30, 71-73. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.30.71 https://doi.org/10.1103/PhysRevLett.30.71
[45]
Davies, P.C.W. (1978) Thermodynamics of Black Holes. Reports on Progress in Physics, 41, 1313-1355. https://doi.org/10.1088/0034-4885/41/8/004 https://iopscience.iop.org/article/10.1088/0034-4885/41/8/004/pdf
[46]
Bekenstein, J.D. (2001) The Limits of Information. Studies in History and Philosophy of Science Part B, 32, 511-524. https://doi.org/10.1016/S1355-2198(01)00020-X
[47]
Fulling, S.A. (1973) Nonuniqueness of Canonical Field Quantization in Riemannian Space-Time. Physical Review D, 7, 2850-2862. https://journals.aps.org/prd/abstract/10.1103/PhysRevD.7.2850 https://doi.org/10.1103/PhysRevD.7.2850
[48]
Davies, P.C.W. (1975) Scalar Production in Schwarzschild and Rindler Metrics. Journal of Physics A: Mathematical and General, 8, 609-616. https://iopscience.iop.org/article/10.1088/0305-4470/8/4/022/pdf https://doi.org/10.1088/0305-4470/8/4/022
[49]
Gibbons, G.W. and Hawking, S.W. (1977) Cosmological Event Horizons, Thermodynamics, and Particle Creation. Physical Review D, 15, 2738-2751 https://journals.aps.org/prd/abstract/10.1103/PhysRevD.15.2738 https://doi.org/10.1103/PhysRevD.15.2738
[50]
Padmanabhan, T. (2010) Thermodynamical Aspects of Gravity: New Insights. Reports on Progress in Physics, 73, Article ID: 046901 https://iopscience.iop.org/article/10.1088/0034-4885/73/4/046901/pdf https://doi.org/10.1088/0034-4885/73/4/046901
[51]
Jacobson, T. and Parentani, R. (2003) Horizon Entropy. Foundations of Physics, 33, 323-348. https://link.springer.com/article/10.1023/A:1023785123428 https://doi.org/10.1023/A:1023785123428
[52]
Davies, P.C.W. and Taylor, J.G. (1974) Do Black Holes Really Explode? Nature, 250, 37-38. https://www.nature.com/articles/250037a0 https://doi.org/10.1038/250037a0
[53]
Helfer, A.D. (2003) Do Black Holes Radiate? Reports on Progress in Physics, 66, 943-1008. https://iopscience.iop.org/article/10.1088/0034-4885/66/6/202/pdf https://doi.org/10.1088/0034-4885/66/6/202
[54]
Padmanabhan, T. (2004) Entropy of Static Spacetimes and Microscopic Density of States. Classical and Quantum Gravity, 21, 4485-4494 https://iopscience.iop.org/article/10.1088/0264-9381/21/18/013/pdf https://doi.org/10.1088/0264-9381/21/18/013
[55]
‘t Hooft, G. (2009) Dimensional Reduction in Quantum Gravity. arXiv:gr-qc/9310026. https://arxiv.org/abs/gr-qc/9310026
[56]
Susskind, L. (1994) The World as a Hologram. Journal of Mathematical Physics, 36, 6377-6396. https://arxiv.org/abs/hep-th/9409089 https://doi.org/10.1063/1.531249
[57]
Chen, H.Z., Fisher, Z., Hernandez, J., Myers, R.C. and Ruan, S.M. (2020) Information Flow in Black Hole Evaporation. Journal of High Energy Physics, 2020, Article No. 152. https://link.springer.com/content/pdf/10.1007/JHEP03(2020)152.pdf https://doi.org/10.1007/JHEP03(2020)152
[58]
Verlinde, E. (2011) On the Origin of Gravity and the Laws of Newton. Journal of High Energy Physics, 2011, Article No. 29. https://link.springer.com/content/pdf/10.1007/JHEP04(2011)029.pdf https://doi.org/10.1007/JHEP04(2011)029
[59]
Smolin, L. (2010) Newtonian Gravity in Loop Quantum Gravity. https://arxiv.org/pdf/1001.3668.pdf
