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Multiscale Entropy and Its Implications to Critical Phenomena, Emergent Behaviors, and Information

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Abstract

Thermodynamics of critical phenomena in a system is well understood in terms of the divergence of molar quantities with respect to potentials. However, the prediction and the microscopic mechanisms of critical points and the associated property anomaly remain elusive. It is shown that while the critical point is typically considered to represent the limit of stability of a system when the system is approached from a homogenous state to the critical point, it can also be considered to represent the convergence of several homogeneous subsystems to become a macro-homogeneous system when the critical point is approached from a macro-heterogeneous system. Through the understanding of statistic characteristics of entropy in different scales, it is demonstrated that the statistic competition of key representative configurations results in the divergence of molar quantities when metastable configurations have higher entropy than the stable configuration. Furthermore, the connection between change of configurations and the change of information is discussed, which provides a quantitative framework to study complex, dissipative systems.

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References

  1. C. Kittel, Introduction to Solid State Physics, Wiley, New York, 2005

    MATH  Google Scholar 

  2. S.W. Hawking, Black Holes and Thermodynamics, Phys. Rev. D, 1976, 13, p 191-197

    Article  ADS  Google Scholar 

  3. F. Ross, S.W. Hawking, and G.T. Horowitz, Entropy, Area, and Black Hole Pairs, Phys. Rev. D, 1995, 51, p 4302-4314

    Article  ADS  MathSciNet  Google Scholar 

  4. C.E. Shannon, A Mathematical Theory of Communication, Bell Syst. Tech. J., 1948, 27, p 623-656

    Article  MathSciNet  MATH  Google Scholar 

  5. S. Pavoine, S. Ollier, and D. Pontier, Measuring Diversity from Dissimilarities with Rao’s Quadratic Entropy: Are Any Dissimilarities Suitable?, Theor. Popul. Biol., 2005, 67, p 231-239

    Article  MATH  Google Scholar 

  6. J. Quijano and H. Lin, Entropy in the Critical Zone: A Comprehensive Review, Entropy, 2014, 16, p 3482-3536

    Article  ADS  MathSciNet  Google Scholar 

  7. M.A. Busa and R.E.A. van Emmerik, Multiscale Entropy: A Tool for Understanding the Complexity of Postural Control, J. Sport Heal. Sci., 2016, 5, p 44-51

    Article  Google Scholar 

  8. Z.K. Liu, Y. Wang, and S.L. Shang, Thermal Expansion Anomaly Regulated by Entropy, Sci. Rep., 2014, 4, p 7043

    Article  Google Scholar 

  9. K.G. Wilson, The Renormalization Group: Critical Phenomena and the Kondo Problem, Rev. Mod. Phys., 1975, 47, p 773-840

    Article  ADS  MathSciNet  Google Scholar 

  10. A. Pelissetto and E. Vicari, Critical Phenomena and Renormalization-Group Theory, Phys. Rep.-Rev. Sect. Phys. Lett., 2002, 368, p 549-727

    MathSciNet  MATH  Google Scholar 

  11. Z.K. Liu and Y. Wang, Computational Thermodynamics of Materials, Cambridge University Press, Cambridge, 2016

    Book  Google Scholar 

  12. M. Hillert, Phase Equilibria, Phase Diagrams and Phase Transformations: Their Thermodynamic Basis, Cambridge University Press, Cambridge, 2008

    MATH  Google Scholar 

  13. J.W. Gibbs, The Collected Works of J. Willard Gibbs: Vol. I, Thermodynamics, Yale University Press, New Haven, 1948

    MATH  Google Scholar 

  14. Z.K. Liu, X.Y. Li, and Q.M. Zhang, Maximizing the Number of Coexisting Phases Near Invariant Critical Points for Giant Electrocaloric and Electromechanical Responses in Ferroelectrics, Appl. Phys. Lett., 2012, 101, p 82904

    Article  Google Scholar 

  15. D. Kondepudi and I. Prigogine, Modern Thermodynamics: From Heat Engines to Dissipative Structures, Wiley, New York, 1998

    MATH  Google Scholar 

  16. J.W. Gibbs, The Collected Works of J. Willard Gibbs: Vol. II, Statistic Mechanics, Yale University Press, New Haven, 1948

    Google Scholar 

  17. S.M. Ross, A First Course in Probability, Pearson, London, 2012

    Google Scholar 

  18. L.D. Landau and E.M. Lifshitz, Statistical Physics, Pergamon Press Ltd., New York, 1980

    MATH  Google Scholar 

  19. M. Asta, R. McCormack, and D. de Fontaine, Theoretical Study of Alloy Stability in the Cd-Mg System, Phys. Rev. B, 1993, 48, p 748

    Article  ADS  Google Scholar 

  20. Y. Wang, S.L. Shang, H. Zhang, L.Q. Chen, and Z.K. Liu, Thermodynamic Fluctuations in Magnetic States: Fe3Pt as a Prototype, Philos. Mag. Lett., 2010, 90, p 851-859

    Article  ADS  Google Scholar 

  21. H.L. Lukas, S.G. Fries, and B. Sundman, Computational Thermodynamics: The CALPHAD Method, Vol 131, Cambridge University Press, Cambridge, 2007

    Book  MATH  Google Scholar 

  22. L. Kaufman and H. Bernstein, Computer Calculation of Phase Diagram, Academic Press Inc., New York, 1970

    Google Scholar 

  23. W. Kohn and L.J. Sham, Self-Consisten Equations Including Exchange and Correlation Effects, Phys. Rev., 1965, 140, p A1133-A1138

    Article  ADS  Google Scholar 

  24. G. Kresse, J. Furthmüller. Vienna Ab-initio Simulation Package (VASP). https://www.vasp.at. Accessed 13 Jan 2019

  25. Quantum Espresso. http://www.quantum-espresso.org/. Accessed 13 Jan 2019

  26. The Extreme Science and Engineering Discovery Environment (XSEDE). https://www.xsede.org/. Accessed 13 Jan 2019

  27. National Energy Research Scientific Computing Center (NERSC). http://www.nersc.gov/. Accessed 13 Jan 2019

  28. Materials Project. http://materialsproject.org/. Accessed 13 Jan 2019

  29. OQMD: An Open Quantum Materials Database. http://oqmd.org. Accessed 13 Jan 2019

  30. AFLOW: Automatic Flow for Materials Discovery. http://www.aflowlib.org. Accessed 13 Jan 2019

  31. A. van de Walle and G. Ceder, The Effect of Lattice Vibrations on Substitutional Alloy Thermodynamics, Rev. Mod. Phys., 2002, 74, p 11-45

    Article  ADS  Google Scholar 

  32. Y. Wang, Z.K. Liu, and L.Q. Chen, Thermodynamic Properties of Al, Ni, NiAl, and Ni3Al from First-Principles Calculations, Acta Mater., 2004, 52, p 2665-2671

    Article  Google Scholar 

  33. S.L. Shang, Y. Wang, D. Kim, and Z.K. Liu, First-Principles Thermodynamics from Phonon and Debye Model: Application to Ni and Ni3Al, Comput. Mater. Sci., 2010, 47, p 1040-1048

    Article  Google Scholar 

  34. X.L. Liu, B.K. Vanleeuwen, S.L. Shang, Y. Du, and Z.K. Liu, On the Scaling Factor in Debye–Grüneisen Model: A Case Study of the Mg-Zn Binary System, Comput. Mater. Sci., 2015, 98, p 34-41

    Article  Google Scholar 

  35. S.L. Shang, Y. Wang, and Z.K. Liu, First-Principles Elastic Constants of α- and θ-Al2O3, Appl. Phys. Lett., 2007, 90, p 101909

    Article  ADS  Google Scholar 

  36. S.L. Shang, H. Zhang, Y. Wang, and Z.K. Liu, Temperature-Dependent Elastic Stiffness Constants of Alpha- and Theta-Al2O3 from First-Principles Calculations, J. Phys. Condens. Matter, 2010, 22, p 375403

    Article  Google Scholar 

  37. Y. Wang, J.J. Wang, H. Zhang, V.R. Manga, S.L. Shang, L.Q. Chen, and Z.K. Liu, A First-Principles Approach to Finite Temperature Elastic Constants, J. Phys. Condens. Matter, 2010, 22, p 225404

    Article  ADS  Google Scholar 

  38. J.M. Sanchez, Cluster Expansion and the Configurational Energy of Alloys, Phys. Rev. B: Condens. Matter, 1993, 48, p R14013-R14015

    Article  ADS  Google Scholar 

  39. A. van de Walle, M. Asta, and G. Ceder, The Alloy Theoretic Automated Toolkit: A User Guide, CALPHAD, 2002, 26, p 539-553

    Article  Google Scholar 

  40. A. Zunger, S.H. Wei, L.G. Ferreira, and J.E. Bernard, Special Quasirandom Structures, Phys. Rev. Lett., 1990, 65, p 353-356

    Article  ADS  Google Scholar 

  41. C. Jiang, C. Wolverton, J. Sofo, L.Q. Chen, and Z.K. Liu, First-Principles Study of Binary bcc Alloys Using Special Quasirandom Structures, Phys. Rev. B, 2004, 69, p 214202

    Article  ADS  Google Scholar 

  42. A. van de Walle, P. Tiwary, M. de Jong, D.L. Olmsted, M. Asta, A. Dick, D. Shin, Y. Wang, L.-Q. Chen, and Z.K. Liu, Efficient Stochastic Generation of Special Quasirandom Structures, CALPHAD, 2013, 42, p 13-18

    Article  Google Scholar 

  43. R. Car and M. Parrinello, Unified Approach for Molecular-Dynamics and Density-Functional Theory, Phys. Rev. Lett., 1985, 55, p 2471-2474

    Article  ADS  Google Scholar 

  44. H.Z. Fang, Y. Wang, S.L. Shang, and Z.K. Liu, Nature of Ferroelectric-Paraelectric Phase Transition and Origin of Negative Thermal Expansion in PbTiO3, Phys. Rev. B, 2015, 91, p 24104

    Article  ADS  Google Scholar 

  45. Z.K. Liu, Ocean of Data: Integrating First-Principles Calculations and CALPHAD Modeling with Machine Learning, J. Phase Equilib. Diffus., 2018, 39, p 635-649

    Article  Google Scholar 

  46. Y. Wang, L.G. Hector, H. Zhang, S.L. Shang, L.Q. Chen, and Z.K. Liu, Thermodynamics of the Ce Gamma-Alpha Transition: Density-Functional Study, Phys. Rev. B, 2008, 78, p 104113

    Article  ADS  Google Scholar 

  47. G. Kresse, J. Furthmuller, J. Furthmüller, J. Furthmueller, J. Furthmuller, and J. Furthmüller, Efficient Iterative Schemes for Ab Initio Total-Energy Calculations Using a Plane-Wave Basis Set, Phys. Rev. B, 1996, 54, p 11169

    ADS  Google Scholar 

  48. Y. Wang, L.G. Hector, H. Zhang, S.L. Shang, L.Q. Chen, and Z.K. Liu, A Thermodynamic Framework for a System with Itinerant-Electron Magnetism, J. Phys. Condens. Matter, 2009, 21, p 326003

    Article  Google Scholar 

  49. L. Kouwenhoven and L. Glazman, Revival of the Kondo Effect, Phys. World, 2001, 14, p 33-38

    Article  Google Scholar 

  50. Z.K. Liu, Y. Wang, and S.-L. Shang, Origin of Negative Thermal Expansion Phenomenon in Solids, Scr. Mater., 2011, 66, p 130

    Article  Google Scholar 

  51. Z.K. Liu, S.L. Shang, and Y. Wang, Fundamentals of Thermal Expansion and Thermal Contraction, Materials (Basel), 2017, 10, p 410

    Article  ADS  Google Scholar 

  52. S.A. Mey, Reevaluation of the Al-Zn System, Z. Met., 1993, 84, p 451-455

    Google Scholar 

  53. Z.K. Liu, Z.G. Mei, Y. Wang, and S.L. Shang, Nature of Ferroelectric–Paraelectric Transition, Philos. Mag. Lett., 2012, 92, p 399-407

    Article  ADS  Google Scholar 

  54. G. Shirane and S. Hoshino, On the Phase Transition in Lead Titanate, J. Phys. Soc. Jpn., 1951, 6, p 265

    Article  ADS  Google Scholar 

  55. S.G. Jabarov, D.P. Kozlenko, S.E. Kichanov, A.V. Belushkin, B.N. Savenko, R.Z. Mextieva, and C. Lathe, High Pressure Effect on the Ferroelectric-Paraelectric Transition in PbTiO3, Phys. Solid State, 2011, 53, p 2300-2304

    Article  ADS  Google Scholar 

  56. D. Damjanovic, Ferroelectric, Dielectric and Piezoelectric Properties of Ferroelectric Thin Films and Ceramics, Rep. Prog. Phys., 1998, 61, p 1267-1324

    Article  ADS  Google Scholar 

  57. J. Chen, X. Xing, C. Sun, P. Hu, R. Yu, X. Wang, and L. Li, Zero Thermal Expansion in PbTiO3-Based Perovskites, J. Am. Chem. Soc., 2008, 130, p 1144-1145

    Article  Google Scholar 

  58. P.-E. Janolin, P. Bouvier, J. Kreisel, P.A. Thomas, I.A. Kornev, L. Bellaiche, W. Crichton, M. Hanfland, and B. Dkhil, High-Pressure PbTiO3: An Investigation by Raman and X-Ray Scattering up to 63 GPa, Phys. Rev. Lett., 2008, 101, p 237601

    Article  ADS  Google Scholar 

  59. N. Sicron, B. Ravel, Y. Yacoby, E.A. Stern, F. Dogan, and J.J. Rehr, Nature of the Ferroelectric Phase-Transition in PbTiO3, Phys. Rev. B, 1994, 50, p 13168-13180

    Article  ADS  Google Scholar 

  60. K. Sato, T. Miyanaga, S. Ikeda, and D. Diop, XAFS Study of Local Structure Change in Perovskite Titanates, Phys. Scr., 2005, 2005, p 359

    Article  Google Scholar 

  61. W. Cochran and R.A. Cowley, Dielectric Constants and Lattice Vibrations, J. Phys. Chem. Solids, 1962, 23, p 447-450

    Article  ADS  Google Scholar 

  62. Y. Wang, J.J. Wang, W.Y. Wang, Z.G. Mei, S.L. Shang, L.Q. Chen, and Z.K. Liu, A Mixed-Space Approach to First-Principles Calculations of Phonon Frequencies for Polar Materials, J. Phys.-Condens. Matter, 2010, 22, p 202201

    Article  ADS  Google Scholar 

  63. Y. Wang, S.L. Shang, H. Fang, Z.K. Liu, and L.Q. Chen, First-Principles Calculations of Lattice Dynamics and Thermal Properties of Polar Solids, Comput. Mater., 2016, 2, p 16006

    Article  Google Scholar 

  64. Y. Wang, J.E. Saal, Z.G. Mei, P.P. Wu, J.J. Wang, S.L. Shang, Z.K. Liu, and L.Q. Chen, A First-Principles Scheme to Phonons of High Temperature Phase: No Imaginary Modes for Cubic SrTiO3, Appl. Phys. Lett., 2010, 97, p 162907

    Article  ADS  Google Scholar 

  65. M.J. Zhou, Y. Wang, Y. Ji, Z.K. Liu, L.Q. Chen, and C.-W. Nan, First-Principles Lattice Dynamics and Thermodynamic Properties of Pre-Perovskite PbTiO3, Acta Mater, 2019, 171, p 146-153

    Article  Google Scholar 

  66. L. Szilard, Uber die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen, Z. Phys., 1929, 53, p 840-856

    Article  ADS  MATH  Google Scholar 

  67. L. Szilard, On the Decrease of Entropy in a Thermodynamic System by the Intervention of Intelligent Beings, Behav. Sci., 1964, 9, p 301-310

    Article  Google Scholar 

  68. C.E. Shannon, Prediction and Entropy of Printed English, Bell Syst. Tech. J., 1951, 30, p 50-64

    Article  MATH  Google Scholar 

  69. L. Brillouin, Physical Entropy and Information. II, J. Appl. Phys., 1951, 22, p 338-343

    Article  ADS  MathSciNet  MATH  Google Scholar 

  70. L. Brillouin, The Negentropy Principle of Information, J. Appl. Phys., 1953, 24, p 1152-1163

    Article  ADS  MATH  Google Scholar 

  71. L. Brillouin, Information Theory and Its Applications to Fundamental Problems in Physics, Nature, 1959, 183, p 501-502

    Article  ADS  Google Scholar 

  72. L. Brillouin, Thermodynamics, Statistics, and Information, Am. J. Phys., 1961, 29, p 318-328

    Article  ADS  MathSciNet  MATH  Google Scholar 

  73. L. Brillouin, Science and Information Theory, Academic Press, New York, 1962

    Book  MATH  Google Scholar 

  74. K. Maruyama, F. Nori, and V. Vedral, Colloquium: The Physics of Maxwell’s Demon and Information, Rev. Mod. Phys., 2009, 81, p 1-23

    Article  ADS  MathSciNet  MATH  Google Scholar 

  75. R. Landauer, Irreversibility and Heat Generation in the Computing Process, IBM J. Res. Dev., 1961, 5, p 183-191

    Article  MathSciNet  MATH  Google Scholar 

  76. R. Landauer, Dissipation and Noise Immunity in Computation and Communication, Nature, 1988, 335, p 779-784

    Article  ADS  Google Scholar 

  77. C.H. Bennett, The Thermodynamics of Computation—A Review, Int. J. Theor. Phys., 1982, 21, p 905-940

    Article  Google Scholar 

  78. S. Toyabe, T. Sagawa, M. Ueda, E. Muneyuki, and M. Sano, Experimental Demonstration of Information-to-Energy Conversion and Validation of the Generalized Jarzynski Equality, Nat. Phys., 2010, 6, p 988-992

    Article  Google Scholar 

  79. A. Bérut, A. Arakelyan, A. Petrosyan, S. Ciliberto, R. Dillenschneider, and E. Lutz, Experimental Verification of Landauer’s Principle Linking Information and Thermodynamics, Nature, 2012, 483, p 187-189

    Article  ADS  Google Scholar 

  80. L. Brillouin, Negentropy and Information in Telecommunications, Writing, and Reading, J. Appl. Phys., 1954, 25, p 595-599

    Article  ADS  MathSciNet  MATH  Google Scholar 

  81. U. Seifert, Stochastic Thermodynamics, Fluctuation Theorems and Molecular Machines, Rep. Prog. Phys., 2012, 75, p 126001

    Article  ADS  Google Scholar 

  82. P. Strasberg, G. Schaller, T. Brandes, and M. Esposito, Quantum and Information Thermodynamics: A Unifying Framework Based on Repeated Interactions, Phys. Rev. X, 2017, 7, p 021003

    Google Scholar 

  83. E. Pop, Energy Dissipation and Transport in Nanoscale Devices, Nano Res, 2010, 3, p 147-169

    Article  Google Scholar 

  84. S. Vinjanampathy and J. Anders, Quantum Thermodynamics, Contemp. Phys., 2016, 57, p 545-579

    Article  ADS  Google Scholar 

  85. S.E. Jørgensen, A New Ecology: Systems Perspective, Elsevier, Amsterdam, 2007

    Google Scholar 

  86. B. Ravel, N. Slcron, Y. Yacoby, E.A. Stern, F. Dogan, and J.J. Rehr, Order-Disorder Behavior in the Phase Transition of PbTiO3, Ferroelectrics, 1995, 164, p 265-277

    Article  Google Scholar 

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Acknowledgments

ZKL is grateful for financial supports from many funding agencies in the United States as listed in the cited references, including the National Science Foundation (NSF with the latest Grant 1825538), the Department of Energy (with the latest Grants being DE-FE0031553 and DE-NE0008757), Army Research Lab, Office of Naval Research (with the latest Grant N00014-17-1-2567), Wright Patterson AirForce Base, NASA Jet Propulsion Laboratory, and the National Institute of Standards and Technology, plus a range of national laboratories and companies that supported the NSF Center for Computational Materials Design, the LION clusters at the Pennsylvania State University, the resources of NERSC supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231, and the resources of XSEDE supported by NSF with Grant ACI-1053575. BL would like to acknowledge the partial financial support from the NSF Grant Number DMS-1713078. The authors thank Prof. Yi Wang at Penn State for stimulating discussions.

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  1. Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA, 16802, USA

    Zi-Kui Liu

  2. Department of Statistics, The Pennsylvania State University, University Park, PA, 16802, USA

    Bing Li

  3. Department of Ecosystem Science and Management, The Pennsylvania State University, University Park, PA, 16802, USA

    Henry Lin

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Liu, ZK., Li, B. & Lin, H. Multiscale Entropy and Its Implications to Critical Phenomena, Emergent Behaviors, and Information. J. Phase Equilib. Diffus. 40, 508–521 (2019). https://doi.org/10.1007/s11669-019-00736-w

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