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Complex Systems and the Evolution of Matter

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Thinking in Complexity

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Abstract

How can order arise from complex, irregular, and chaotic states of matter? In classical antiquity philosophers tried to take the complexity of natural phenomena back to first principles. Astronomers suggested mathematical models in order to reduce the irregular and complex planetary orbits as they are experienced to regular and simple movements of spheres. Simplicity was understood, still for Copernicus, as a feature of truth (Sect. 2.1). With Newton and Leibniz something new was added to the theory of kinetic models. The calculus allows scientists to compute the instaneous velocity of a body and to visualize it as the tangent vector of the body’s trajectory. The velocity vector field has become one of the basic concepts in dynamical systems theory. The cosmic theories of Newton and Einstein have been described by dynamical models which are completely deterministic (Sect. 2.2).

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References

  1. For historical sources of Sect. 2.1 compare Mainzer, K.: Symmetries in Nature. De Gruyter: New York (1994) (German original 1988 ) Chapter 1

    Google Scholar 

  2. Diels, H.: Die Fragmente der Vorsokratiker, 6th ed., revised by W. Kranz, 3 vol. Berlin (1960/1961) (abbrev.: Diels-Kranz), 12 A 10 (Pseudo-Plutarch)

    Google Scholar 

  3. Diels-Kranz 13 A 5, B 1

    Google Scholar 

  4. Diels-Kranz 22 B 64, B 30

    Google Scholar 

  5. Heisenberg, W: Physik und Philosophie. Ullstein: Frankfurt (1970) 44

    Google Scholar 

  6. Diels-Kranz 22 B8

    Google Scholar 

  7. Diels-Kranz 31 B8

    Google Scholar 

  8. Heisenberg, W: Die Plancksche Entdeckung und die philosophischen Grundlagen der Atomlehre, in: Heisenberg, W: Wandlungen in den Grundlagen der Naturwissenschaften. S. Hirzel: Stuttgart (1959) 163

    Google Scholar 

  9. Cf. also Hanson, N.R.: Constellations and Conjectures. Boston (1973) 101

    Google Scholar 

  10. Hanson, N.R. (see Note 9, 113) carried out corresponding calculations. 2. 11 Bohr, H.: Fastperiodische Funktionen. Berlin (1932)

    Google Scholar 

  11. Forke, A.: Geschichte der alten chinesischen Philosophie. Hamburg (1927) 486;

    Google Scholar 

  12. Fêng Yu-Lan: A History of Chinese Philosophy vol. 2: The Period of Classical Learning. Princeton NJ (1953) 120

    Google Scholar 

  13. Mainzer, K.: Geschichte der Geometrie. B. I. Wissenschaftsverlag: Mannheim/ Wien/Zürich (1980) 83;

    MATH  Google Scholar 

  14. Edwards, C.H.: The Historical Development of the Calculus. Springer: Berlin (1979) 89

    Book  MATH  Google Scholar 

  15. Mainzer, K.: Geschichte der Geometrie (see Note 13) 100;

    Google Scholar 

  16. Abraham, R.H./Shaw, C.D.: Dynamics — The Geometry of Behavior Part 1. Aerial Press: Santa Cruz (1984) 20

    Google Scholar 

  17. Audretsch, J./Mainzer, K. (eds.): Philosophie und Physik der Raum-Zeit. B.I. Wissenschaftsverlag: Mannheim (1988) 30

    Google Scholar 

  18. Audretsch, J./Mainzer, K. (eds.): Philosophie und Physik der Raum-Zeit (see Note 15 ) 40;

    Google Scholar 

  19. Weyl, H.: Raum, Zeit, Materie. Vorlesung über Allgemeine Relativitätstheorie. Wissenschaftliche Buchgesellschaft: Darmstadt (1961) (Reprint of the 5th Edition (1923)) 141

    Google Scholar 

  20. Mach, E.: Die Mechanik. Historisch-kritisch dargestellt. Wissenschaftliche Buchgesellschaft: Darmstadt (1976) (Reprint of the 9th Edition (1933)) 149;

    Google Scholar 

  21. Abraham, R.H./Shaw, C.D.: Dynamics — The Geometry of Behavior (see Note 14 ) 57

    Google Scholar 

  22. Ruelle, D.: Small random pertubations of dynamical systems and the definition of attractors. Commun. Math. Phys. 82 (1981) 137–151;

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. Abraham, R.H./Shaw, C.D.: Dynamics — The Geometry of Behavior (see Note 14 ) 45

    Google Scholar 

  24. For an analytical elaboration cf. Stauffer, D., Stanley, H.E.: From Newton to Mandelbrot. A Primer in Theoretical Physics. Springer: Berlin (1990) 26

    Google Scholar 

  25. Nicolis, G./Prigogine, I.: Die Erforschung des Komplexen (see Chapter 1, Note 3 ) 132;

    Google Scholar 

  26. Abraham, R.H./Shaw, C.D.: Dynamics — The Geometry of Behavior (see Note 14) 168, 174

    Google Scholar 

  27. For an analytical elaboration cf. Mainzer, K.: Symmetries in Nature (see Note 1) Chapter 3. 31;

    Google Scholar 

  28. Stauffer, D./Stanley, H.E.: From Newton to Mandelbrot (see Note 19 ) 24

    Google Scholar 

  29. Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer: Berlin (1978);

    MATH  Google Scholar 

  30. Davies, P.C.W.: The Physics of Time Asymmetry. Surrey University Press: London (1974);

    Google Scholar 

  31. Penrose, R.: The Emperor’s New Mind. Oxford University Press: Oxford (1989) 181

    Google Scholar 

  32. Lichtenberg, A.J./Liebermann, M.A.: Regular and Stochastic Motion. Springer: Berlin (1982);

    Google Scholar 

  33. Schuster, H.G.: Deterministic Chaos. An Introduction. Physik-Verlag: Weinheim (1984) 137

    MATH  Google Scholar 

  34. Poincaré, H.: Les Méthodes Nouvelles de la Méchanique Céleste. Gauthier-Villars: Paris (1892)

    Google Scholar 

  35. Arnold, V.I.: Small Denominators II, Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally-periodic motions under a small perturbation of the Hamiltonian, Russ. Math. Surveys 18 (1963) 5;

    ADS  Google Scholar 

  36. Kolmogorov, A.N.: On Conservation of Conditionally-Periodic Motions for a Small Change in Hamilton’s Function, Dokl. Akad. Nauk. USSR 98 (1954) 525;

    MathSciNet  Google Scholar 

  37. Moser, J.: Convergent series expansions of quasi-periodic motions, Math. Anm 169 (1967) 163

    Google Scholar 

  38. Cf. Arnold, V.I.: Mathematical Methods of Classical Mechanics (see Note 22);

    Google Scholar 

  39. Schuster, H.G.: Deterministic Chaos (see Note 23), 141

    Google Scholar 

  40. Hénon, M./Heiles, C.: The applicability of the third integral of the motion: Some numerical experiments, Astron. J. 69 (1964) pp. 73;

    Article  ADS  Google Scholar 

  41. Schuster, H.G.: Deterministic Chaos (see Note 23), 150;

    Google Scholar 

  42. Figures 2.16a-d from M.V. Berry in S. Jorna (ed.), Topics in nonlinear dynamics, Am. Inst. Phys. Conf. Proc. vol. 46 (1978)

    Google Scholar 

  43. For mathematical details compare, e.g. Staufer, D., Stanley, H.E.: From Newton to Mandelbrot (see Note 19 ), 83

    Google Scholar 

  44. Mainzer, K.: Symmetrien der Natur (see Note 1), 423;

    Google Scholar 

  45. Primas, H./Müller-Herold, U.: Elementare Quantenchemie. Teubner: Stuttgart (1984) with an elementary introduction to the Galileo-invariant quantum mechanics (Chapter 3)

    Google Scholar 

  46. Audretsch, J./Mainzer, K. (eds.): Wieviele Leben hat Schrödingers Katze? B. I. Wissenschaftsverlag: Mannheim (1990)

    Google Scholar 

  47. Gutzwiller, M.C.: Chaos in Classical and Quantum Mechanics. Springer: Berlin (1990)

    MATH  Google Scholar 

  48. Friedrich, H.: Chaos in Atomen, in: Mainzer, K., Schirmacher, W. (eds.): Quanten, Chaos und Dämonen (see Note 1 of Chapter 1);

    Google Scholar 

  49. Friedrich, H./Wintgen, D.: The hydrogen atom in a uniform magnetic field, Physics Reports 183 (1989) 37–79

    Article  MathSciNet  ADS  Google Scholar 

  50. Birkhoff, G.D.: Nouvelles recherches sur les systèmes dynamiques, Mem. Pont. Acad. Sci. Novi Lyncaei 1 (1935) 85

    Google Scholar 

  51. Enz, C.P.: Beschreibung nicht-konservativer nicht-linearer Systeme I-II, Physik in unserer Zeit 4 (1979) 119–126, 5 (1979) 141–144 (II)

    Google Scholar 

  52. Lorenz, E.N.: Deterministic nonperiodic flow, J. Atoms. Sci. 20 (1963) 130;

    Article  ADS  Google Scholar 

  53. Schuster, H.G.: Deterministic Chaos (see Note 23) 9

    Google Scholar 

  54. Eckmann, J.P.: Roads to turbulence in dissipative dynamical systems, Rev. Mod. Phys. 53 (1981) 643;

    Article  MathSciNet  ADS  MATH  Google Scholar 

  55. Computer simulation of Fig. 2.21 from Lanford, O.E., Turbulence Seminar, in: Bernard, P., Rativ, T. (eds.): Lecture Notes in Mathematics 615, Springer: Berlin (1977) 114

    Google Scholar 

  56. Mandelbrot, B.B.: The Fractal Geometry of Nature, Freeman: San Fransisco (1982);

    MATH  Google Scholar 

  57. Grassberger, P.: On the Hausdorff dimension of fractal attractors, J. Stat. Phys. 19 (1981) 25;

    Google Scholar 

  58. Lichtenberg, A.J./Liebermann, M.A.: Regular and Stochastic Motions (see Note 23)

    Google Scholar 

  59. Collet, E./Eckmann, J P.: Iterated Maps of the Interval as Dynamical Systems, Birkhäuser: Boston (1980) (see Figures 2. 22–24 )

    Google Scholar 

  60. Großmann, S./Thomae, E.: Invariant distributions and stationary correlation functions of one-dimensional discrete processes, Z. Naturforsch. 32 A (1977) 353;

    Google Scholar 

  61. Feigenbaum, M.J.: Quantitative universality for a class of nonlinear transformations, J. Stat. Phys. 19 (1978) 25

    Article  MathSciNet  ADS  MATH  Google Scholar 

  62. Mainzer, K.: Symmetrien der Natur (see Note 1)

    Google Scholar 

  63. Cf. Nicolis, G./Prigogine, I.: Die Erforschung des Komplexen (see Note 3, Chapter 1 ) 205

    Google Scholar 

  64. Cf. Prigogine, I.: From Being to Becoming — Time and Complexity in Physical Sciences, Freemann: San Fransisco (1980);

    Google Scholar 

  65. Cf. Prigogine, I.: Introduction to Thermodynamics of Irreversible Processes, Wiley: New York (1961)

    Google Scholar 

  66. Fig. 2.26 from Feynman, R.P./Leighton, R.B./Sands, M.: The Feynman Lectures of Physics vol. II., Addison-Wesley (1965)

    Google Scholar 

  67. Haken, H.: Synergetics (see Note 4, Chapter 1 ) 5

    Google Scholar 

  68. Haken, H.: Synergetics (see Note 4, Chapter 1 ) 202;

    Google Scholar 

  69. Haken, H.: Advanced Synergetics. Instability Hierarchies of Self-Organizing Systems and Devices. Springer: Berlin (1983) 187;

    MATH  Google Scholar 

  70. Weinberg, S.: Gravitation and Cosmology. Principles and Applications of the General Theory of Relativity. Wiley: New York (1972)

    Google Scholar 

  71. Cf. Mainzer, K.: Symmetrien der Natur (see Note 1) Chapter 4

    Google Scholar 

  72. Curie, P.: Sur la Symétrie dans les Phénomènes Physiques, Journal de Physique 3 (1894) 3

    Google Scholar 

  73. Audretsch, J./Mainzer, K. (eds.): Vom Anfang der Welt. C.H. Beck: München (1990);

    Google Scholar 

  74. Mainzer, K.: Symmetrien der Natur (see Note 1) 515;

    Google Scholar 

  75. Fritzsch, H.: Vom Urknall zum Zerfall. Die Welt zwischen Anfang and Ende. Piper: München (1983) 278

    Google Scholar 

  76. Hawking, S.: A Brief History of Time. From the Big Bang to Black Holes. Bantam Press: London (1988);

    Google Scholar 

  77. Hoyle, F./Burbridge, G./Narlikar, J.V.: A quasi-steady state cosmological model with creation of matter. Astrophys. Journal 410 (1993) 437457

    Google Scholar 

  78. Hartle, J.B./Hawking, S.W.: Wave function in the universe. Physical Review D 28 (1983) 2960–2975;

    Article  MathSciNet  ADS  Google Scholar 

  79. Mainzer, K.: Hawking. Herder: Freiburg (2000)

    Google Scholar 

  80. Audretsch, J./Mainzer, K. (eds.): Vom Anfang der Welt (see Note 48 ) 165

    Google Scholar 

  81. Greene, B.: The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory. W.W. Norton & Co: New York (1999);

    Google Scholar 

  82. Hawking, S.W.: The Universe in a Nutshell. Bantam Books: New York (2001);

    Google Scholar 

  83. Mainzer, K.: The Little Book of Time. Copernicus Books: New York (2002)

    Google Scholar 

  84. Whitesides, G.M./Mathias, J.P./Seto, C.T.: Molecular self-assembly and nano-chemistry: A chemical strategy for the synthesis of nanostructures. Science 254 (1991) 1312–1319

    Article  ADS  Google Scholar 

  85. Feynman, R.: There’s plenty of room at the bottom. Miniaturization 282 (1961) 295–296

    Google Scholar 

  86. Drexler, K.E.: Nanotechnology summary. Encyclopedia Britannica Science and the Future Yearbook 162 (1990);

    Google Scholar 

  87. Drexler, K.E.: Nanosystems: Molecular Machinery, Manufacturing, and Computation. John Wiley & Sons: New York (1992)

    Google Scholar 

  88. Whitesides, G.M.: The once and future nanomachine. Scientific American 9 (2001) 78–83

    Article  Google Scholar 

  89. Newkome, G.R. (ed.): Advances in Dendritic Macromolecules. JAI Press: Greenwich, Conn. (1994)

    Google Scholar 

  90. Curl, R.F./Smalley, R.E.: Probing Cho. Science 242 (1988) 1017–1022;

    Google Scholar 

  91. Smalley, R.W.: Great balls of carbon: The story of Buckminsterfullerene. The Sciences 31 (1991) 22–28

    Google Scholar 

  92. Müller, A.: Supramolecular inorganic species: An expedition into a fascinating rather unknown land mesoscopia with interdisciplinary expectations and discoveries, J. Molecular Structure 325 (1994) 24;

    Article  Google Scholar 

  93. Angewandte Chemie (International Edition in English) 34 (1995) 2122–2124;

    Google Scholar 

  94. Müller, A./Mainzer, K.: From molecular systems to more complex ones. In: Müller, A., Dress, A., Vögtle, F. (Eds.): From Simplicity to Complexity in Chemistry–and Beyond. Vieweg: Wiesbaden (1995) 1–11

    Google Scholar 

  95. Fig. 2.32 with drawings of Bryan Christie: Spektrum der Wissenschaft Spezial 2 (2001) 22

    Google Scholar 

  96. Dry, C.M.: Passive smart materials for sensing and actuation. Journal of Intelligent Materials Systems and Structures 4 (1993) 415

    Article  Google Scholar 

  97. Amato, I.: Animating the material world. Science 255 (1992) 284–286

    Article  ADS  Google Scholar 

  98. Joy, B.: Why the future doesn’t need us. Wired 4 (2000)

    Google Scholar 

  99. Smalley, R.E.: Of chemistry, love and nanobots. Scientific American 9 (2001) 76–77

    Article  Google Scholar 

  100. Abarbanel, H.D.I.: Analysis of Observed Data. Springer: New York (1996);

    Book  MATH  Google Scholar 

  101. Kanz, H./Schreiber, T.: Nonlinear Time Series Analysis. Cambridge University Press: Cambridge (1997)

    Google Scholar 

  102. Takens, F.: Detecting strange attractors in turbulence. In: Rand, D.A., Young, L.S. (eds.): Dynamical Systems and Turbulence. Springer: Berlin (1981) 336–381

    Google Scholar 

  103. Kaplan, D./Glass, L.: Understanding Nonlinear Dynamics. Springer: New York (1995) 310 (Fig. 6. 20 )

    Google Scholar 

  104. Kaplan, D./Glass, L.: Understanding Nonlinear Dynamics (see Note 67) 310 (Fig. 6.21), 311 (Fig. 6. 22 )

    Google Scholar 

  105. Kaplan, D./Glass, L.: Understanding Nonlinear Dynamics (see Note 67) 316 (Fig. 6.26), 317 (Fig. 6. 28 )

    Google Scholar 

  106. Grassberger, P./Procaccia I.: Characterization of strange attractors. Physical Review Letters 50 (1983) 346–349

    Article  MathSciNet  ADS  Google Scholar 

  107. Deco, G./Schürmann, B.: Information Dynamics: Foundations and Applications. Springer: New York (2001) 17 (Fig. 2. 6 )

    Google Scholar 

  108. Chen, G./Moiola, J.L.: An overview of bifurcation, chaos and nonlinear dynamics in control systems. In: Chua, L.O. (ed.): Journal of the Franklin Institute Engineering and Applied Mathematics: Philadelphia (1995) 838

    Google Scholar 

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  1. Lehrstuhl für Philosophie und Wissenschaftstheorie, Institut für Interdisziplinäre Informatik, Universität Augsburg, Universitätsstrasse 10, 86135, Augsburg, Germany

    Professor Dr. Klaus Mainzer

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Mainzer, K. (2004). Complex Systems and the Evolution of Matter. In: Thinking in Complexity. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-05364-5_2

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