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Self-organized Periodic Processes: From Macro-layers to Micro-world of Diffusion and Down to the Quantum Aspects of Light

  • Jaroslav ŠestákEmail author
  • Pavel Hubík
  • Jiří J. Mareš
  • Jiří Stávek
Chapter
Part of the Hot Topics in Thermal Analysis and Calorimetry book series (HTTC, volume 11)

Abstract

Self-similarity and the orderly crystal (often dendritic) growth are an important parts of nature as well as the source of solid-state thermal chemistry under nonequilibrium (undercooling) conditions providing theoretical roots of chemical swinging clock. Such oscillation processes known in chemistry and biology apply for systems far from equilibrium involving special cases of oscillations extending from the self-organized periodic chemical reactions (such as Liesegang’s or Belousov–Zhabotinsky’s reactions) to ordered solid-state processes, from liquids to atmosphere, from macro to micro, indispensable in biology. The chapter deals with a remarkable problem of thermal physics, unresolved for more than 70 years, concerning class of diffusion-controlled periodic chemical reactions, where macroscopically observed diffusion action attains, with appreciable accuracy, the value of Planck’s quantum. Because the classical and quantum diffusions are processes, which are indistinguishable in the configuration space, a quantum criterion in terms of diffusion constants is valid. This criterion enables one to find out conditions under which the quantum behaviour of self-organized periodic reactions can be observed. Examples are shown for the subcritical and critical oscillatory regimes; a special kind of self-organized Liesegang’s rings—annual growth rings of a trunk of larch tree is discussed. The text even involves a thinkable hypothesis of the light self-organization based on the previously analysed principle on least time (Fermat) and of the least action (Maupertuis). It was already noticed by Galileo who opened this problem aware that the cycloid curve yields the quickest descent leading to the so-called brachistochrone. The chapter contains 130 references.

Keywords

Diffusion Constant Brownian Particle Diffusion Action Autocatalytic Reaction Instant Speed 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

The results were developed and the book realized thanks to the resources made available by the financial support of the CENTEM project, reg. no. CZ.1.05/2.1.00/03.0088, jointly funded by the ERDF (within the OP RDI program of the Czech Ministry of Education, Youth and Sports).

References

  1. 1.
    von Foerster H (1960) On self-organizing systems and their environments. In: Yovits MC, Cameron S (eds), Self-organizing systems, Pergamon Press, London, pp 31–50; and (1992) Cybernetics. In: Skapiro SC (ed) The encyclopedia of artificial intelligence, 2nd edn, Wiley, New York, pp 309–312Google Scholar
  2. 2.
    Nicolis G, Prigogine I (1977) Self-organization in nonequilibrium systems: from dissipative structures to order through fluctuations. Wiley, New YorkGoogle Scholar
  3. 3.
    Georgiev GY, Georgiev IY (2002) The least action and the metric of an organized system. Open Syst Inform Dynam 9:371–380CrossRefGoogle Scholar
  4. 4.
    Gauss CF (1829) Über ein neues allgemeines Grundgesetz der Mechanik. Crelle’s J 4:232CrossRefGoogle Scholar
  5. 5.
    Hertz H (1896) Principles of mechanics miscellaneous papers III. Macmillan, New YorkGoogle Scholar
  6. 6.
    De Sapio V, Khatib O, Delp S (2008) Least action principles and their application to constrained and task-level problems in biomechanics. Multibody Syst Dyn 19:303–322CrossRefGoogle Scholar
  7. 7.
    Ashby WR (1947) Principles of the self-organizing dynamic system. J General Psych 37:125–128CrossRefGoogle Scholar
  8. 8.
    Heylighen F (1990) Classical and non-classical representations in physics. Cyber Syst 21:423–444; and Heylighen F, Joslyn C (2001) Cybernetics and second order cybernetics. Encyclop Phys Sci Technol 4:155–1701Google Scholar
  9. 9.
    Heylighen F, Dewaele JM (1996) Complexity and self-organization. Cambridge Dictionary of Philosophy, 784–785Google Scholar
  10. 10.
    Kauffman SA (1993). The origins of order. Oxford University Press; and (2005) At home in the universe: the search for laws of self-organization and complexity. Viking, LondonGoogle Scholar
  11. 11.
    Bar-Yam Y (1997) Dynamics of complex systems: studies in nonlinearity. Westview PressGoogle Scholar
  12. 12.
    Waldrop MM (1992) Complexity: the emerging science at the edge of order and chaos. Viking, LondonGoogle Scholar
  13. 13.
    Prigogine I, Stengers I (1984) Order out of chaos. Bantam Books, New YorkGoogle Scholar
  14. 14.
    Haken H (2000) Information and self-organization: a macroscopic approach to complex systems. Springer, New YorkGoogle Scholar
  15. 15.
    Tyson J (1976) The Belousov-Zhabotinsky reactions. Springer, HeidelbergCrossRefGoogle Scholar
  16. 16.
    Tockstein A, Treindl L (1986) Chemické oscilace (Chemical oscillations). Academia, Praha, (in Czech); and Gray P, Scot S (1990) Chemical oscillations and instabilities. Oxford PressGoogle Scholar
  17. 17.
    Šesták J (2004) Power laws, fractals and deterministic chaos: or how nature is smart. Chapter 13 in his book ‘Heat, thermal analysis and society’ Nucleus, Hradec Králové, pp 219–235Google Scholar
  18. 18.
    Glicksmann ME (2011) Principles of solidification. Springer, Berlin/HeidelbergCrossRefGoogle Scholar
  19. 19.
    Chvoj Z, Kožíšek Z, Šesták J (1989) Non-equilibrium processes of melt solidification and metastable phase formation. Monograph as a special issue of Thermochim Acta, vol 153, Elsevier, AmsterdamGoogle Scholar
  20. 20.
    Glicksman ME, (1976) Curvature, composition and supercooling kinetics at dendrite growth. Metall Trans A A7:1747 and (1984) Mater Sci Eng 65:45–57Google Scholar
  21. 21.
    Gravner J (2009) Modeling snow crystal growth: a three-dimensional mesoscopic approach. Phys Rev 79:011601CrossRefGoogle Scholar
  22. 22.
    Belousov BP, (1959) Collection of Short Papers on Radiation Medicine. Medical Publications, Moscow, pp 145–9; and (1985) In: Field RJ, Burger M (eds) Oscillations and travelling waves in chemical systems, Wiley, New York, pp 605–13Google Scholar
  23. 23.
    Zaikin AN, Zhabotinsky AM (1970) Concentration wave propagation in two-dimensional liquid-phase self-oscillating system. Nature 225:535CrossRefGoogle Scholar
  24. 24.
    Winfree AT (1980) The geometry of biological time. Springer, New York; and (1987) The timing of biological clocks. Freeman, San FranciscoGoogle Scholar
  25. 25.
    Runge FF, Liesegang RE, Belousov BP, Zhabotinsky AM (1987) In: Kuhnert L, Niedersen U (eds) Selbsorganisation chemischer Strukturen. Ostwald’s Klassiker, Verlag H. Deutsch, Frankfurt am MainGoogle Scholar
  26. 26.
    Camazine D, Sneyd TB (2003) Self-organization in biological systems. Princeton University PressGoogle Scholar
  27. 27.
    Fechner GT (1831) Maassbestimmungen über die galvanische Kette. Leipzig, F.A, BrockhausGoogle Scholar
  28. 28.
    Ord WM (1879) On the influence of colloids upon crystalline form and cohesion. E. Stanford, LondonGoogle Scholar
  29. 29.
    Pringsheim N (1895) Über chemische Niederschläge in Gallerte. Z Phys Chem 17:473Google Scholar
  30. 30.
    Liesegang RE (1896) Ueber einige Eigenschaften von Gallerten. Naturwiss. Wochenschrift 11 353; and Michaleff P, Nikiforoff VK, Schemjakin FM (1934) Uber eine neue Gesetzmässigkeit für periodische Rektionen in Gelen. Kolloid-Z. 66:197–200Google Scholar
  31. 31.
    Leduc S (1912) Das Leben in seinem physikalisch-chemischen Zusammenhang. L. Hofstetter Verlag, HalleGoogle Scholar
  32. 32.
    Runge FF (1855) Der Bildungstrieb der Stoffe: veranschaulicht in selbständig gewachsenen Bildern. Selbstverlag, Oranienburg, p 32Google Scholar
  33. 33.
    Nikiforov VK (1931) lectures at the Mendeleev chemical congress. MoscowGoogle Scholar
  34. 34.
    Maupertuis PLM (1768) Oevres de Maupertuis. Alyon: Paris, vol IV, p 36Google Scholar
  35. 35.
    Fermat P (1662) Synthesis ad reflexiones—a latter to de la Chambre in Oeuveres de P. Fermat, Tom 1, Paris 1891, p 173Google Scholar
  36. 36.
    Priogine I, Nicoli G, Baylogantz A (1972) Über chemische Niederschläge in Gallerte. Physics Today 25Google Scholar
  37. 37.
    Stávek J, Šípek M, Šesták J (2002) Application of the principle of least action to some self-organized chemical reactions. Thermochim Acta 388:440CrossRefGoogle Scholar
  38. 38.
    Stávek J, Šípek M, Šesták J (2002) On the mechanism and mutual linking of some self-organized chemical reactions. Proc/Acta Western Bohemian University Pilsen 3:1Google Scholar
  39. 39.
    Šesták J (2004) The principle of lest action and self-organization of chemical reactions. Chapter 15 in his book ‘Heat, thermal analysis and society’ Nucleus, Hradec Králové, pp 260–273Google Scholar
  40. 40.
    Küster E (1931) Űber Zonenbildung in kolloiden Medien. JenaGoogle Scholar
  41. 41.
    Mikhalev P, Nikiforov VK, Schemyakin FM (1934) Űber eine neue Gesetzmässigkeit für periodische Reaktionen in Gelen. Kolloid Z 66:197CrossRefGoogle Scholar
  42. 42.
    Christiansen JA, Wulf I (1934) Untersuchungen űber das Liesegang-Phänomen. Z Phys Chem B 26:187Google Scholar
  43. 43.
    C Raman V, Ramaiah KS (1939) Studies on Liesegang rings, Proc Acad Sci India 9A:455–478Google Scholar
  44. 44.
    Schaafs W (1952) Untersuchungen an Liesegangschen Ringen. Kolloid Z. 128:92CrossRefGoogle Scholar
  45. 45.
    Joos G, Enderlein HD, Schädlich H (1959) Zur Kenntnis der rhythmischen Fällungen Liesegang-Ringe. Z Phys Chem (Frankfurt) 19:397CrossRefGoogle Scholar
  46. 46.
    Stávek J, Šípek M (1995) Interpretation of periodic precipitation pattern formation by the concept of quantum mechanics. Cryst Res, Tech 30Google Scholar
  47. 47.
    Stávek J (2003) Diffusion action of chemical waves. Apeiron 10:183–193Google Scholar
  48. 48.
    Lafever R (1968) Dissipative structures in chemical systems. J Chem Phys 49:4977CrossRefGoogle Scholar
  49. 49.
    de Broglie L (1926) Ondes et mouvements. Gauthier-Villars et Cie, Paris, p 1Google Scholar
  50. 50.
    LaViolette PA (1994) Subquantum kinetics. Staslane, New YorkGoogle Scholar
  51. 51.
    Mareš JJ, Stávek J, Šesták J (2004) Quantum aspects of self-organized periodic chemical reaction. J Chem Phys 121:1499CrossRefGoogle Scholar
  52. 52.
    Mareš JJ, Šesták J (2000) An attempt at quantum thermal physics. J Thermal Anal Calor 80:681Google Scholar
  53. 53.
    Mareš JJ, Šesták J Stávek J, Ševčíková H, Hubík P (2005) Do periodic chemical reactions reveal Fürth’s quantum diffusion limit? Physica E 29:145Google Scholar
  54. 54.
    Marek M, Ševčíková H (1988) From chemical to biological organization. Springer, BerlinGoogle Scholar
  55. 55.
    Stávek J, Mareš JJ, Šesták J (2000) Life cycles of Belousov-Zhabotinsky waves in closed systems. Proc/Acta Western Bohemian Uni Pilsen 1:1Google Scholar
  56. 56.
    Deneubourg C, Sneyd F, Bonabeau T (2003) Self-organization in biological systems. Princeton University PressGoogle Scholar
  57. 57.
    Ebeling W, Feistel R (2011) Physics of self-organization and evolution. Wiley, WeinheimGoogle Scholar
  58. 58.
    Epstein IR, Pojman JA (1998) An introduction to nonlinear chemical dynamics: oscillations, waves, patterns, and chaos. Oxford University Press, USA, p 3Google Scholar
  59. 59.
    Glendinning P (1994) Stability, instability, and chaos. Cambridge Press, New YorkCrossRefGoogle Scholar
  60. 60.
    Field RJ, Schneider FW (1989) Oscillating chemical reactions and non-linear dynamics. J Chem Educ 66:195CrossRefGoogle Scholar
  61. 61.
    Gulick L (1992) Encounters with chaos. McGraw-Hill, New YorkGoogle Scholar
  62. 62.
    Leblond J-M L, Balibar F (1990) Quantics-rudiments of quantum physics. North-Holland, AmsterdamGoogle Scholar
  63. 63.
    de la Peña L, Cetto AM (1996) The quantum dice—an introduction to stochastic electrodynamics. Academic, DordrechtGoogle Scholar
  64. 64.
    Mareš JJ, Šesták J, Hubík P (2013) Transport constitutive relations, quantum diffusion and periodic reactions. Chapter 14 in book In: J. Šesták, J. Mareš, P. Hubík (eds) Glassy, amorphous and nano-crystalline materials: thermal physics, analysis, structure and properties. pp 227–245. Springer, Berlin (ISBN 978-90-481-2881-5)Google Scholar
  65. 65.
    Stávek J, Šípek M, Šesták J (2001) Diffusion action of waves occurring in the Zhabotinsky-Belousov kind of chemical reactions. Proc/Acta Western Bohemian Uni Pilsen 2:1Google Scholar
  66. 66.
    Kalva Z, Šesták J, Mareš JJ, Stávek J (2009) Transdisciplinarity of diffusion including aspects of quasiparticles, quantum diffusion and self-organized transport, chapter 20 in the book. In: Šesták J, Holeček M, Málek J (eds) Some thermodynamic, structural and behavior aspects of materials accentuating non-crystalline states, OPS-ZČU Plzen, pp 128–151 (ISBN 978-80-87269-20-6)Google Scholar
  67. 67.
    Kalva Z, Šesták J (2004) Transdiciplinary aspects of diffusion and magnetocaloric effect. J Thermal Anal Calor 76:1CrossRefGoogle Scholar
  68. 68.
    Sommerfeld AJW (1929) Wave-mechanics: supplementary volume to atomic structure and spectral lines (trans: Henry L. Brose), DuttonGoogle Scholar
  69. 69.
    Fűrth R (1933) Űbereinige Beziehungenzwischen klassischer Statistik und Quantummechanik. Z Physik 81:143CrossRefGoogle Scholar
  70. 70.
    Einstein A (1956) Investigations on the theory of the Brownian movement. In: Fürth R (ed), Dover Publications, New YorkGoogle Scholar
  71. 71.
    Smoluchowski MV (1916) DreiVorträgeüber Diffusion, BrownischeMolekularbewegung und Koagulation von Kolloidteilchen. Physik Zeitschr 17:557Google Scholar
  72. 72.
    Chung KL, Zhao Z (1995) From Brownian motion to Schrödinger´s equation. Springer Verlag, BerlinCrossRefGoogle Scholar
  73. 73.
    Łuczka J, Rudnick R, Hanggi P (2005) The diffusion in the quantum Smoluchowski equation. Phys A 351:60–68CrossRefGoogle Scholar
  74. 74.
    Field RJ, Noyes RM (1974) Oscillations in chemical systems. IV. Limit cycle behavior in a model of a real chemical reaction. J Chem Phys 60:1877–1884CrossRefGoogle Scholar
  75. 75.
    Rebane KK (1992) Possibility of self-organization in photosynthetic light harvesting antennae. J Phys Chem 96:9583–9585CrossRefGoogle Scholar
  76. 76.
    Surrey T, Elowitz MB, Wolf PE (1998) Chromophore-assisted light on activation and self-organization of microtubules and motors. Proc National Acad Sci USA 95:4293–4298CrossRefGoogle Scholar
  77. 77.
    Lotka AJ (1910) Contribution to the Theory of Periodic Reaction. J Phys Chem 14:271–274CrossRefGoogle Scholar
  78. 78.
    Chance B, Chost AK, Pye EK, Hess B (1973) Biological and biochemical oscillations. Academic, New YorkGoogle Scholar
  79. 79.
    Feltz B, Crommelinck M, Goujon P (eds) (2006) Self-organization and emergence in life sciences. Springer, HeidelbergGoogle Scholar
  80. 80.
    Lehn J-M (2002) Toward self-organization and complex matter. Science 295:2400–2403CrossRefGoogle Scholar
  81. 81.
    Turing AM (1952) The chemical basis of morphogenesis. Phil Trans R Soc Lond B 237:37–72CrossRefGoogle Scholar
  82. 82.
    Ranvier L (1871) Contribution á l’histologie et á la physiologie des nerfs périphériques. C R Acad Sci 73:1168–1171Google Scholar
  83. 83.
    McFadden J, Al-Khalili J (2014) Life on the edge: the coming of age of quantum biology: a readable intro to the relation between quantum physics and biological processes. Dreamscape MediaGoogle Scholar
  84. 84.
    Silverman A (2015) The vital question: energy evolution and the origins of complex life. Dreamscape MediaGoogle Scholar
  85. 85.
    Šesták J, Mareš JJ, Hubík P, Proks I (2009) Contribution by both the Lazare and Sadi Carnots to the caloric theory of heat: its inspirative role in thermodynamics. J Thermal Anal Calor 97:679–683CrossRefGoogle Scholar
  86. 86.
    Feng De-Jun, Lau Ka-Sing (2014) Geometry and analysis of fractals. Springer, Berlin/HeidelbergCrossRefGoogle Scholar
  87. 87.
    Ozao R, Ochiai M (1993) Fractal nature and thermal analysis of powders. J Thermal Anal 40:1331CrossRefGoogle Scholar
  88. 88.
    Galwey AK, Brown ME (1999) Thermal decomposition of ionic solids. Elsevier, Amsterdam; and Brown ME, Dollimore D, Galwey AK (1980) Reactions in the solid state. Elsevier, AmsterdamGoogle Scholar
  89. 89.
    Jesenák V (1985) Philosophy of the mechanism of diffusion controlled processes; Thermochim. Acta 92:39; and (1985) Thermal effects of oscillating solid-state reactions; Thermochim. Acta 95: 91Google Scholar
  90. 90.
    Logvinkov SM, Semchenko GD, Kobyzeva DA. (1998) On the self-mechanism of reversible chemical solid-phase reactions in the MgO–Al2O3–SiO2 system, Russ. Ogneup. Tekh. Keram 8: 29–34; and (1999) Conjugated processes in the MgO–Al2O3–SiO2 system and the oscillatory, autocatalized evolution of phase composition, Russ. Ogneup. Tekh. Keram, 9:6–13Google Scholar
  91. 91.
    Osmialowski B, Kolehmainen E, Dobosz R, Rissanen K (2010) Self-organization of 2-Acylaminopyridines in the solid state and in solution. J Phys Chem A 114:10421–10426CrossRefGoogle Scholar
  92. 92.
    Avramov I, Hoche T, Russel C (1999) Is there a crystallization pendulum? J Chem Phys 110:8676CrossRefGoogle Scholar
  93. 93.
    Stávek J, Ulrich J (1994) Interpretation of crystal growth and dissolution by the reaction fractal dimensions. Cryst Res Technol 29:763–785; and Chubynsky MV, Thorpe MF (2001) Self-organization and rigidity in network glasses. Curr Opin Solid State Mater Sci 5:525–532Google Scholar
  94. 94.
    Usychenko VG (2006) Electron self-organization in electronic devices in the light of principles of mechanics and thermodynamics. Russ. Zhurnal Tekhnicheskoĭ Fiziki, 76: 17–25; translated in Theoret. Mat Phys 51:409–417Google Scholar
  95. 95.
    Sze SM (1969) Physics of Semiconductor Devises. Wiley, London; and Teichert C (2002) Self-organization of nanostructures in semiconductor heteroepitaxy: a review. Phys Rep 365: 335Google Scholar
  96. 96.
    Janek J (2000) Oscillatory kinetics at solid/solid phase boundaries in ionic crystals. Solid State Ionics 131:129–142CrossRefGoogle Scholar
  97. 97.
    Liu Ruey-Tarng, Liaw Sy-Sang, Maini PK (2007) Oscillatory Turing patterns in a simple reaction-diffusion systems. J Korean Phys Soc 50:234–238CrossRefGoogle Scholar
  98. 98.
    Ren Jie, Zhang Xiaoyan, Jinzhang Gao, Yang Wu (2013) The application of oscillating chemical reactions to analytical determinations. Cent Eur J Chem 11:1023–1031Google Scholar
  99. 99.
    Betzler SB, Wisnet A, Breitbach B, Mitterbauer C, Weickert J, Schmidt-Mende L, Scheu C (2014) Template-free synthesis of novel, highly-ordered 3D hierarchical Nb3O7(OH) superstructures with semiconductive and photoactive properties. J Mater Chem A 2:12005CrossRefGoogle Scholar
  100. 100.
    Šesták J, Barta Č (2001) Invited plenary lecture: thermophysical research under microgravity: kinetic phase diagrams determination inspace lab. CD—Proceedings of the 3rd IPMM (Intell Process Manu Mater J Mech), Vancouver, CanadaGoogle Scholar
  101. 101.
    Brandt L, Loiseau J-Ch (2015) General introduction to hydrodynamic instabilities. KTH Mechanics; and Curry JA, Webster PJ (1999) Thermodynamics of atmosphere. Academic, New YorkGoogle Scholar
  102. 102.
    Epstein IR, Showalter K (1996) Nonlinear chemical dynamics: oscillations and chaos. J Phys Chem 100:13132–13147CrossRefGoogle Scholar
  103. 103.
    Orosz G, Wilson RE, Krauskopf B (2004) Global bifurcation investigation of an optimal velocity traffic model with driver reaction time. Phys Rev E 70:026207CrossRefGoogle Scholar
  104. 104.
    Michelson AA, Morley EW (1886) Influence of motion of the medium on the velocity of light. Am J Sci 31: 377–386; and (1887) On the relative motion of the earth and the luminiferous aether. Am J Sci 34: 333–345Google Scholar
  105. 105.
    Caghill RT, Kitto K (2003) Michelson-Morley experiment revised and the cosmic background radiation preferred frame 10:104–117Google Scholar
  106. 106.
    Bernoulli J (1696) The brachistochrone problem for the shrewdest mathematicians. Acta EruditorumGoogle Scholar
  107. 107.
    Erlichson H (1999) Johann Bernoulli’s brachistochrone solution using Fermat’s principle of least time. Eur J Phys 20:299–304CrossRefGoogle Scholar
  108. 108.
    Haws L, Kiser T (1995) Exploring the brachistochrone problem. Amer Math Monthly 102:328–336CrossRefGoogle Scholar
  109. 109.
    Stávek J (2014) On the brachistochrone problem in the Michelson-Morley experiment, Apeiron, unpublished as yetGoogle Scholar
  110. 110.
    Gjurchinovsi A (2004) Einstein´s mirror and Fermat´s principle of least time. J Phys 72:1325–1327Google Scholar
  111. 111.
    Whittaker RT (1910) History of theories of Aether and electricity. Lobngman, DublinGoogle Scholar
  112. 112.
    Veričák V (1910) Anwendung der Lobatschewki Geometrie in der Relativitätstheorie. Phys Zeit 11:93–96Google Scholar
  113. 113.
    Barett JF (2010) The hyperbolic theory of special relativity. https://arxiv.org/ftp/arxiv/papers/1102/1102.0462.pdf
  114. 114.
    de Saxce G, Vallee C (2016) Galilean mechanics and thermodynamics of continua. Wiley-ISTE, LondonGoogle Scholar
  115. 115.
    Feynman R (2010) Feynman Lectures vol III. http://www.feynmanlectures.caltech.edu/III_toc.html; and (1985) QED—the strange theory of light and matter. Princeton University Press, Princeton
  116. 116.
    Stávek J (2006) Evaluation of self-organized photon waves. Apeiron 13:102–117; and (2010) Doppler-Voigt-Einstein self-organization–the mechanism for information transfer. Apeiron 17:214–222Google Scholar
  117. 117.
    Stávek J (2004) Diffusion of individual Brownian particles through Young´s double-slits. Apeiron 11:752–186; and (2005) Possible solar microwave background radiation. Galilean Electrodyn.16:31–38; and (2007) On the photon information constant. Apeiron, unpublished as yetGoogle Scholar
  118. 118.
    Einstein A (1905) Zur Elektrodynamikbewegter Körper. Ann der Physik 322:891–921CrossRefGoogle Scholar
  119. 119.
    Sommerfeld A (1909) On the composition of velocities in the theory of relativity. Verh der DPG 21:577–582Google Scholar
  120. 120.
    Rybczyk JA (2008) Alternative versions of the relativistic acceleration composition formula, http://www.mrelativity.net/
  121. 121.
    Aryal PR (2014) A Study of the behavior of Brownian motion under brachistochrone-type metrics. New Mexico State University, Las CrucesGoogle Scholar
  122. 122.
    Rezakhani AT, Kuo W-J, Hamma A, Lidar D.A, Zanardi P (2009) Quantum adiabatic brachistochrone. Phys Rev Lett 103: 080502Google Scholar
  123. 123.
    Mareš JJ, Hubík P, Šesták J, Špička V, Krištofik J, Stávek J (2008) Phenomenological approach to the caloric theory of heat. Thermochim Acta 474:16–24CrossRefGoogle Scholar
  124. 124.
    Ashby N, Brittin WE, Love WF, Wyss W (1975) Brachistochrone with coulomb friction. Am J Physics 43(10):902CrossRefGoogle Scholar
  125. 125.
    Golubev YuF (2012) Brachistochrone with dry and arbitrary viscous friction. J. Comput Syst Sci Int 51:22–37CrossRefGoogle Scholar
  126. 126.
    Benson DC (1969) An elementary solution of the brachistochrone problem. Am Mathem Monthly 76:889–890Google Scholar
  127. 127.
    Jeremić S, Šalinić A, Obradović Z, Mitrović Z (2011) On the brachistochrone of a variable mass particle in general force fields. Mathemat Comput Model 54:2900–2912CrossRefGoogle Scholar
  128. 128.
    Manor G, Rimon E (2014) The speed graph method: time optimal navigation among obstacles subject to safe braking constraint. IEEE Int Conf Robot Automat 1155–1160Google Scholar
  129. 129.
    Perlick V (1991) The brachistochrone problem in a stationary space-time. J Math Phys 32:3148CrossRefGoogle Scholar
  130. 130.
    Maleki M, Hadi-Vencheh A (2010) Combination of non-classical pseudospectral and direct methods for the solution of brachistochrone problem. Inter J Mathem Comp 87:1847–1856CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  • Jaroslav Šesták
    • 1
    Email author
  • Pavel Hubík
    • 2
  • Jiří J. Mareš
    • 2
  • Jiří Stávek
    • 3
  1. 1.New Technologies Research Centre (NTC-ZČU)University of West BohemiaPilsenCzech Republic
  2. 2.Division of Solid-State PhysicsInstitute of Physics, Czech Academy of SciencesPragueCzech Republic
  3. 3.Laboratory of Diffusion ProcessesPragueCzech Republic

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