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Statistics of Fractal Systems

  • Oldrich Zmeskal
  • Stanislav Nespurek
  • Michal Vesely
  • Petr Dzik
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 289)

Abstract

Distribution functions are used for the description of energy distribution of elementary particles, atoms, and molecules in dynamic systems. These distribution functions depend on the energy of the system and on its properties. The paper focuses on the generalization of the relationships commonly used to study the statistical properties of particles in 3D space so that they become generally applicable onto an E-dimensional space. These relationships can then be applied e.g. for studying the properties of the particles in 2D and in 1D space.

Two approaches are discussed to describe the classic (Maxwell Boltzmann) and quantum (Fermi-Dirac, Einstein-Bose) distribution functions. The first approach is based on standard theory of probability, the second one on the fractal theory. We have shown that both approaches lead to the same results for defined boundary conditions. But the validity of the second one, i.e. the fractal approach, is much more general.

Keywords

fractal physics classic and quantum statistics classical theory of statistics fractal theory of statistics 

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References

  1. 1.
    Beiser, A.: Perspectives of Modern Physics. MacGraw-Hill, New York (1969)Google Scholar
  2. 2.
    Maxwell Boltzmann distribution, in Wikipedia, http://en.wikipedia.org/wiki/Maxwell-Boltzmann_distribution
  3. 3.
    Mandelbrot, B.B.: Fractal Geometry of Nature. W. H. Freeman and Co., New York (1983)Google Scholar
  4. 4.
    Zmeskal, O., Nezadal, M., Buchnicek, M.: Fractal–Cantorian Geometry, Hausdorff Dimension and the Fundamental Laws of Physics. Chaos, Solitons & Fractals 17, 113–119 (2003)CrossRefMATHGoogle Scholar
  5. 5.
    Zmeskal, O., Nezadal, M., Buchnicek, M.: Field and potential of fractal–Cantorian structures and El Naschie’s ε(∞) theory. Chaos, Solitons & Fractals 19, 1013–1022 (2004)CrossRefMATHGoogle Scholar
  6. 6.
    Zmeskal, O., Buchnicek, M., Vala, M.: Thermal Properties of bodies in fractal and cantorian physics. Chaos, Solitons & Fractals 25, 941–954 (2005)CrossRefMATHGoogle Scholar
  7. 7.
    Zmeskal, O., Vala, M., Weiter, M., Stefkova, P.: Fractal-cantorian geometry of space-time. Chaos, Solitons & Fractals 42, 1878–1892 (2009)CrossRefGoogle Scholar
  8. 8.
    Zmeskal, O., Nespurek, S., Weiter, M.: Space-charge-limited currents: An E-infinity Cantorian approach. Chaos, Solitons & Fractals 34, 143–156 (2007)CrossRefMATHGoogle Scholar
  9. 9.
    Zmeskal, O., Dzik, P., Vesely, M.: Entropy of fractal systems, Computers and mathematics with applications (2013), doi:10.1016/j.camwa.2013.01.017Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Oldrich Zmeskal
    • 1
  • Stanislav Nespurek
    • 1
  • Michal Vesely
    • 1
  • Petr Dzik
    • 1
  1. 1.Faculty of ChemistryBrno University of TechnologyBrnoCzech Republic

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