Phase Diagrams and Catastrophes

  • K. Keller
  • G. Dangelmayr
  • H. Eikemeier
Conference paper
Part of the Springer Series in Synergetics book series (SSSYN, volume 4)

Abstract

Maxwell sets of catastrophe polynomials are put into correspondence with thermodynamic phase diagrams near higher-order critical points. An abstract lattice model is set up which, using scaling principles, provides a theoretical basis for making catastrophe-theoretic techniques capable of reproducing empirical critical exponents.

Keywords

Phenol Helium Pyridine Paral Ferro 

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References

  1. 1.
    Haken, H.: Synergetics, Springer 1978MATHGoogle Scholar
  2. 1.
    Haken, H.: Cooperative phenomena in systems far from thermal equilibrium and in nonphysical systems, Rev. Mod. Physics 47 (1975), 67MathSciNetADSCrossRefGoogle Scholar
  3. 2.
    Falk, G.: Theoretische Physik II: Thermodynamik, Springer 1968Google Scholar
  4. 3.
    Thom, R.: Structural Stability and Morphogenesis, Benjamin 1975MATHGoogle Scholar
  5. 4.
    Arnold, V. : Critical Points of Smooth Functions, Proc. of the International Congress of Mathematicians, Vol. 1, Vancouver 1974Google Scholar
  6. 5.
    Dangelmayr, G.: M.S. thesis, Tübingen 1975Google Scholar
  7. 6.
    Keller, K.: M.S. thesis, Tübingen 1975Google Scholar
  8. 7.
    Vendrik, M.C.M.: A Classification of Phase Diagrams by Means of Catastrophe Theory, thesis, Nijmegen 1977Google Scholar
  9. 8.
    Poston, T., J. Stewart: Catastrophe Theory and its Applications, Pitman 1978MATHGoogle Scholar
  10. 9.
    Kittel, C.: Physik der Wärme, Oldenbourg 1973Google Scholar
  11. 10.
    Fowler, D.H.: The Riemann-Hugeniot catastrophe and van der Waals equation, in: Towards a Theoretical Biology, Vol. 4 (ed. H. Waddington), Edinburgh 1972Google Scholar
  12. 11.
    Haken, H.: Laserlicht - Ein neues Beispiel für eine Phasenumwandlung? in: Festkörperprobleme X, Pergamon, Vieweg 1970Google Scholar
  13. 12.
    Kohnstamm, P.: Thermodynamik der Gemische, Handbuch der Physik X, Springer 1926Google Scholar
  14. 13.
    Ma, S.K.: Modern Theory of Critical Phenomena, Benjamin 1976Google Scholar
  15. 14.
    Griffiths, R.B.: Thermodynamic models for tricritical points in ternary and quaternary fluid mixtures, Journal of Chemical Physics 60, 195 (1974)ADSCrossRefGoogle Scholar
  16. 15.
    Jahn, I.R., E. Neumann: Phase diagram of the solid state solution system NH4Cl1-xBrx as determined by optical methods, Solid State Communications 12, 721 (1973)ADSCrossRefGoogle Scholar
  17. 16.
    Fatuzzo, E., W.J. Merz: Ferroelectricity, in: Selected Topics in Solid State Physics, Vol. 7, (ed. E.P. Wolfarth), North-Holland 1967Google Scholar
  18. 17.
    Stanley, H.E.: Phase Transitions and Critical Phenomena, Clarendon Press 1971Google Scholar
  19. 18.
    Griffiths, R.B., B. Widom: Multicomponent-fluid tricritical points, Phys. Rev. A 8, 2173 (1973)ADSCrossRefGoogle Scholar
  20. 19.
    Dangelmayr, G., K. Keller: Thermodynamic Example for the Star Catastrophe (in preparation)Google Scholar
  21. 20.
    Imry, Y., D.J. Scalapino, L. Gunther: Phase Transitions in Systems with Coupled Order Parameters, preprints, University of California, Santa Barbara 1973 and Tufts University, Medford 1973Google Scholar
  22. 21.
    Obermayer, E.: private communication, 1978Google Scholar
  23. 22.
    Kadanoff, L.P.: Critical behavior, universality and scaling, in: Proceedings of the Int. School of Phys. ‘Enrico Fermi,’ Varenna 1970 (ed. M.S. Green), Academic Press 1971Google Scholar
  24. 23.
    Keller, K.: Ph.D. thesis, Tübingen 1979Google Scholar
  25. 24.
    Benguigui, L., L.S. Schulman: Topological classification of phase transitions, Phys. Letters 45 A, 315 (1973)ADSGoogle Scholar
  26. 25.
    Stanley, H.E., A. Hankey, M.H. Lee: Scaling, Transformation Methods and Universality, Academic Press 1971Google Scholar
  27. 26.
    Mandelbrot, B.B.: Fractals. Form, Chance and Dimension, Freeman 1977MATHGoogle Scholar
  28. 27.
    Berry, M.V.: Catastrophe and fractal regimes in random waves, and Distribution of modes in fractal resonators, this volumeGoogle Scholar
  29. 28.
    Zeeman, E.C.: The Umbilic Bracelet and the Double-Cusp Catastrophe, in: Structural Stability, the Theory of Catastrophes and Application in the Sciences, (ed. P. Hilton), Lecture Notes in Math. 525, Springer 1976Google Scholar
  30. 29.
    Callahan, J.: Special Bifurcations of the Double Cusp, preprint, University of Warwick 1978Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1979

Authors and Affiliations

  • K. Keller
    • 1
  • G. Dangelmayr
    • 1
  • H. Eikemeier
    • 1
  1. 1.Institute for Information SciencesUniversity of TübingenTübingenFed. Rep. of Germany

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