Phase Transitions and Hysteresis

  • Martin Brokate
  • Jürgen Sprekels
Part of the Applied Mathematical Sciences book series (AMS, volume 121)


In the previous chapters, the occurrence of hysteresis has been discussed from a mainly phenomenological and mathematical point of view. The attempt to formalize the input-output behaviour of hysteresis loops gave rise to introduce the notion of hysteresis operators, and we studied their properties and differential equations in which they appear. In this approach, we did not pay much attention to the physical circumstances. In particular, we entirely ignored the fact that in nature hysteresis effects are often caused by phase transitions which are accompanied by abrupt changes of some of the involved physical quantities, as well as by the absorption or release of energy in the form of latent heat. The area of the hysteresis loop itself gives a measure for the amount of energy that has been dissipated or absorbed during the phase transformation.


Phase Transition External Field Free Energy Density Stefan Problem Total Free Energy 
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  1. 1.
    For a more detailed treatment of the theory of phase transitions, we refer the reader to the monographs of Stanley (1971) and Ma (1976).Google Scholar
  2. 2.
    See Devonshire (1954).Google Scholar
  3. 4.
    In particular, the second principle of thermodynamics and the so-called principle of material frame indifference impose strong restrictions on the form that the constitutive equations may have. For a detailed discussion, we refer to Müller (1985).Google Scholar
  4. 5.
    For the definition of these quantities, we refer to Müller (1985).Google Scholar
  5. 10.
    For a detailed treatment of the Landau approach to temperature-induced first-order phase transitions, we refer to Chapter IV in Tolédano-Tolédano (1987).Google Scholar
  6. 11.
    Cf. Devonshire (1954).Google Scholar
  7. 13.
    In this respect we refer to the works of Modica (1987), Owen (1989) and Fonseca-Tartar (1989), for instance.Google Scholar
  8. 14.
    See Cahn-Allen (1977).Google Scholar
  9. 15.
    This equation was first considered in Cahn-Hilliard (1958).Google Scholar
  10. 16.
    See Remark 6.4.4 at the end of Chapter 6.Google Scholar
  11. 19.
    Actually, in Alt-Pawlow (1990,1992a) a more general form has been assumed for both the fluxes j and q; see also de Groot-Mazur (1984).Google Scholar
  12. 20.
    Other examples to which this theory applies are the lattice gas or the metallic order-disorder transition, see Penrose-Fife (1990).Google Scholar
  13. 24.
    Cf. Caginalp (1986,1989,1991), Caginalp-Lin (1987) and Caginalp-Fife (1988), for non-conserving dynamics, and Caginalp (1990), for conserving dynamics.Google Scholar
  14. 25.
    Problems of this type are usually referred to as free boundary problems, with the unknown interface Γ(t) playing the role of the free boundary. For surveys on the existing mathematical theory of free boundary problems, we refer the reader to Fasano-Primicerio (1983a, 1983b), Bossavit-Damlamian-Frémond (1985a, 1985b), Hoffmann-Sprekels (1990a, 1990b) and Chadam-Rasmussen (1993a, 1993b, 1993c).Google Scholar
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    Cf. Gibbs (1948).Google Scholar
  16. 28.
    Recently, some of these formal arguments have been made rigorous for certain cases; in this respect, we refer the reader to Stoth (1996a, 1996b) and Soner (1994).Google Scholar
  17. 29.
    For the explicit form of the constant c 0, we refer to Caginalp (1990).Google Scholar
  18. 30.
    For recent results, see Luckhaus (1990), Chen-Reitich (1992) and Soner (1994).Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1996

Authors and Affiliations

  • Martin Brokate
    • 1
  • Jürgen Sprekels
    • 2
  1. 1.Mathematisches SeminarChristian-Albrechts-Universität zu KielKielGermany
  2. 2.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany

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