The Ginzburg-Landau Equations for Superconductivity with Random Fluctuations

  • Andrei Fursikov
  • Max Gunzburger
  • Janet Peterson
Part of the International Mathematical Series book series (IMAT, volume 10)

Thermal fluctuations and material inhomogeneities have a large effect on superconducting phenomena, possibly inducing transitions to the non-superconducting state. To gain a better understanding of these effects, the Ginzburg—Landau model is studied in situations for which the described physical processes are subject to uncertainty. An adequate description of such processes is possible with the help of stochastic partial differential equations. The boundary value problem of Neumann type for the stochastic Ginzburg—Landau equations with additive and multiplicative white noise is investigated. We use white noise with minimal restriction on its independence property. The existence and uniqueness of weak and strong statistical solutions are proved. Our approach is based on using difference schemes for the Ginzburg— Landau equations.


White Noise Weak Solution Wiener Process Landau Equation Stochastic Problem 
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  1. 1.
    Bardeen, J., Cooper, L., Schrieffer, J.: Theory of superconductivity. Phys. Rev. 108,1175–1204 (1957)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bensoussan, A., Temam, R.: Equations aux derivees partielle stochastiques non lin-earies (1) Isr. J. Math. 11, 95–129 (1972)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bensoussan, A., Temam, R.: Equations stochastiques du type Navier—Stokes. J. Funct.Ana. 13, 195–222 (1973)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chapman, J., Du, Q., Gunzburger, M.: A Ginzburg—Landau type model of superconducting/normal junctions including Josephson junctions. Europ. J. Appl. Math.6,97–114 (1995)MATHMathSciNetGoogle Scholar
  5. 5.
    Chapman, J., Du, Q., Gunzburger, M.: A model for variable thickness superconducting thin films. Z. Angew. Math. Phys. 47, 410–431 (1996)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Chapman, J., Du, Q., Gunzburger, M., Peterson, J.: Simplified Ginzburg—Landau type models of superconductivity in the high kappa, high field limit. Adv. Math. Sci. Appl.5, 193–218 (1995)MATHMathSciNetGoogle Scholar
  7. 7.
    Chapman, J., Howinson, S., Ockendon, J.: Macroscopic models for superconductivity.SIAM Review 34, 529–560 (1992)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Daletskii, Yu.: Infinite dimensional elliptic operators and connected with them parabolic equations (in Russian). Uspekhi Mat. Nauk 22, 3–54 (1967)Google Scholar
  9. 9.
    Deang, J., Du, Q., Gunzburger, M.: Stochastic dynamics of Ginzburg—Landau vortices in superconductors. Phys. Rev. B 64, 052506 (2001)CrossRefGoogle Scholar
  10. 10.
    Deang, J., Du, Q., Gunzburger, M.: Modeling and computation of random thermal fluctuations and material defects in the Ginzburg—Landau model for superconductivity.J. Comput. Phys. 181, 45–67 (2002)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Dorsey, A., Huang, M., Fisher M.: Dynamics of the normal to superconducting vortex-glass transition: Mean-field theory and fluctuations. Phys. Rev. B 45, 523–526 (1992)CrossRefGoogle Scholar
  12. 12.
    Du, Q., Gray, P.: High-kappa limit of the time dependent GinzburgLandau model for superconductivity. SIAM J. Appl. Math. 56, 1060–1093 (1996)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Du, Q., Gunzburger, M., Peterson, J.: Analysis and approximation of the GinzburgLandau model of superconductivity. SIAM Review 34, 54–81 (1992)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    E, W., Mattingly, J., Sinai, Y.: Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equation. Commun. Math. Phys. 224, 83–106 (2001)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Filippov, A., Radievsky, A., Zelster, A.: Nucleation at the fluctuation induced first order phase transition to superconductivity. Phy. Lett. A 192, 131–136 (1994)CrossRefGoogle Scholar
  16. 16.
    Flandoli, F., Maslowski, B.: Ergodicity of the 2D Navier-Stokes equation under random perturbations. Commun. Math. Phys. 172., 119–141 (1995)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Fursikov, A.: Optimal Control of Distributed Systems. Theory and Applications. Am.Math. Soc, Providence, RI (1999)Google Scholar
  18. 18.
    Gikhman, I., Skorokhod, A.: Introduction to the Theory of Random Processes. Dover,New York (1969)Google Scholar
  19. 19.
    Gikhman, I., Skorokhod, A.: The Theory of Stochastic Processes. Springer, Berlin (1974)MATHGoogle Scholar
  20. 20.
    Ginzburg, V., Landau, L.: On the theory of superconductivity (in Russian). Zh.Eksperim. Teor. Fiz. 20, 1064–1082 (1950); English transl.: In: ter Haar, D. (ed.) Men of Physics: L. D. Landau, pp. 138–167. Pergamon, Oxford (1965)Google Scholar
  21. 21.
    Gor–kov, L.: Microscopic derivation of the Ginzburg—Landau equations in the theory of superconductivity (in Russian). Zh. Eksperim. Teor. Fiz. 36, 1918-1923 (1959); English transl.: Sov. Phys.—JETP 9, 1364–1367 (1959)Google Scholar
  22. 22.
    Hohenberg, P., Halperin, B.: Theory of dynamic critical phenomena. Rev. Mod. Phy.49, 435–479 (1977)CrossRefGoogle Scholar
  23. 23.
    Kamppeter, T., Mertens, F., Moro, E., Sanchez, A., Bishop, A.: Stochastic vortex dynamics in two-dimensional easy-plane ferromagnets: Multiplicative versus additive noise. Phys. Rev. B 59, pp.11 349–11 357 (1991)Google Scholar
  24. 24.
    Krylov, N.: Introduction to the Theory of Diffusion Processes. Am. Math. Soc, Providence, RI (1996)Google Scholar
  25. 25.
    Krylov, N.: SPDEs in L q((0, t], L p) spaces. Electron. J. Probab. 5, 1–29 (2000)MathSciNetGoogle Scholar
  26. 26.
    Krylov, N.: Introduction to the Theory of Random Processes. Am. Math. Soc, Providence, RI(2002)Google Scholar
  27. 27.
    Krylov, N., Rozovskii, B.: Stochastic evolution equation. J. Sov. Math. 16, 1233–1277 (1981)MATHCrossRefGoogle Scholar
  28. 28.
    Kuksin, S.: Randomly Forced Nonlinear PDEs and Statistical Hydrodynamics in 2 Space Dimensions. Zurich Lectures in Advanced Mathematics, European Math. Soc (EMS), Zurich (2006)Google Scholar
  29. 29.
    Kuksin, S., Shirikyan, A.: Stochastic dissipative PDE's and Gibbs measures. Commun.Math. Phys. 213, 291–330 (2000)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Larkin, A.: Effect of inhomogeneities on the structure of the mixed state of superconductors. Sov. Phys. JETP 31, 784–786 (1970)Google Scholar
  31. 31.
    Lions, J.-L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications I. Dunod, Paris (1968)Google Scholar
  32. 32.
    Mikulevicius, R., Rozovskii, B.: Stochastic Navier—Stokes Equations for Turbulent Flows. Preprint (2003)Google Scholar
  33. 33.
    Pardoux, E.: Sur le equations aux derivees partielles stochastiques monotones. C. R.Acad. Sci. 275, A101–A103 (1972)MathSciNetGoogle Scholar
  34. 34.
    Rozovskii, B.: Stochastic evolution systems. Kluwer, Dordrecht (1990)MATHGoogle Scholar
  35. 35.
    Sasik, R., Bettencourt, L., Habib, S.: Thermal vortex motion in a two-dimensional condensate. Phys. Rev. B 62, 1238–1243 (2000)CrossRefGoogle Scholar
  36. 36.
    Shirikyan, A.: Ergodicity for a class of Markov processes and applications to randomly forced PDE's. I. Russ. J. Math. Phys. 12, 81–96 (2005)MATHMathSciNetGoogle Scholar
  37. 37.
    Shirikyan, A.: Ergodicity for a class of Markov processes and applications to randomly forced PDE's II. Discrete Contin. Dyn. Syst. Ser. B6, 911–926 (2006)MATHMathSciNetGoogle Scholar
  38. 38.
    Shirikyan, A.: Exponential mixing for randomly forced PDE's: method of coupling.In: Bardos, C., Fursikov, A. Eds. Instability of Models Connected with Fluid Flows II,pp. 155–188. Springer, New York (2008)CrossRefGoogle Scholar
  39. 39.
    Skocpol, W., Tinkham, M.: Fluctuations near superconducting phase transitions. Rep.Progr. Phys. 38, 1049–1097 (1975)CrossRefGoogle Scholar
  40. 40.
    Sobolev, S.L.: Some Applications of Functional Analysis in Mathematical Physics (in Russian). 1st ed. Leningrad State Univ., Leningrad (1950); 3rd ed. Nauka, Moscow (1988); English transl. of the 1st Ed.: Am. Math. Soc., Providence, RI (1963); English transl. of the 3rd Ed. with comments by V. P. Palamodov: Am. Math. Soc., Providence,I (1991)Google Scholar
  41. 41.
    Tinkham, M.: Introduction to Superconductivity. McGraw-Hill, New York (1975)Google Scholar
  42. 42.
    Troy, R., Dorsey, A.: Transport properties and fluctuations in type-II superconductors near Hc2. Phys. Rev. B 47, 2715–2724 (1993)CrossRefGoogle Scholar
  43. 43.
    Ullah, S., Dorsey, A.: The effect of fluctuations on the transport properties of type-II superconductors in a magnetic field. Phys. Rev. B 44, 262–273 (1991)CrossRefGoogle Scholar
  44. 44.
    Vishik, M., Fursikov, A.: Mathematical Problems of Statistical Hydromechanics.Kluwer, Boston (1988)MATHGoogle Scholar
  45. 45.
    Vishik, M., Komech, A,: On solvability of the Cauchy problem for the direct Kol-mogorov equation corresponding to the stochastic Navier—Stokes type equation (in Russian). In: Complex Analysis and its Applications, pp. 126–136. Nauka, Moscow (1978)Google Scholar

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© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Moscow State UniversityMoscowRussia
  2. 2.School of Computational SciencesFlorida State UniversityTallahasseeUSA

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