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.
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References
Bardeen, J., Cooper, L., Schrieffer, J.: Theory of superconductivity. Phys. Rev. 108,1175–1204 (1957)
Bensoussan, A., Temam, R.: Equations aux derivees partielle stochastiques non lin-earies (1) Isr. J. Math. 11, 95–129 (1972)
Bensoussan, A., Temam, R.: Equations stochastiques du type Navier—Stokes. J. Funct.Ana. 13, 195–222 (1973)
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)
Chapman, J., Du, Q., Gunzburger, M.: A model for variable thickness superconducting thin films. Z. Angew. Math. Phys. 47, 410–431 (1996)
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)
Chapman, J., Howinson, S., Ockendon, J.: Macroscopic models for superconductivity.SIAM Review 34, 529–560 (1992)
Daletskii, Yu.: Infinite dimensional elliptic operators and connected with them parabolic equations (in Russian). Uspekhi Mat. Nauk 22, 3–54 (1967)
Deang, J., Du, Q., Gunzburger, M.: Stochastic dynamics of Ginzburg—Landau vortices in superconductors. Phys. Rev. B 64, 052506 (2001)
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)
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)
Du, Q., Gray, P.: High-kappa limit of the time dependent GinzburgLandau model for superconductivity. SIAM J. Appl. Math. 56, 1060–1093 (1996)
Du, Q., Gunzburger, M., Peterson, J.: Analysis and approximation of the GinzburgLandau model of superconductivity. SIAM Review 34, 54–81 (1992)
E, W., Mattingly, J., Sinai, Y.: Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equation. Commun. Math. Phys. 224, 83–106 (2001)
Filippov, A., Radievsky, A., Zelster, A.: Nucleation at the fluctuation induced first order phase transition to superconductivity. Phy. Lett. A 192, 131–136 (1994)
Flandoli, F., Maslowski, B.: Ergodicity of the 2D Navier-Stokes equation under random perturbations. Commun. Math. Phys. 172., 119–141 (1995)
Fursikov, A.: Optimal Control of Distributed Systems. Theory and Applications. Am.Math. Soc, Providence, RI (1999)
Gikhman, I., Skorokhod, A.: Introduction to the Theory of Random Processes. Dover,New York (1969)
Gikhman, I., Skorokhod, A.: The Theory of Stochastic Processes. Springer, Berlin (1974)
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)
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)
Hohenberg, P., Halperin, B.: Theory of dynamic critical phenomena. Rev. Mod. Phy.49, 435–479 (1977)
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)
Krylov, N.: Introduction to the Theory of Diffusion Processes. Am. Math. Soc, Providence, RI (1996)
Krylov, N.: SPDEs in L q((0, t], L p) spaces. Electron. J. Probab. 5, 1–29 (2000)
Krylov, N.: Introduction to the Theory of Random Processes. Am. Math. Soc, Providence, RI(2002)
Krylov, N., Rozovskii, B.: Stochastic evolution equation. J. Sov. Math. 16, 1233–1277 (1981)
Kuksin, S.: Randomly Forced Nonlinear PDEs and Statistical Hydrodynamics in 2 Space Dimensions. Zurich Lectures in Advanced Mathematics, European Math. Soc (EMS), Zurich (2006)
Kuksin, S., Shirikyan, A.: Stochastic dissipative PDE's and Gibbs measures. Commun.Math. Phys. 213, 291–330 (2000)
Larkin, A.: Effect of inhomogeneities on the structure of the mixed state of superconductors. Sov. Phys. JETP 31, 784–786 (1970)
Lions, J.-L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications I. Dunod, Paris (1968)
Mikulevicius, R., Rozovskii, B.: Stochastic Navier—Stokes Equations for Turbulent Flows. Preprint (2003)
Pardoux, E.: Sur le equations aux derivees partielles stochastiques monotones. C. R.Acad. Sci. 275, A101–A103 (1972)
Rozovskii, B.: Stochastic evolution systems. Kluwer, Dordrecht (1990)
Sasik, R., Bettencourt, L., Habib, S.: Thermal vortex motion in a two-dimensional condensate. Phys. Rev. B 62, 1238–1243 (2000)
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)
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)
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)
Skocpol, W., Tinkham, M.: Fluctuations near superconducting phase transitions. Rep.Progr. Phys. 38, 1049–1097 (1975)
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)
Tinkham, M.: Introduction to Superconductivity. McGraw-Hill, New York (1975)
Troy, R., Dorsey, A.: Transport properties and fluctuations in type-II superconductors near Hc2. Phys. Rev. B 47, 2715–2724 (1993)
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)
Vishik, M., Fursikov, A.: Mathematical Problems of Statistical Hydromechanics.Kluwer, Boston (1988)
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)
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Fursikov, A., Gunzburger, M., Peterson, J. (2009). The Ginzburg-Landau Equations for Superconductivity with Random Fluctuations. In: Isakov, V. (eds) Sobolev Spaces in Mathematics III. International Mathematical Series, vol 10. Springer, New York, NY. https://doi.org/10.1007/978-0-387-85652-0_2
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