We introduce and discuss discrete two-dimensional models for XY spin systems and screw dislocations in crystals. We prove that, as the lattice spacing E tends to zero, the relevant energies in these models behave like a free energy in the complex Ginzburg-Landau theory of superconductivity, justifying in a rigorous mathematical language the analogies between screw dislocations in crystals and vortices in superconductors. To this purpose, we introduce a notion of asymptotic variational equivalence between families of functionals in the framework of Gamma-convergence. We then prove that, in several scaling regimes, the complex Ginzburg-Landau, the XY spin system and the screw dislocation energy functionals are variationally equivalent. Exploiting such an equivalence between dislocations and vortices, we can show new results concerning the asymptotic behavior of screw dislocations in the vertical bar log epsilon vertical bar(2) energetic regime.

Variational Equivalence Between Ginzburg-Landau, XY Spin Systems and Screw Dislocations Energies / Roberto, Alicandro; Marco, Cicalese; Ponsiglione, Marcello. - In: INDIANA UNIVERSITY MATHEMATICS JOURNAL. - ISSN 0022-2518. - STAMPA. - 60:1(2011), pp. 171-208. [10.1512/iumj.2011.60.4339]

Variational Equivalence Between Ginzburg-Landau, XY Spin Systems and Screw Dislocations Energies

PONSIGLIONE, Marcello
2011

Abstract

We introduce and discuss discrete two-dimensional models for XY spin systems and screw dislocations in crystals. We prove that, as the lattice spacing E tends to zero, the relevant energies in these models behave like a free energy in the complex Ginzburg-Landau theory of superconductivity, justifying in a rigorous mathematical language the analogies between screw dislocations in crystals and vortices in superconductors. To this purpose, we introduce a notion of asymptotic variational equivalence between families of functionals in the framework of Gamma-convergence. We then prove that, in several scaling regimes, the complex Ginzburg-Landau, the XY spin system and the screw dislocation energy functionals are variationally equivalent. Exploiting such an equivalence between dislocations and vortices, we can show new results concerning the asymptotic behavior of screw dislocations in the vertical bar log epsilon vertical bar(2) energetic regime.
2011
analysis of microstructure; calculus of variations; crystals; discrete-to-continuum limits; topological singularities
01 Pubblicazione su rivista::01a Articolo in rivista
Variational Equivalence Between Ginzburg-Landau, XY Spin Systems and Screw Dislocations Energies / Roberto, Alicandro; Marco, Cicalese; Ponsiglione, Marcello. - In: INDIANA UNIVERSITY MATHEMATICS JOURNAL. - ISSN 0022-2518. - STAMPA. - 60:1(2011), pp. 171-208. [10.1512/iumj.2011.60.4339]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/378697
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