Ginzburg-Landau、XY自旋系统与螺型位错能的变分等价性
Variational equivalence between Ginzburg-Landau, XY spin systems and screw dislocations energies
摘要：
我们介绍并讨论了晶体中XY自旋系统和螺旋位错的离散二维模型。我们证明，作为晶格间距$\e$趋于零，在这些模型中的相关能量表现得像一个自由能在复杂的金斯堡-朗道超导理论，证明在一个严格的数学语言之间的类比晶体中的螺旋位错和超导体中的涡旋。为此，我们引入了一个概念的渐近变分等价的泛函之间的家庭的$Γ$收敛的框架。然后，我们证明，在几个标度制度，复杂的金斯堡-朗道，XY自旋系统和螺旋位错能量泛函变分等价。利用位错和涡旋之间的这种等价性，我们可以得到关于螺旋位错在$|\log\e|^2$能量区域中的渐近行为的新结果。
英文摘要：
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 $Γ$-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 $|\log\e|^2$ energetic regime.
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