In this paper we consider the $XY$ ($N$-dimensional possibly anisotropic) spin type model and, by comparison with a Ginzburg-Landau type functional, we perform a variational analysis in the limit when the number of particles diverges. In particular we show how the appearance of vortex-like singularities can be described by properly scaling the energy of the system through a $\Gamma$-convergence procedure. We also address the problem in the case of long range interactions and solve it in two-dimensions.
Variational analysis of the asymptotics of the XY model / Alicandro, R.; Cicalese, Marco. - In: ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS. - ISSN 0003-9527. - STAMPA. - 192:3(2009), pp. 501-539. [10.1007/s00205-008-0146-0]
Variational analysis of the asymptotics of the XY model
R. ALICANDRO;CICALESE, MARCO
2009
Abstract
In this paper we consider the $XY$ ($N$-dimensional possibly anisotropic) spin type model and, by comparison with a Ginzburg-Landau type functional, we perform a variational analysis in the limit when the number of particles diverges. In particular we show how the appearance of vortex-like singularities can be described by properly scaling the energy of the system through a $\Gamma$-convergence procedure. We also address the problem in the case of long range interactions and solve it in two-dimensions.| File | Dimensione | Formato | |
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