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Variational description of bulk energies for bounded and unbounded spin systems

Abstract : We study the asymptotic behaviour of a general class of discrete energies defined on functions $u:\alpha \in\epsilon \Z^N\cap\Omega\mapsto u(\alpha)\in\mathbb{R}^m$ of the form $E_\epsilon(u)=\sum_{\alpha,\beta \in \epsilon \mathbb{Z}^N\cap\Omega} \epsilon^N g_\epsilon(\alpha,\beta,u(\alpha),u(\beta))$, as the mesh size $\epsilon$ goes to $0$. We prove that under general assumptions, that cover the case of bounded and unbounded spin systems in the thermodynamic limit, the variational limit of $E_\epsilon$ has the form $E(u)=\int_{\Omega}g(x,u(x))dx$. The cases of homogenization and of non-pairwise interacting systems (e.g. multiple-exchange spin-systems) are also discussed.
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https://hal.inria.fr/hal-00766738
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Submitted on : Tuesday, December 18, 2012 - 5:42:33 PM
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Roberto Alicandro, Marco Cicalese, Antoine Gloria. Variational description of bulk energies for bounded and unbounded spin systems. Nonlinearity, IOP Publishing, 2008, 21, pp.1881-1910. ⟨10.1088/0951-7715/21/8/008⟩. ⟨hal-00766738⟩

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