Variational description of bulk energies for bounded and unbounded spin systems

Roberto Alicandro (1) , Marco Cicalese (2) , Antoine Gloria (3)

1 DAEIMI - Dipartimento di Automazione Elettromagnetismo Ingegneria dell'Informazione Matematica Industriale
2 Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”
3 SIMPAF - SImulations and Modeling for PArticles and Fluids

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.

Domaines: Analyse numérique [math.NA]

Roberto Alicandro, Marco Cicalese, Antoine Gloria. Variational description of bulk energies for bounded and unbounded spin systems. Nonlinearity, 2008, 21, pp.1881-1910. ⟨10.1088/0951-7715/21/8/008⟩. ⟨hal-00766738⟩