*Published Paper*

**Inserted:** 20 jul 2007

**Last Updated:** 25 jul 2021

**Journal:** Nonlinearity

**Volume:** 21

**Number:** 8

**Pages:** 1881-1910

**Year:** 2008

**Abstract:**

We study the asymptotic behaviour of a general class of discrete energies defined on functions $u:\alpha\in\varepsilon Z^N\cap\Omega\mapsto u(\alpha)\in R^m$ of the form $E_\varepsilon(u)=\sum_{\alpha,\beta \in \varepsilon Z^N\cap\Omega} \varepsilon^N g_\varepsilon(\alpha,\beta,u(\alpha),u(\beta))$, as the mesh size $\varepsilon$ goes to $0$. We prove that under general assumptions, that cover the case of bounded and unbounded spin system in the thermodynamic limit, the variational limit of $E_\varepsilon$ has the form $E(u)=\int_{\Omega}g(x,u(x))dx$. The case of homogenization and that of non-pairwise interacting systems (e.g. multiple-exchange spin-systems) is also discussed.

**Keywords:**
Homogenization, $\Gamma$-convergence, spin systems

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