Abstract We consider energies on a periodic set ℒ {\mathcal{L}} of the form ∑ i , j ∈ ℒ a i ⁢ j ε ⁢ | u i - u j | {\sum_{i,j\in\mathcal{L}}a^{\varepsilon}_{ij}\lvert u_{i}-u_{j}\rvert} , defined on spin functions u i ∈ { 0 , 1 } {u_{i}\in\{0,1\}} , and we suppose that the typical range of the interactions is R ε {R_{\varepsilon}} with R ε → + ∞ {R_{\varepsilon}\to+\infty} , i.e., if | i - j | ≤ R ε {\lvert i-j\rvert\leq R_{\varepsilon}} , then a i ⁢ j ε ≥ c > 0 {a^{\varepsilon}_{ij}\geq c>0} . In a discrete-to-continuum analysis, we prove that the overall behavior as ε → 0 {\varepsilon\to 0} of such functionals is that of an interfacial energy. The proof is performed using a coarse-graining procedure which associates to scaled functions defined on ε ⁢ ℒ {\varepsilon\mathcal{L}} with equibounded energy a family of sets with equibounded perimeter. This agrees with the case of equibounded R ε {R_{\varepsilon}} and can be seen as an extension of coerciveness result for short-range interactions, but is different from that of other long-range interaction energies, whose limit exits the class of surface energies. A computation of the limit energy is performed in the case ℒ = ℤ d {\mathcal{L}=\mathbb{Z}^{d}} .

中文翻译：

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长程晶格系统中粗粒度的紧凑性

摘要 我们考虑周期集合 ℒ {\mathcal{L}} 上的能量，形式为 ∑ i , j ∈ ℒ ai ⁢ j ε ⁢ | ui - uj | {\sum_{i,j\in\mathcal{L}}a^{\varepsilon}_{ij}\lvert u_{i}-u_{j}\rvert} ，定义在自旋函数 ui ∈ { 0 , 1 } {u_{i}\in\{0,1\}} ，我们假设相互作用的典型范围是 R ε {R_{\varepsilon}} 与 R ε → + ∞ {R_{\varepsilon}\ to+\infty} ，即如果 | 我 - j | ≤ R ε {\lvert ij\rvert\leq R_{\varepsilon}} ，则 ai ⁢ j ε ≥ c > 0 {a^{\varepsilon}_{ij}\geq c>0} 。在离散到连续的分析中，我们证明了这种泛函 ε → 0 {\varepsilon\to 0} 的整体行为是界面能的行为。该证明是使用粗粒度程序执行的，该程序将定义在具有等界能量的 ε ⁢ ℒ {\varepsilon\mathcal{L}} 上的缩放函数与等界周长的集合相关联。这与等界 R ε {R_{\varepsilon}} 的情况一致，可以看作是短程相互作用的矫顽力结果的扩展，但与其他长程相互作用能的极限不同，后者的极限退出类的表面能。在 ℒ = ℤ d {\mathcal{L}=\mathbb{Z}^{d}} 的情况下执行极限能量的计算。