We consider extended slow-fast systems of N interacting diffusions. The typical behavior of the empirical density is described by a nonlinear McKean–Vlasov equation depending on $$\varepsilon $$ ε , the scaling parameter separating the time scale of the slow variable from the time scale of the fast variable. Its atypical behavior is encapsulated in a large N Large Deviation Principle with a rate functional $${{\mathscr {I}}}^\varepsilon $$ I ε . We study the $$\Gamma $$ Γ -convergence of $${\mathscr {I}}^\varepsilon $$ I ε as $$\varepsilon \rightarrow 0$$ ε → 0 and show it converges to the rate functional appearing in the Macroscopic Fluctuations Theory for diffusive systems.

中文翻译：

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从波动动力学到波动流体动力学：A $$\Gamma $$-大偏差泛函方法的收敛

我们考虑扩展的 N 个相互作用扩散的慢-快系统。经验密度的典型行为由非线性 McKean-Vlasov 方程描述，该方程取决于 $$\varepsilon $$ ε ，该标度参数将慢变量的时间尺度与快变量的时间尺度分开。它的非典型行为被封装在一个大 N 大偏差原则中，其速率函数为 $${{\mathscr {I}}}^\varepsilon $$ I ε 。我们研究了 $$\mathscr {I}}^\varepsilon $$ I ε 的 $$\Gamma $$ Γ -收敛为 $$\varepsilon \rightarrow 0$$ ε → 0 并表明它收敛到速率泛函出现在扩散系统的宏观涨落理论中。