The notion of Energy-Dissipation-Principle convergence (EDP-convergence) is used to derive effective evolution equations for gradient systems describing diffusion in a structure consisting of several thin layers in the limit of vanishing layer thickness. The thicknesses of the sublayers tend to zero with different rates and the diffusion coefficients scale suitably. The Fokker–Planck equation can be formulated as gradient-flow equation with respect to the logarithmic relative entropy of the system and a quadratic Wasserstein-type gradient structure. The EDP-convergence of the gradient system is shown by proving suitable asymptotic lower limits of the entropy and the total dissipation functional. The crucial point is that the limiting evolution is again described by a gradient system, however, now the dissipation potential is not longer quadratic but is given in terms of the hyperbolic cosine. The latter describes jump processes across the thin layers and is related to the Marcelin-de Donder kinetics.

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通过广义梯度系统和EDP收敛在薄结构中有效扩散

能量耗散原理收敛（EDP收敛）的概念用于为梯度系统导出有效的演化方程，该方程描述了在由薄层组成的结构中在消失层厚度的极限范围内的扩散。子层的厚度以不同的速率趋于零，并且扩散系数适当地缩放。关于系统和二次Wasserstein型梯度结构的对数相对熵，可以将Fokker-Planck方程公式化为梯度流方程。通过证明熵和总耗散函数的合适的渐近下限来显示梯度系统的EDP收敛。关键点是，极限演化再次由梯度系统描述，但是，现在，耗散势不再是二次的，而是根据双曲余弦给出的。后者描述了跨薄层的跃迁过程，并且与Marcelin-de Donder动力学有关。