We revisit the classical problem of diffusion of a scalar (or heat) released in a two-dimensional medium with an embedded periodic array of impermeable obstacles such as perforations. Homogenization theory provides a coarse-grained description of the scalar at large times and predicts that it diffuses with a certain effective diffusivity, so the concentration is approximately Gaussian. We improve on this by developing a large-deviation approximation which also captures the non-Gaussian tails of the concentration through a rate function obtained by solving a family of eigenvalue problems. We focus on cylindrical obstacles and on the dense limit, when the obstacles occupy a large area fraction and non-Gaussianity is most marked. We derive an asymptotic approximation for the rate function in this limit, valid uniformly over a wide range of distances. We use finite-element implementations to solve the eigenvalue problems yielding the rate function for arbitrary obstacle area fractions and an elliptic boundary-value problem arising in the asymptotics calculation. Comparison between numerical results and asymptotic predictions confirms the validity of the latter.

中文翻译：

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在一系列障碍中扩散：超越同质化

我们重新审视了二维介质中释放的标量（或热量）扩散的经典问题，其中嵌入了不透水障碍物（例如穿孔）的周期性阵列。均质化理论提供了对大量时间标量的粗粒度描述，并预测它以一定的有效扩散率扩散，因此浓度近似为高斯。我们通过开发大偏差近似来改进这一点，该近似还通过求解一系列特征值问题获得的速率函数捕获浓度的非高斯尾部。我们关注圆柱形障碍物和密集极限，当障碍物占据大面积部分且非高斯性最为明显时。我们推导出该极限中速率函数的渐近近似，在很宽的距离范围内一致有效。我们使用有限元实现来解决产生任意障碍物面积分数的速率函数的特征值问题和渐近计算中出现的椭圆边界值问题。数值结果和渐近预测之间的比较证实了后者的有效性。