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Markov Models of Coarsening in Two-Dimensional Foams with Edge Rupture
Journal of Nonlinear Science ( IF 2.6 ) Pub Date : 2021-03-31 , DOI: 10.1007/s00332-021-09696-3
Joseph Klobusicky

We construct Markov processes for modeling the rupture of edges in a two-dimensional foam. We first describe a network model for tracking topological information of foam networks with a state space of combinatorial embeddings. Through a mean-field rule for randomly selecting neighboring cells of a rupturing edge, we consider a simplified version of the network model in the sequence space \(\ell _1({\mathbb {N}})\) which counts total numbers of cells with \(n\ge 3\) sides (n-gons). Under a large cell limit, we show that number densities of n-gons in the mean field model are solutions of an infinite system of nonlinear kinetic equations. This system is comparable to the Smoluchowski coagulation equation for coalescing particles under a multiplicative collision kernel, suggesting gelation behavior. Numerical simulations reveal gelation in the mean-field model, and also comparable statistical behavior between the network and mean-field models.



中文翻译:

带有边缘破裂的二维泡沫粗化的马尔可夫模型

我们构建马尔可夫过程,以对二维泡沫中的边缘破裂进行建模。我们首先描述一种网络模型,用于跟踪具有组合嵌入状态空间的泡沫网络的拓扑信息。通过均值场规则随机选择破裂边缘的相邻单元,我们考虑了序列空间\(\ ell _1({\ mathbb {N}})\)中网络模型的简化版本,该序列计算了总数带有\(n \ ge 3 \)面(n-边)的像元。在较大的像元限制下,我们表明n的数密度平均场模型中的-角是无限个非线性动力学方程组的解。该系统可与Smoluchowski凝聚方程式相比较,该方程用于在乘法碰撞核下聚结颗粒,这表明了凝胶行为。数值模拟显示了平均场模型中的胶凝作用,以及网络模型和平均场模型之间的可比统计行为。

更新日期:2021-03-31
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