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.

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。