An open question in studying normal grain growth concerns the asymptotic state to which microstructures converge. In particular, the distribution of grain topologies is unknown. We introduce a thermodynamiclike theory to explain these distributions in two- and three-dimensional systems. In particular, a bendinglike energy $E_i$ is associated to each grain topology $t_i$, and the probability of observing that particular topology is proportional to $[1/s(t_i)]e^{-\beta E_i}$, where $s(t_i)$ is the order of an associated symmetry group and $\beta$ is a thermodynamiclike constant. We explain the physical origins of this approach and provide numerical evidence in support.

中文翻译：

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晶粒生长微结构中拓扑类型的分布。

研究正常晶粒生长的一个悬而未决的问题涉及微观结构收敛到的渐近状态。特别地，晶粒拓扑的分布是未知的。我们引入一种类似热力学的理论来解释二维和三维系统中的这些分布。特别是弯曲的能量${Ë}_{\mathrm{一世}}$ 与每个晶粒拓扑相关 ${Ť}_{\mathrm{一世}}$，并且观察到特定拓扑的概率与 $[\mathrm{1个}/s（{Ť}_{\mathrm{一世}}）]{Ë}^{-\beta {Ë}_{\mathrm{一世}}}$，在哪里 $s（{Ť}_{\mathrm{一世}}）$ 是关联对称组的顺序，并且 $\beta$是类似热力学的常数。我们解释了这种方法的物理起源，并提供了数值证据来支持。