Department of Chemistry, Princeton Center for Theoretical Science, Princeton Institute for the Science and Technology of Materials, and Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544, USA.
J Chem Phys. 2012 Feb 7;136(5):054106. doi: 10.1063/1.3679861.
We show analytically that the [0, 1], [1, 1], and [2, 1] Padé approximants of the mean cluster number S for both overlapping hyperspheres and overlapping oriented hypercubes are upper bounds on this quantity in any Euclidean dimension d. These results lead to lower bounds on the percolation threshold density η(c), which become progressively tighter as d increases and exact asymptotically as d → ∞, i.e., η(c) → 2(-d). Our analysis is aided by a certain remarkable duality between the equilibrium hard-hypersphere (hypercube) fluid system and the continuum percolation model of overlapping hyperspheres (hypercubes). Analogies between these two seemingly different problems are described. We also obtain Percus-Yevick-like approximations for the mean cluster number S in any dimension d that also become asymptotically exact as d → ∞. We infer that as the space dimension increases, finite-sized clusters become more ramified or "branch-like." These analytical estimates are used to assess simulation results for η(c) up to 20 dimensions in the case of hyperspheres and up to 15 dimensions in the case of hypercubes. Our analysis sheds light on the radius of convergence of the density expansion for S and naturally leads to an analytical approximation for η(c) that applies across all dimensions for both hyperspheres and oriented hypercubes. Finally, we describe the extension of our results to the case of overlapping particles of general anisotropic shape in d dimensions with a specified orientational probability distribution.
我们分析证明，在任何欧几里得维度 d 中，重叠超球体和重叠有向超立方体的平均簇数 S 的 [0, 1]、[1, 1] 和 [2, 1] Padé 逼近是该数量的上界。这些结果导致了渗流阈值密度 η(c) 的下界，随着 d 的增加，下界变得越来越紧，并且在 d → ∞ 时精确渐近，即 η(c) → 2(-d)。我们的分析得到了平衡硬超球体（超立方体）流体系统与重叠超球体（超立方体）的连续渗流模型之间的某种显著对偶性的帮助。描述了这两个看似不同问题之间的相似之处。我们还获得了任何维度 d 中的平均簇数 S 的 Percus-Yevick 型逼近，该逼近也随着 d → ∞ 变得渐近精确。我们推断，随着空间维度的增加，有限大小的簇变得更加分支或“分支状”。这些分析估计用于评估高达 20 维超球体和高达 15 维超立方体的 η(c)的模拟结果。我们的分析揭示了 S 的密度展开的收敛半径，并自然导致适用于超球体和有向超立方体的所有维度的 η(c)的解析逼近。最后，我们描述了我们的结果在具有指定定向概率分布的 d 维一般各向异性形状的重叠粒子情况下的扩展。
