论文标题
高空维度中凸体的排除体积：病毒系数和连续性渗透的应用
Exclusion Volumes of Convex Bodies in High Space Dimensions: Applications to Virial Coefficients and Continuum Percolation
论文作者
论文摘要
使用凸几何的混合体积和QuermassIntegrals的概念，我们得出了适用于任何空间维度的一般凸体的排除量的精确公式，包括旋转平均的排除量和相同方向。我们表明，对于任何$ d $，$ v（k）$是$ k $的体积，我们在所有凸面中的无尺寸排除卷$ v_ {ex}（k）/v（k）$，无论是随机定向还是均匀定向，其中$ v（k）$是$ k $的量。当身体具有相同的方向时，单纯胶将任何$ d $的无量纲排除量最大化，而$ 2 $ d $渐近缩放的行为为$ 2^{2d}/d^{3/2} $，这与面向那样的$ 2^d $形成了鲜明对比。我们为许多非球形凸体以及较低维体的物体提供明确的QuermassIntegrals公式。这些结果被用来确定尺寸2至12的这些形状的旋转平均排除量。尽管球体是具有最小无量纲的最小排除体积的形状，而在认为足够紧凑的凸体中，单纯X具有最大尺寸的尺寸排除体积，具有缩放量的最大尺寸排除体积，其缩放为$ 2^^1.6666666666666666666666666666666666666666666666666666666666666666618; ldots。我们还确定了上述硬粒子的相应第二个病毒系数$ b_2（k）$，并确定作者先前先前衍生出的连续性渗透阈值$η_c$的计算估计值。我们推测，重叠的球体具有$η_c$的最大价值，在所有相同的非零体积凸凸框中，以$ d \ ge 2 $（随机或均匀为导向）重叠，并且在所有相同的，定向的非零量式凸孔的均值$ plapping siblice $ d $ d $ d. $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $中。
Using the concepts of mixed volumes and quermassintegrals of convex geometry, we derive an exact formula for the exclusion volume for a general convex body that applies in any space dimension, including both the rotationally-averaged exclusion volume and with the same orientation. We show that the sphere minimizes the dimensionless exclusion volume $v_{ex}(K)/v(K)$ among all convex bodies, whether randomly oriented or uniformly oriented, for any $d$, where $v(K)$ is the volume of $K$. When the bodies have the same orientation, the simplex maximizes the dimensionless exclusion volume for any $d$ with a large-$d$ asymptotic scaling behavior of $2^{2d}/d^{3/2}$, which is to be contrasted with the scaling of $2^d$ for the sphere. We present explicit formulas for quermassintegrals for many nonspherical convex bodies as well as as well as lower-dimensional bodies. These results are utilized to determine the rotationally-averaged exclusion volume for these shapes for dimensions 2 through 12. While the sphere is the shape possessing the minimal dimensionless exclusion volume, among the convex bodies considered that are sufficiently compact, the simplex possesses the maximal dimensionless exclusion volume with a scaling behavior of $2^{1.6618\ldots d}$. We also determine the corresponding second virial coefficient $B_2(K)$ of the aforementioned hard hyperparticles and compute estimates of the continuum percolation threshold $η_c$ derived previously by the authors. We conjecture that overlapping spheres possess the maximal value of $η_c$ among all identical nonzero-volume convex overlapping bodies for $d \ge 2$, randomly or uniformly oriented, and that, among all identical, oriented nonzero-volume convex bodies, overlapping simplices have the minimal value of $η_c$ for $d\ge 2$.
