Geometric measures of uniaxial solids of revolution in ${\mathbb{R}^{4}}$ and their relation to the second virial coefficient

TL;DR

This paper derives analytical formulas for the second virial coefficients of four-dimensional convex solids of revolution, linking particle shape and aspect ratio to excluded volume, with implications for understanding particle interactions in higher dimensions.

Contribution

It provides new analytical expressions for the second virial coefficient of 4D convex solids of revolution using geometric measures, extending prior work to higher dimensions and complex shapes.

Findings

Analytical expressions for second virial coefficients in 4D solids of revolution.

Dependence of excluded volume on aspect ratio and shape details.

Symmetry property of reduced second virial coefficients for ellipsoids of revolution.

Abstract

We provide analytical expressions for the second virial coefficients of hard, convex, monoaxial solids of revolution in ${\mathbb{R}^{4}}$. The excluded volume per particle and thus the second virial coefficient is calculated using quermassintegrals and rotationally invariant mixed volumes based on the Brunn-Minkowski theorem. We derive analytical expressions for the mutual excluded volume of four-dimensional hard solids of revolution in dependence on their aspect ratio $\nu$ including the limits of infinitely thin oblate and infinitely long prolate geometries. Using reduced second virial coefficients $B_2^{\ast}=B_2/V_{\mathrm{P}}$ as size-independent quantities with $V_{\mathrm{P}}$ denoting the $D$-dimensional particle volume, the influence of the particle geometry to the mutual excluded volume is analyzed for various shapes. Beyond the aspect ratio $\nu$, the detailed particle shape influences the reduced second virial coefficients $B_2^{\ast}$. We prove that for $D$-dimensional spherocylinders in arbitrary-dimensional Euclidean spaces ${\mathbb{R}^{D}}$ their excluded volume solely depends on at most three intrinsic volumes, whereas for different convex geometries $D$ intrinsic volumes are required. For $D$-dimensional ellipsoids of revolution, the general parity $B_2^{\ast}(\nu)=B_2^{\ast}(\nu^{-1})$ is proven.