Noura Dawass

2.3. S HAPE EFFECTS OF K IRKWOOD –B UFF I NTEGRALS 2 33 grating G ∞ / σ 3 (Eq. (1.25) ). Therefore, w ( r ) has a universal behavior at small distances which leads to the universality of KBIs in Figure 2.4 (a). Using a Taylor expansion around r = 0, one can show that w ( r ) can be written as function of A / V [81] , w ( r ) ≈ µ 1 − r 4 A V + O ( r 2 ) ¶ (2.7) In Figure 2.4 (b), we plot the ratio of w ( r ) using Eq. (2.7) to numerical w ( r ) for a sphere, cube, spheroid with a = 2, and cuboid with a = 2. A subvolume with L = 1 is used. At small distances ( r < 0.1), the ratio is practically 1 for all shapes considered. Eq. (2.7) shows that the function w ( r ) depends on the size r and the ratio A / V . The shape contribution originates from the term O ( r 2 ). There- fore, properties of large subvolumes are independent of shape. This is referred to as the so-called shape thermodynamics limit, where properties of the subvol- ume are dependent on the size but not the shape of the subvolume [123] . In the conventional thermodynamic limit, properties are independent of both size and shape of the subvolume. It is important to note that Eq. (2.7) provides a physical reason for the poor convergence of truncated KBIs (i.e. truncating the integral of Eq. (1.3) ). If we consider a subvolume V with zero surface area A = 0, this will yield the weight function c ( r ) w ( r ) = 4 π r 2 . Substituting this in the expression of KBIs of finite subvolumes (Eq. (1.25) ), one arrives at an expression of KBIs truncated to finite distances, ˆ G αβ ( R ) = Z R 0 d r 4 π r 2 h g ∞ αβ ( r ) − 1 i (2.8) Therefore, truncated KBIs correspond to the nonphysical case of finite–size KBIs with subvolumes V and zero surface area. We verified numerically that findings from computing KBIs using different 3 D shapes apply to KBIs computed using 2 D shapes as well. In Figure 2.5, we show G V / σ 2 vs. σ / L for a circle and a square. For a circle, we compute the KBIs using the analytic function for w ( x ), and also using the numerically function w ( x ). Both functions result in identical values of KBIs for all sizes of the sub- volume. Using circle or square for the shape of the subvolume results in KBIs that converge to the same value of G ∞ / σ 2 .