Noura Dawass

1 12 I NTRODUCTION V r V ’ ( r ) Figure 1.3: Schematic representation of the definition of the function τ ( r ) (Eq. (1.22) ), the overlap volume between two spheres separated by a distance r . r depends on r 1 . In this case, the double volume integrals in Eq. (1.18) are re- duced to a single radial integral by rewriting the left hand side (L.H.S) of Eq. (1.18) as G V αβ = Z ∞ 0 d r c ( r ) w ( r ) ( g αβ ( r ) − 1) (1.19) where d r c ( r ) is a hyperspherical volume element and w ( r ) is a purely geometric function characteristic of the volume V defined as: w ( r ) ≡ 1 V Z V d r 1 Z V d r 2 δ ( r − | r 1 − r 2 | ) (1.20) Once the function w ( r ) is known, the 2D dimensional integral of Eq. (1.18) re- duces to the one-dimensional integral of Eq. (1.19) , and the expression for KBIs for finite subvolumes is obtained. For the calculation of w ( r ), we first rewrite the L.H.S of Eq. (1.18) as G V αβ = 1 V Z V d r τ ( r )( g αβ ( r ) − 1) (1.21) where the integral is over all of space and τ ( r ) ≡ Z V d r 1 Z V d r 2 δ ( r − r 1 + r 2 ) (1.22) The function τ ( r ) has a simple geometrical interpretation: it is the overlap be- tween the subvolume V and the same subvolume V shifted by r . This may be seen by making the variable substitution r 0 2 = r 2 + r which yields τ ( r ) ≡