Noura Dawass

4.1. I NTRODUCTION 4 67 F ∞ ≈ Z ∞ 0 [ g ( r ) − 1]4 π r 2 µ − 3 2 r ¶ d r (4.3) It is important to note that the structure of Eq. (4.3) is similar to KBIs in the ther- modynamic limit (Eq. (1.3) ). So, analogous to Eq. (1.28) , F V can be defined as, F V ( L ) = F ∞ + C L (4.4) where C is a constant. For finite systems, F V can be computed using F V ≈ Z L 0 [ g ( r ) − 1] µ − 3 2 r ¶ w ( x )d r (4.5) where the function w ( x ) is provided in Table 1.1 for a spherical subvolume. The similarity between the expression for KBIs (Eq. (1.25) ) and surface term (Eq. (4.3) ) in the thermodynamic limit allows for deriving an estimation for surface effects as in Eq. (4.1) . Using Eq. (4.1) , and Eq. (4.3) an explicit expression for surface effects in the thermodynamic limit, denoted here by F ∞ 2 , is obtained from F ∞ 2 ≈ Z L 0 [ g ( r ) − 1] µ − 3 2 r ¶ u 2 ( r )d r (4.6) with u 2 ( r ) defined in Eq. (4.2) . An alternative method to extrapolate KBIs G V to the thermodynamic limit is to use the scaling of LG V with L , rather than the scaling of G V with 1/ L . The scaling of G V in Eq. (1.28) can be rewritten as LG V ( L ) = G ∞ L + F ∞ (4.7) By fitting the linear part of the scaling of LG V with L , it is possible to obtain G ∞ and F ∞ . Finding the slope and intercepts of a straight line is easier than extrap- olating the linear regime of the scaling of G V with 1/ L . Another advantage of this approach is that an estimation of the surface effects is automatically computed. This estimation can be compared to other available methods for computing F ∞ . In summary, it is shown that three methods are available for estimating G ∞ from integrals of finite subvolumes: 1. Using the scaling of G V (Eq. (1.25) ) with 1/ L . To estimate G ∞ , the linear regime of the scaling is extrapolated to the limit 1/ L → 0. 2. Using the direct extrapolation formula G 2 (Eq. (4.1) ) combined with the function u 2 ( r ) (Eq. (4.2) ). This will converge to G ∞ for large L .