Noura Dawass

3.3. F INITE - SIZE EFFECTS OF RADIAL DISTRIBUTION FUNCTIONS 3 45 finite subvolumes effects. For the finite effects of RDFs, a correction based on a relation from the work of Ben-Naim [13] is used, that strictly only applies when r →∞ , g αβ ( r ) = g ∞ αβ ( r ) − 1 L 3 box µ δ αβ ρ α + G ∞ αβ ¶ (3.7) The application of Eq. (3.7) implies that the difference between g αβ ( r ) and g ∞ αβ ( r ) in Eq. (3.7) is independent of r (for all values of r ). In the results section, we compare the validation of all RDF corrections over the whole range of r . When including the RDF correction (Eq. (3.7) ) in the L.H.S of Eq. (1.18) the following expression for the finite KBIs is obtained [83] , G αβ ( V , L 3 box ) = 1 V Z V Z V h g ∞ αβ ( r 12 ) − 1 i d r 1 d r 2 − V L 3 box µ δ αβ ρ α + G ∞ αβ ¶ (3.8) The effect of the finite size of the subvolume, V , is accounted for by considering the boundary effects considered through the function Q (Eq. (1.27) ). The dou- ble integral in Eq. (3.8) , R V R V , is expanded to account for the other integration domains, R V R L 3 box and R V R L 3 box − V . As explained earlier in sections 1.3.2 and 3.2, particles in a layer outside V in the volume L 3 box − V contribute to the double in- tegral R V R V . This contribution scales with the surface area, S , of the subvolume. Considering the finite subvolume effect and using S / V ∝ 1/ V 1/3 , we have G αβ ( V , L 3 box ) = 1 V Z V Z L 3 box h g ∞ αβ ( r 12 ) − 1 i d r 1 d r 2 − V L 3 box µ δ αβ ρ α + G ∞ αβ ¶ + C αβ V 1/3 (3.9) where C αβ is a constant that is unique for each thermodynamic state of the sys- tem. Cortes-Huerto et al. [83] restrict the volume V between V ζ and L 3 box , where V ζ = 4 πζ 3 /3. As a result of the values of r being always larger than ζ , the value of g ∞ αβ ( r 12 ) is set to one. Additionally, it is assumed that the system is transition- ally invariant and the transformation r 2 → r = r 1 − r 2 applies which transforms the integrals in Eq. (3.9) to the ones in the original KBIs expression (Eq. (1.25) ). Applying these assumptions, the following expression for KBIs for finite subvol- umes was derived [83] , G V αβ = G ∞ αβ µ 1 − V V box ¶ − V V box δ αβ ρ α + C αβ V 1/3 (3.10) C αβ is a constant originating from the scaling of G V αβ with A / V , and it is specific to each thermodynamic state. By defining λ = ( V / V box ) 1/3 , we can write: