Noura Dawass

2.4. C ONCLUSIONS 2 37 2.4. C ONCLUSIONS We have introduced a method for computing KBIs for finite subvolumes of arbi- trary convex shapes. This requires a numerical method to obtain the geometrical function w ( x ), which is needed when computing KBIs from RDFs. We showed that w ( x ) is related to the probability of finding two particles inside a subvolume V at a certain distance, and we presented a numerical scheme based on um- brella sampling MC for this. The numerical method was verified by comparing the results with analytic expressions for hyperspherical subvolumes in 1 D (line), 2 D (circle), and 3 D (sphere). The method was used to compute the function w ( x ) for the following subvolumes: square, cube, and spheroids and cuboids with different aspect ratios. These functions are tabulated in a data repository (see Ref. [118] ). We computed KBIs for subvolumes with different shapes, using an analytic RDF model representing an isotropic liquid. In the thermodynamic limit, KBIs are independent of the shape of the subvolume, and the approach to the thermodynamic limit only depends on the area over volume ratio, and not the shape of the subvolume. This is due to the observation that for small r , w ( r ) is only a function of r and the surface to volume ratio of the subvolume, and in- dependent of the shape of the subvolume. At small r , a universal expression for w ( r ) was derived and used to show how truncated KBIs (Eq. (2.8) ) correspond to the nonphysical case of a finite volume with zero surface area. It would be interesting to investigate whether or not these findings are applicable to non- isotropic liquids.