Noura Dawass

2.2. N UMERICAL COMPUTATION OF w ( x ) 2 25 and for spheroids and cuboids with different aspect ratios. From this, finite vol- ume KBIs for a liquid with a model RDF are computed for various shapes, both in 2 D and 3 D . Our findings are summarized in section 2.4. 2.2. N UMERICAL COMPUTATION OF w ( x ) In this section, we present a numerical method to compute the function w ( x ) for convex subvolumes. Table 2.1 shows a schematic representation of the shapes studied here. For cuboids and spheroids, w ( x ) depends on the aspect ratio, a , and the function w ( x ) is computed for each a . To find w ( x ), we first compute w ( r ) and then normalize the distance r us- ing the maximum distance between two points in the subvolume, L max (see Ta- ble 2.1) , so x = r / L max . The function w ( r ) is proportional to the probability dis- tribution function p ( r ) for finding two points inside the subvolume, V , sepa- rated by distance r [74] . Therefore, by construction we obtain w ( r = 0) = 1 and w ( r = L max ) = 0, so consequently w ( x = 0) = 1 and w ( x = 1) = 0. To compute the probability distribution function p ( r ) numerically, distances between two points inside the subvolume are divided into N bins ( i 1 , i 2 , ... i N ) of equal sizes, sep- arated by ∆ r . Each bin contains all distances between i ∆ r and ( i − 1) ∆ r . As a result of this discretization, we sample the probability distribution p ( i ), which is then used to compute w ( i ). The value of ∆ r has to be chosen such that the func- tion p ( i ) is properly sampled. We find that a small value of ∆ r results in poor statistics, especially in the first few bins. We recommend setting ∆ r to L /100 ( L is the characteristic length of the subvolume, see Table 2.1) . To further improve the statistics, umbrella sampling [22, 119] is implemented for computing w ( x ). This introduces a weightfunction W ( i ) which modifies the sampled distribution of distances.