Noura Dawass

3.4. R ESULTS AND DISCUSSION 3 53 each plot a dashed line is added to indicate values of c 2 and difference% that we consider acceptable. Combinations of x and y that fall below the line are con- sidered to give poor estimations of the KBIs. Specifically, the following threshold values are considered, 8 for − ln(1 − c 2 ) and 5 for − ln(difference) corresponding to c 2 = 0.9997 and difference = 0.7%, respectively. Using a system with L box = 40 provides sufficient linear regime that results in very low values of the difference % between KBIs computed and KBIs from very large systems. Figure 3.5 shows that choosing lower x and y is favorable. The same observations are made when studying L box = 20 (Figure 3.6) . Using a large value of y could result in an extrapolation range that includes the diverging part of the G V and 1/ R scaling. Figure 3.5 and 3.6 s how that, for a given x , the values of − ln(1 − c 2 ) and − ln(difference) start decreasing after a specific y value. In general, y should not be larger than 4 σ . Finally, for L box = 10, Figure 3.7 indicates that there is a range that is sufficiently linear. However, very few possibilities of x and y combinations provide low differences, i.e. the linear regime extends to a value that deviates from the integral at the thermodynamic limit. Besides examining the linear range for each system size individually, it is possible to investigate the effect of the system size by comparing Figure 3.5, 3.6 and 3.7. Mainly, larger box sizes provide a longer linear regime and smaller dif- ferences between KBIs computed from these boxes and integrals from very large boxes. When performing the study for the other integrals as well, G V 11 and G V 12 , a few general findings could be observed. First, it is recommended to choose x larger than 1, to avoid any fluctuations at small distances. This corresponds to discarding at least 1 σ from the G V and 1/ V scaling line. As for the end of the ex- trapolation range, y , taking short distances ensures that the extrapolation range does not extend to the end of the line, where the values of G V αβ are diverging. We find that not exceeding b = a + 4 σ ensures that the linear fit is acceptable. This applies to boxes with L box larger than 20. Finally, while it is important to use lin- ear ranges with high values of the correlation coefficient, c 2 that does not always result in correct estimations of the KBIs at the thermodynamic limit, especially for small simulation boxes. In the following section, inaccuracies in KBIs result- ing from finite sizes related to RDFs are discussed.