Noura Dawass

4.3. R ESULTS 4 71 Table 4.3: KBIs in the thermodynamic limit G ∞ for a WCA system at T = 2 and ρ = 0.6 (dimen- sionless units). Values of G ∞ are computed from systems with various number of particles N and using the different methods listed in Table 4.1. N Scaling of G V with 1/ L Direct estimation of G 2 Scaling of LG V with L 500 − 1.5063 ± 0.0003 n/a − 1.5057 ± 0.0008 1000 − 1.5027 ± 0.0000 n/a − 1.5028 ± 0.0002 5000 − 1.5012 ± 0.0000 − 1.5017 ± 0.0004 − 1.5013 ± 0.0002 10000 − 1.5012 ± 0.0000 − 1.5015 ± 0.0004 − 1.5012 ± 0.0001 30000 − 1.5004 ± 0.0001 − 1.5007 ± 0.0007 − 1.5003 ± 0.0006 50000 − 1.4999 ± 0.0001 − 1.5002 ± 0.0009 − 1.500 ± 0.001 ure 4.3 (b) shows that this is not true for all system sizes. In fact, the values of G ∞ can be accurately estimated for systems with a minimum number of par- ticles of 5000, which is larger than the minimum size required in the previous extrapolation method (Figure 4.2) . The third method to find G ∞ is to use the scaling of LG V with L (Eq. (1.28) ). Figure 4.4 shows that plotting the integrals of finite subvolumes as LG V vs L results in a clear linear regime that is easily iden- tified. The value of the slope of the fitted line corresponds to the value of G ∞ . For this method, systems with number of particles equal to or larger than 500 can already be used to compute G ∞ . In principle, all methods for estimating G ∞ should result in the same answer in the thermodynamic limit. In Table 4.3, val- ues of KBIs G ∞ obtained using the three methods studied in this work are listed. For KBIs reported in Table 4.3, only uncertainties larger than 0.01 % are shown. The values are obtained from systems with various sizes. For each system size, a linear range was used to compute G ∞ . In chapter 3, guidelines were provided for selecting a range for the extrapolation of G V vs. 1/ L . Essentially, the first few molecular diameters after r = 0, and distances beyond L /2 should be avoided. Fitting lines of the scaling of LG V vs. L is more convenient. In general, fitting regions are chosen such that the coefficient of determination is equal to or very close to 1. The values of G ∞ in Table 4.3 show that the three methods provide very similar estimations with statistical uncertainties below 0.1 %. Moreover, the results in Table 4.3 show that computing G ∞ using the direct estimation of G 2 requires larger systems compared to the other methods. This was found to be true for other densities as well as for systems with LJ particles (see the Support- ing Information of Ref. [128] ). From studying other systems, it was found that the scaling of LG V with L is the easiest method to apply.