10 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.5 1 1.5 2 (y) y Gaussian fit G$_{4}$ G$_{2H}$ WCA  =0.375 (a) 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.5 1 1.5 2 (y) y Gaussian fit G$_{4}$ G$_{2H}$ WCA  =0.480 (b) -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0 0.5 1 1.5 2 2.5 3 (y) y Gaussian fit $_{G$_{4}$}$ -  G$_{2H}$ G$_{4}$ -  L $_{L}$ -  G$_{2H}$ WCA =0.375 (c) -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0 0.5 1 1.5 2 2.5 3 (y) y Gaussian fit $_{G$_{4}$}$ -  G$_{2H}$ G$_{4}$ -  L $_{L}$ -  G$_{2H}$ WCA =0.480 (d) FIG. 5: Real space kernel of monatomic WCA fluids as predicted by Gaussian fits of the reciprocal space kernel data, Eq. (30): (a) Kernels obtained from two- and four-term Gaussian fits for a WCA fluid at ρ$_{a}$ = 0 . 375 and T = 0 . 765; (b) Kernels obtained from two- and four-term Gaussian fits for a WCA fluid at ρ$_{a}$ = 0 . 480 and T = 0 . 765; (c) Differences between the kernels for a WCA fluid at ρ$_{a}$ = 0 . 375 and T = 0 . 765; (d) Differences between the kernels for a WCA fluid at ρ$_{a}$ = 0 . 480 and T = 0 . 765. 0 1 2 3 4 5 6 7 8 9 0 0.5 1 1.5 2 2.5 3 G 4 (y) y 4-term Gaussian fit WCA  =0.375 WCA  =0.480 WCA  =0.840 LJ  =0.840 Cl$_{2}$  =1.088 (a) 0 1 2 3 4 5 6 7 8 9 0 0.5 1 1.5 2 2.5 3 L (y) y Lorentzian-type fit WCA  =0.375 WCA  =0.480 WCA  =0.840 LJ  =0.840 Cl$_{2}$  =1.088 (b) FIG. 6: Real space viscosity kernels of monatomic and diatomic fluids, WCA ( ρ$_{a}$ = 0 . 375, T = 0 . 765), WCA ( ρ$_{a}$ = 0 . 480, T = 0 . 765), WCA ( ρ$_{a}$ = 0 . 840, T = 1 . 0), LJ ( ρ$_{a}$ = 0 . 840, T = 1 . 0), chlorine ( ρ$_{a}$ = 1 . 088, T = 0 . 97): (a) Kernels obtained from the four-term Gaussian functional form Eq. (30); (b) Kernels obtained numerically from the Lorentzian-type functional form Eq. (28). tural normalization factor where g ( r ) is the radial distribution function (RDF). Eq. (32) is a measure of the range over which the cor- relation function decays to zero and therefore could be regarded as a correlation length of the radial distribu- tion function. The RDF (or structure factor in reciprocal space) can be defined either in terms of the vector norm r$_{ij}$ between the atoms i and j or between the centres of mass of molecules i and j : g ( r ) = 〈 ∑ N i $_{=1}$∑ N j> 1 δ ( | r − r $_{ij}$| ) 4 πr $^{2}$Nρ 〉 , where N is the total number of atoms or molecules, and ρ is the atomic or molecular number density. The radial distribution functions and normalization factors are presented in figure 7. We can see that the RDF are typical monatomic and diatomic Lennard-Jones pair correlation functions. ξ$_{g}$ generally increases as we in- crease the density and temperature from 0 . 605 at state ξ$_{g}$ = ∞ ∫ 0 r [ g ( r ) − 1] $^{2}$dr ∞ 0 [ g ( r ) − 1] $^{2}$dr (32)