8 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 5 10 15 20 25 30 35 ~$^{ }$ (k y ) k$_{y}$ kernel data WCA ( =0.375, T=0.765) WCA ( =0.480, T=0.765) EMD NEMD (a) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5 10 15 20 25 30 35 ~ L (k y ) / $^{η}$ ~ L  (0) k$_{y}$ Lorentzian-type fit WCA ( =0.375, T=0.765) WCA ( =0.480, T=0.765) (b) 0 1 2 3 4 5 6 0 5 10 15 20 25 30 35 ~$^{ }$ (k y ) k$_{y}$ kernel data WCA ( =0.840, T=1.0) LJ ( =0.840, T=1.0) Cl$_{2}$  ( =1.088, T=0.97) (c) 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 30 35 ~ L (k y ) / $^{η}$ ~ L  (0) k$_{y}$ Lorentzian-type fit WCA ( =0.840, T=1.0) LJ ( =0.840, T=1.0) Cl$_{2}$ ( =1.088, T=0.97) (d) FIG. 3: Reciprocal-space kernels of monatomic and diatomic fluids: (a) Kernel data of a WCA fluid at two different densities ( T = 0 . 765, N$_{a}$ = 2048); (b) Best fit of normalized kernel data (a) to Lorentzian-type function Eq. (28); (c) Kernel data of a WCA fluid and LJ fluid at the same state point ( ρ$_{a}$ = 0 . 840, T = 1 . 0, N$_{a}$ = 2048), Chlorine at ( ρ$_{a}$ = 1 . 088, T = 0 . 97, N$_{a}$ = 1728); (d) Best fit of the normalized kernel data (c) to the Lorentzian-type function Eq. (28). the fitting parameters are significantly affected by tem- perature, the resulting kernels vary weakly over the range of temperatures chosen here. This was also observed by Hansen et al. for WCA monatomic fluids. sian function the inverse Fourier cosine transform is N$_{G}$ The viscosity kernel in reciprocal space is an even func- tion since it is symmetric about the origin; thus the the real space kernel is symmetric because the Fourier trans- form keeps the even properties of the function. This means that the viscosity kernel in physical space can be found via an inverse Fourier cosine transform, F − 1 c [ . . . ] (a special case of the continuous Fourier transform arising naturally when attempting to transform an even func- tion), of the viscosity kernel in reciprocal space. Since the integral is being computed over an interval symmet- ric about the origin (i.e. - ∞ to + ∞ ), the second integral must vanish to zero, and the first may be simplified to give: Though the Lorentzian-type function given in Eq. (28) fulfills the criteria for having an inverse Fourier transform η$_{L}$ ( y ) (i.e. the function is absolutely integrable, square integrable and the function and its derivative are piece- wise continuous), the integral in Eq. (29) is not readily obtained analytically in the general case. However, the integral can be evaluated numerically. In this work, a Simpson method has been employed for this purpose. The real space kernels for atomic and diatomic flu- ids at zero frequency are presented in figure 4, 5 and 6. Figure 4(a) shows the resulting kernels for chlorine and figure 4(b) shows the resulting kernels for a LJ fluid ex- tracted from two-term and four-term Gaussian functions Eq. (30), and inverse Fourier transform of Eq. (28). We find very little difference between the kernels obtained via two- and four-term Gaussians for these systems. Fig- ures 4(c) and 4(d) show the differences between all three fits. It can be seen that there exists a significant differ- ence (almost 25%) between the kernels extracted from Gaussian and Lorentzian type functions for small y . The discrepancy decreases rapidly as y increases and becomes approximately zero for y ≥ 1 . 5. The width of the kernel for chlorine is roughly 4-6 atomic diameters, figure 4(a), and 3-5 atomic diameters for monatomic LJ and WCA The inverse Fourier cosine transform of the Gaussian function, Eq. (27), exists [49] and it is even possible to obtain an analytical expression. For an N$_{G}$ term Gaus- F − 1 c [˜ η ( k$_{y}$ )] = η ( y ) = √ 2 π ∞ ∫ 0 ˜ η ( k$_{y}$ ) cos( k$_{y}$y ) dk$_{y}$. (29) η$_{G}$ ( y ) = η$_{0}$ √$_{2}$$_{π}$ ∑ j A$_{j}$σ$_{j}$ exp[ − ( σ$_{j}$y ) $^{2}$/ 2] A$_{j}$, σ$_{j}$ ∈ R$_{+}$ . (30) B. Viscosity kernels in physical space