Finite-size estimates of Kirkwood-Buff and similar integrals

Recently, Krüger and Vlugt [Phys. Rev. E 97, 051301(R) (2018)] have proposed a method to approximate an improper integral ${\int }_0^{\infty }drF\left(r\right)$, where $F\left(r\right)$ is a given oscillatory function, by a finite-range integral ${\int }_0^{L}drF\left(r\right)W(r/L)$ with an appropriate weight function $W\left(x\right)$. The method is extended here to an arbitrary (embedding) dimensionality $d$. A study of three-dimensional Kirkwood-Buff integrals, where $F\left(r\right)=4\pi r^{2}h\left(r\right)$, and static structure factors, where $F\left(r\right)=(4\pi /q)rsin\left(qr\right)h\left(r\right)$, $h\left(r\right)$ being the pair correlation function, shows that, in general, a choice $d\ne 3$ (e.g., $d=7$) for the embedding dimensionality may significantly reduce the error of the approximation ${\int }_0^{\infty }drF\left(r\right)\simeq {\int }_0^{L}drF\left(r\right)W(r/L)$.

Figure 1

Plot of the weight functions $W_d^{\left(k\right)}\left(x\right)$ for embedding dimensionalities $d=3$, 5, 7, and 9 and indices $k=2$ and 3.

Figure 2

Plot of the relative error $\left|{\mathcal{I}}_{L,d}^{\left(k\right)}\left[F\left(r\right)\right]/\mathcal{I}\left[F\left(r\right)\right]-1\right|$ vs $L$, where $F\left(r\right)=4\pi r^{2}h\left(r\right)$ and $h\left(r\right)$ is given by Eq. (26). Panels (a) and (b) correspond to $\chi =2$ and 20, respectively. Only the values corresponding to $L=\text{integer}$ are shown.

Figure 3

Plot of the relative error $\left|{\mathcal{I}}_{L,d}^{\left(k\right)}\left[F\left(r\right)\right]/\mathcal{I}\left[F\left(r\right)\right]-1\right|$ vs $\chi$, where $F\left(r\right)=4\pi r^{2}h\left(r\right)$ and $h\left(r\right)$ is given by Eq. (26). Panels (a) and (b) correspond to $L=5$ and 20, respectively.

Figure 4

Plot of the relative error $\left|{\mathcal{I}}_{L,d}^{\left(k\right)}\left[F\left(r\right)\right]/\mathcal{I}\left[F\left(r\right)\right]-1\right|$ vs $L$, where $F\left(r\right)=4\pi r^{2}h\left(r\right)$ and $h\left(r\right)$ is the exact solution of the PY integral equation for hard spheres. Panels (a) and (b) correspond to $\varphi =0.2$ and 0.5, respectively. Only the values corresponding to $L=\text{integer}$ are shown.

Figure 5

Plot of the relative error $\left|{\mathcal{I}}_{L,d}^{\left(k\right)}\left[F\left(r\right)\right]/\mathcal{I}\left[F\left(r\right)\right]-1\right|$ vs $\varphi$, where $F\left(r\right)=4\pi r^{2}h\left(r\right)$ and $h\left(r\right)$ is the exact solution of the PY integral equation for hard spheres. Panels (a) and (b) correspond to $L=5$ and 10, respectively.

Figure 6

Plot of ${\mathcal{I}}_{L,d}^{\left(k\right)}\left[F\left(r\right)\right]$ vs $L^{-3}$ ($L\ge 5$), where $F\left(r\right)=2h\left(r\right)$ and $h\left(r\right)$ is the exact pair correlation function for a 1D system of hard rods at a packing fraction $\varphi =0.8$. The inset is a magnification of the small framed region ($L\ge 10$) in the main figure.

Figure 7

Plot of the relative error $\left|{\mathcal{I}}_{L,d}^{\left(k\right)}\left[F\left(r\right)\right]/\mathcal{I}\left[F\left(r\right)\right]-1\right|$ vs $q$, where $F\left(r\right)=(4\pi /q)rsin\left(qr\right)h\left(r\right)$ and $h\left(r\right)$ is the exact solution of the PY integral equation for hard spheres at a packing fraction $\varphi =0.5$. Panels (a) and (b) correspond to $L=5$ and 10, respectively.