Three-point susceptibilities $\chi_n(k;t)$ and $\chi_n^s(k;t)$: mode-coupling approximation

Recently, it was argued that a three-point susceptibility equal to the density derivative of the intermediate scattering function, $\chi_n(k;t) = d F(k;t)/d n$, enters into an expression for the divergent part of an integrated four-point dynamic density correlation function of a colloidal suspension [Berthier \textit{et al.}, J. Chem. Phys. \textbf{126}, 184503 (2007)]. We show that, within the mode-coupling theory, the equation of motion for $\chi_n(k;t)$ is essentially identical as the equation of motion for the $\mathbf{q}\to 0$ limit of the three-point susceptibility $\chi_{\mathbf{q}}(\mathbf{k};t)$ introduced by Biroli \textit{et al.} [Phys. Rev. Lett. \textbf{97}, 195701 (2006)]. We present a numerical solution of the equation of motion for $\chi_n(k;t)$. We also derive and numerically solve an equation of motion for the density derivative of the self-intermediate scattering function, $\chi_n^s(k;t) = d F^s(k;t)/d n$. We contrast the wave vector dependence of $\chi_n(k;t)$ and $\chi_n^s(k;t)$.