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

TL;DR

This paper explores three-point susceptibilities within mode-coupling theory, deriving and numerically solving their equations of motion, and compares their wave vector dependencies to advance understanding of dynamic correlations in colloidal suspensions.

Contribution

It establishes the equivalence of the equations of motion for hi_n(k;t) and the qa0a0limit of hi_q(k;t), and derives a new equation for hi_n^s(k;t), providing numerical solutions and analysis.

Findings

Equation of motion for hi_n(k;t) matches the qa0a0limit of hi_q(k;t).

Numerical solutions for hi_n(k;t) and hi_n^s(k;t) are presented.

Wave vector dependence of hi_n(k;t) and hi_n^s(k;t) are contrasted.

Abstract

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 $\chi_n^s(k;t)$.