The Van Hove function describes the time dependent pair correlation between atoms,
\[G(r, t) = \frac{1}{4\pi \rho_0Nr^2}\sum_{i,j}\delta(r  \vec{r}_i(0)  \vec{r}_j(t))\]\(i\) and \(j\) runs over all the \(N\) atoms in the system. \(\vec{r}_i\) refers to the position of the atom \(i\) at time \(t\). The following formula describes the fundamental construction of the Van Hove function from the dynamic structure factor \(S(Q, E)\),
\[G(r, t) = \frac{1}{2\pi\rho_0r}\int S(Q, E)sin(Qr)e^{i\omega t}Q\,dQ\,dE\]For more details and references about the formulation, refer to Ref. [1, 2]. From the above formulation, there are two main immediate derivations to reiterate,

If we integrate out the energy (as mathematically shown in Eqn. 6 in Ref. [1]), as for a normal pair distribution function (PDF) measurement without discriminating the incoming and outgoing energy, that is just to set \(t = 0\), in the formula above – the exponential term in the Fourier transform of the energy part accordingly becomes \(1\), and we can pull out the energy relevant terms to have Eqn. 6. in Ref. [1]. Such a mathematical result indicates that in a normal PDF measurement, we are actually measuring the equaltime correlation (i.e., simultaneously catch atoms with respect to the pair distance – since we can only measure pair distances without knowing sample position directly due to the phase problem). In contrast, the nonequaltime correlation (i.e., \(t \neq 0\)) refers to something like catching atom A at \(t = 0\) but catching atom B at \(t=t_1\), again, with respect to the pair distance, in which case we then get the evolution of the realspace correlation with time.

If we can somehow measure only the elastic scattering, that means the \(S(Q, E)\) function is a Dirac \(\delta\) function in the energy space and we know that the Fourier transform of a Dirac \(\delta\) function is a constant in the coupled space. Therefore, such a measurement will yield a static picture of the atomic configuration, i.e., correlation is constant in time space, or, atoms ‘not moving’ in the measured picture.