in an online video lecture,(around 38min, where the exactly statement is at 38min28secs.) i got one question, suppose we have a system of $N$ particles, $\left\{ {{{\vec r}_i}(t)} \right\}i = 1, \cdot \cdot \cdot ,N$ are the position vectors of the particles. I was told in the lecture that the so-called self intermediate scattering function is defined as. $${F_s}(k,t) = \frac{1}{N}\left\langle {\sum\limits_{i = 1}^N {{e^{i\vec k \cdot [{{\vec r}_i}(t) - {{\vec r}_i}(0)]}}} } \right\rangle.$$ (for homogeneous system, it only depends on the absolute value of $\vec k$.)
Furthrmore, also defined is the overlap funtion $${Q_s} = \frac{1}{N}\left\langle {\sum\limits_{i = 1}^N {\theta (a - \left| {{{\vec r}_i}(t) - {{\vec r}_i}(0)} \right|)} } \right\rangle$$. where $\theta$ is the Heaviside step function and $a$ is a scaler at constant value
it is said that ${F_s}(k,t)$ and ${Q_s}$ are equivalent and have similar physical content if $2\pi/k$ is replaced by $a$. But I really did not see why. Could anybody give me some help on it. Thanks!
Regard $X_j:=r_j(t)-r_j(0)$ as a random variable. Since $\left\langle\theta(a-|X_j|)\right\rangle=\Bbb P(|X_j|\le a)$, $Q_s$ is the average across $j$ of the CDFs of the $|X_j|$. By homogeneity, this characterizes the joint distribution of the $X_j$. Similarly, $F_s$ is the average across $j$ of the characteristic functions of the $X_j$.