Spatial Correlation Function and Ensemble average

Question

Well, I was reading the Statistical Mechanics book by Pathria, to understand the concepts of the correlation function. I want to quote some lines.

Spatial correlation functions are based on n-particle densities. The one-body number density is defined by the average quantity \begin{equation} n_1(\vec{r})=\langle \sum_{i}\delta(\vec{r}-\vec{r_j})\rangle \end{equation} This defines the local number density in which $n_1(\vec{r})d\vec{r}$ is a measure of the probability of finding a particle inside an elemental volume dr located at position r.

Now my question is about the averaging. Is it not the ensemble average? Because particle number density at a given point inside material is truely a random variable. So we need some distribution function to be average. So my question is, what kind of average was that?

Answer 1

Yes, that is an ensemble average. You create many realisations of your system and count how many particles are "around" each point $\vec{r}$. Or, if the system is ergodic, you just take the time average of the same quantity.