(a) In quantum statistical mechanics, define the one-particle density matrix in the r-representation where $\mathbf{r}$ is the position of the particle.
(b) For a system of $N$ identical free bosons, let
$$
\rho_1(\mathbf{r})=\frac{1}{V} \sum_{\mathbf{k}}\left\langle N_k\right\rangle e^{i \mathbf{k} \cdot \mathbf{r}},
$$
where $\left\langle N_k\right\rangle$ is the thermal averaged number of particles in the momentum state $\mathbf{k}$. Discuss the limiting behavior of $\rho_1(\mathbf{r})$ as $r \rightarrow \infty$, when the temperature $T$ passes from $T>T_c$ to $T<T_c$, where $T_c$ is the Bose-Einstein condensation temperature. In the case $\lim _{r \rightarrow \infty} \rho_1(r)$ approaches zero, can you describe how it approaches zero as $r$ becomes larger and larger?