I have grand canonical partition function $Z_{\lambda}$. From the statistical mechanics, we can calculate average number of particles $\langle N \rangle$ as
$$\langle N \rangle = \lambda \frac{\partial \ln{Z_{\lambda}}}{\partial \lambda}.$$
However, it is just an average number of particles in the system. How can we obtain local particle density?
Any ideas are welcome.
The local particle density is a function of the point. Therefore, to get it from some derivative, one needs to introduce a derivative with respect to a function of the point.
This is something already done by mathematicians, and it is a functional derivative. If one does not know about functional derivatives, it would be too long to provide a minimum of the underlying theory here. However, when we are equipped with functional derivatives, it is relatively simple to show that if we add to a many-body Hamiltonian the interaction with an external one-particle field $u(\bf{r})$, the one-particle density $n(\bf{r})$ can be obtained from a generalization of the formula for the average number: $$ n({\bf r}) = \frac{\delta \ln Z }{\delta \ln z^*({\bf r}) }, $$ where $Z$ is the grand canonical partition function as a functional of $$ z^*({\bf r}) = \frac{e^{\beta (\mu - u(\bf{r}))}}{\Lambda^3}, $$ where $\beta = \frac{1}{k_B T}$, $\mu$ is the chemical potential, and $\Lambda$ the de Broglie thermal length. All details can be found wherever there is an introduction to classical density functional theory.
