# Minimize grand potential functional for density matrix

• I
Gold Member
TL;DR Summary
##\rho## is the density matrix
##\Omega## is the grand potential
##\text{Tr}## stands for 'trace'
I'd like to show that, by minimizing this functional
$$\Omega[\hat \rho] = \text{Tr} \hat \rho \left[ \hat H - \mu \hat N + \frac 1 {\beta} \log \hat \rho \right]$$
I get the well known expression
$$\Omega[\hat \rho_0] = - \frac 1 {\beta} \log \text{Tr} e^{-\beta (\hat H - \mu \hat N )}$$

I'm familiar with minimizing a functional of the form ##F[g] = \int dx f(g(x))##, but this notations for operators eludes me.

Thanks,
Ric