 #1
dRic2
Gold Member
 878
 225
 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
$$\Omega[\hat \rho] = \text{Tr} \hat \rho \left[ \hat H  \mu \hat N + \frac 1 {\beta} \log \hat \rho \right]$$
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