Statistical Operator: Explaining Temperature in Physics

[LagrangeEuler]

I have a question about statistical operator. In statistical physics you deal with temperature. So for example $\hat{\rho}=\frac{1}{Z}e^{-\beta \hat{H}}$ where $\beta=\frac{1}{k_BT}$. In definition there is temperature. And also equivalent definition is
$\hat{\rho}=\sum_i w_i|\psi_i\rangle \langle \psi_i|$ where in definition isn't temperature. I'm confused about this. Can you give me some explanation?

 

[daveyrocket]

Both your definitions are the same, the second is just the general expansion of a one-body operator. $w_i$ is the probability of being in state i, given by $\langle\psi_i| \tfrac{1}{Z} e^{-\beta \hat{H}} |\psi_i\rangle$. It's the same as the first definition.