Key equations from quantum statistical tools

Equilibrium

Expectation values

$A=\mathrm{T}\mathrm{r}[{\hat{\rho }}_{\mathrm{e}\mathrm{q}}\hat{A}]$

Density matrix of fermions in equilibrium

${\hat{\rho }}_{\mathrm{e}\mathrm{q}}=\sum _{\alpha }f(E_{\alpha }-\mu )|E_{\alpha }⟩⟨E_{\alpha }|$

Nonequilibrium

• Expectation values:

• Current operator:

• Spin torque operator: