Key equations from quantum statistical tools

Equilibrium

Expectation values

$A=\mathrm {Tr} [{\hat {\rho }}_{\mathrm {eq} }{\hat {A}}]$

Density matrix of fermions in equilibrium

${\hat {\rho }}_{\mathrm {eq} }=\sum _{\alpha }f(E_{\alpha }-\mu )|E_{\alpha }\rangle \langle E_{\alpha }|=f({\hat {H}}-\mu {\hat {I}})$

• Fermi-Dirac distribution function:
• Hamiltonian and its spectral decomposition: <math> \hat{H} = \sum_\alpha E_\alpha |E_\alpha \rangle \langle E_\alpha|
• Function of Hamiltonian: F(\hat{H}) = \sum_\alpha F(E_\alpha) |E_\alpha \rangle \langle E_\alpha|

Nonequilibrium

• Expectation values:

• Current operator:

• Spin torque operator: