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: ${\hat {H}}=\sum _{\alpha }E_{\alpha }|E_{\alpha }\rangle \langle E_{\alpha }|$
• 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: