In the Keldysh framework for nonequilibrium dynamics of quantum systems we learn that there are essentially two Green's functions that characterize a system: the retarded Green's function $G^R(\omega)$ (and its partner $G^A(\omega)$), and the Keldysh Green's function $G^K(\omega)$. These are furthermore related by the relation $$ G^K(\omega) = F(\omega) [G^R(\omega) - G^A(\omega) ] , $$ which serves to define the occupation function $F(\omega)$.
In equilibrium we know that for bosons (fermions) $F(\omega)$ is fixed by the fluctuation dissipation relation to obey $F(\omega) = \coth(\beta\omega/2)$ ($F(\omega ) = \tanh(\beta\omega/2)$). For bosons this obeys $| F(\omega) | > 1$ and for fermions, $|F(\omega) | < 1$.
My question is whether, in a general nonequilibrium setting, the function $F(\omega)$ may develop a region where $ | F(\omega) | <1 $ for bosons (or $|F(\omega)| > 1$ for fermions)? In particular, could $F(\omega) = 0 $ for a bosonic system or $F(\omega) \to \infty$ for a fermionic system, even for just a single frequency? If yes, is there an example? If no, is there a theorem which constrains these functions out of equilibrium?
