  Owing to the Kubo relation, the shear viscosities of pionic and nucleonic
components have been evaluated from their corresponding retarded correlators of
viscous stress tensor in the static limit, which become non-divergent only for
the non-zero thermal widths of the constituent particles. In the real-time
thermal field theory, the pion and nucleon thermal widths have respectively
been obtained from the pion self-energy for different meson, baryon loops and
the nucleon self-energy for different pion-baryon loops. We have found a
non-monotonic momentum distributions of pion and nucleon thermal widths, which
have been integrated out by their respective Bose-enhanced and Pauli-blocked
phase space factors during evaluation of their shear viscosities. The viscosity
to entropy density ratio for this mixed gas of pion-nucleon system decreases
and approaches toward its lower bound as the temperature and baryon chemical
potential increase within the relevant domain of hadronic matter.
