  The self energy of $\Delta$-baryon is evaluated at finite temperature and
density using the real time formalism of thermal field theory. The
Dyson-Schwinger equation is used to get the exact thermal propagator followed
by the spectral function of $\Delta$. The $\pi N$ scattering cross section
obtained using explicit $\Delta$ exchange is normalized to the experimental
data in vacuum and its medium modification is implemented by means of the exact
thermal propagator. A significant suppression of the peak of the cross-section
is observed at higher temperature and baryon density. Effects on the mean
relaxation time of nucleons and the temperature dependence of the shear
viscosity of a pion nucleon gas are demonstrated.
