  We compare two approaches in modeling repulsive interactions among hadrons:
the excluded volume approximation and the S-matrix formalism. These are applied
to study the thermodynamics of the $\pi N \Delta$ system. It is shown that the
introduction of an extraneous repulsion between pions and nucleons via the
excluded volume approach, in addition to the interaction that generates the
$\Delta$-resonance, is incompatible with the analysis based on the physical
phase shift of pion-nucleon scattering in the $P_{33}$ channel. This finding
suggests that the repulsive force between hadrons is interaction-channel
dependent and is hence unlikely to be captured by a single phenomenological
parameter. The S-matrix approach employed here can be used, however, to provide
useful estimates of the magnitude of the effective eigenvolume.
