S-matrix formulation of thermodynamics with N-body scatterings

We apply a phase space expansion scheme to incorporate the N-body scattering processes in the S-matrix formulation of statistical mechanics. A generalized phase shift function suitable for studying the thermal contribution of $N \rightarrow N$ processes is motivated and examined in various models. Using the expansion scheme, we revisit how the hadron resonance gas model emerges from the S-matrix framework, and consider an example of structureless scattering in which the phase shift function can be exactly worked out. Finally we analyze the influence of dynamics on the phase shift function in a simple example of 3- and 4-body scattering.