In the first part of the talk I will introduce the S-matrix formalism for computing thermodynamic quantities. The second part will be devoted to the study of IR divergences arising from 3 to 3 forward scattering, and how to find a suitable set of scattering processes that admits an IR safe sum.
After a detailed study within a simplified model, I will comment about extensions to thermal QCD and possible implications for the stability of its perturbative series.