Kinetical foundations of non-conventional statistics

After considering the kinetical interaction principle (KIP) introduced by Kaniadakis (Physica A 296 (2001) 405), we study in the Boltzmann picture, the evolution equation and the H-theorem for non-extensive systems. The q-kinetics and the κ-kinetics are studied in detail starting from the most general non-linear Boltzmann equation compatible with the KIP.