  Kinetic models for polyatomic gases have two temperatures for the two
different types of degrees of freedom, the translational and the internal
energy degrees of freedom. Therefore, in the case of BGK models one expects two
types of relaxations, a relaxation of the distribution function to a Maxwell
distribution and a relaxation of the two temperatures to an equal value. The
speed for the first type of relaxation may be faster or slower than the second
type of relaxation. Models found in the literature often allow only for one of
these two cases. We believe that a model should allow for both cases. That is
why we derive a new multi-species polyatomic BGK model which allows for both
regimes. For this new model we prove conservation properties, positivity of the
temperatures, the H-theorem and characterize the equilibrium as a Maxwell
distribution with equal temperatures. Moreover, we prove the convergence rate
to equilibrium and that we can actually capture both regimes of the relaxation
processes.
