  Kinetic models of polyatomic gas typically account for the internal degrees
of freedom at the level of the two-particle distribution function. However,
close to the hydrodynamic limit, the internal (rotational) degrees of freedom
tend to be well represented just by rotational kinetic energy density. We
account for the rotational energy by augmenting the Ellipsoidal-statistical BGK
(ES-BGK) model, an extension of the Bhatnagar-Gross- Krook (BGK) model, at the
level of the single-particle distribution function with an
advection-diffusion-relaxation equation for the rotational energy. This reduced
model respects the H theorem and recovers the compressible hydrodynamics for
polyatomic gases as its macroscopic limit. As required for a polyatomic gas
model, this extension of the ES-BGK model has not only correct specific heat
ratio but also allows for three independent tunable transport coefficients:
thermal conductivity, shear viscosity, and bulk viscosity. We illustrate the
effectiveness of the model via a lattice Boltzmann method implementation.
