  We study the entropy production estimate for the polyatomic ellipsoidal BGK
model, which is a relaxation type kinetic model describing the time evolution
of polyatomic particle systems. An interesting dichotomy is observed between
$0<\theta\leq 1$ and $\theta=0$: In each case, a distinct target Maxwellians
should be chosen to estimate the entropy production functional from below by
the relative entropy. The time asymptotic equilibrium state toward which the
distribution function stabilizes bifurcates accordingly.
