We know that, in non-relativistic case, for Maxwell-Statistics, the local equilibrium distribution is like the following form $$ C\exp\left(-\lambda_{1}|\vec{u}-\vec{U}|^2-\lambda_{2}\vec{\xi}^2\right), $$ where $\vec{u}$ denotes the microscopic gas particle velocity, and $\vec{U}$ denotes the macroscopic velocity, and $\vec{\xi}$ denotes the internal variable.
In three dimension, $\vec{\xi}$ is zero-dimensional for monatomic molecule, and two-dimensional for diatomic molecule.
And in relativistic case, for monatomic molecule, the local equilibrium distribution is like the following form $$ C\exp(-\lambda_{1}U_{\alpha}p^{\alpha}), $$ where $p^{\alpha}$ denotes the momentum 4-vector of particle, and $\vec{U}$ denotes the macroscopic velocity 4-vector.
But what the internal variable and the local equilibrium distribution is like for diatomic molecule or polyatomic molecule?
