  The Bhatnagar-Gross-Krook (BGK) Boltzmann equation with the
Maxwellian-Boltzmann-type equilibrium state leads to the set of
thermo-hydrodynamic equations such as the continuity, the Navier-Stokes, and
the heat-transfer equations in the scaling limit. With its efficient and
promising framework handling multi-scale physics, the collision model has been
studied with both of theoretical and numerical approaches to apply it for
extensive flow conditions such as the flexible choices of the Prandtl number.
In this study, using an analytic technique of the kinetic generator, we employ
a non-local formulation for the equilibrium state leading to the
thermo-hydrodynamic equations with flexible choices of transport coefficients
and the equation of state (EOS). The equilibrium state includes the
quasi-equilibrium state intrinsically, being formulated with the non-local
macroscopic quantities so that the longer-range interaction is explicitly
involved. According to the new formulation, the consistency with conservation
laws, the Boltzmann H-theorem, and the fluctuation-dissipation theorem are
examined.
