  In recent works it has been demonstrated that using an appropriate rescaling,
linear Boltzmann-type equations give rise to a scalar fractional diffusion
equation in the limit of a small mean free path. The equilibrium distributions
are typically heavy-tailed distributions, but also classical Gaussian
equilibrium distributions allow for this phenomena if combined with a
degenerate collision frequency for small velocities. This work aims to an
extension in the sense that a linear BGK-type equation conserving not only
mass, but also momentum and energy, for both mentioned regimes of equilibrium
distributions is considered. In the hydrodynamic limit we obtain a fractional
diffusion equation for the temperature and density making use of the Boussinesq
relation and we also demonstrate that with the same rescaling fractional
diffusion cannot be derived additionally for the momentum. But considering the
case of conservation of mass and momentum only, we do obtain the incompressible
Stokes equation with fractional diffusion in the hydrodynamic limit for
heavy-tailed equilibria.
