  In [Phys. Rev. 94 (1954), 511-525], P.L. Bhatnagar, E.P. Gross and M. Krook
introduced a kinetic equation (the BGK equation), effective in physical
situations where the Knudsen number is small compared to the scales where
Boltzmann's equation can be applied, but not enough for using hydrodynamic
equations. In this paper, we consider the stochastic particle system
(inhomogeneous Kac model) underlying Bird's direct simulation Monte Carlo
method (DSMC), with tuning of the scaled variables yielding kinetic and/or
hydrodynamic descriptions. Although the BGK equation cannot be obtained from
pure scaling, it does follow from a simple modification of the dynamics. This
is proposed as a mathematical interpretation of some arguments in [Phys. Rev.
94 (1954), 511-525], complementing previous results in [Arch. Ration. Mech.
Anal. 240 (2021), 785-808] and [Kinet. Relat. Models 16 (2023), 269-293].
