We extend models of our Galaxy based on distribution functions ( df s ) that are analytic functions of the action integrals to extended distribution functions ( edf s ) , which have an analytic dependence on metallicity as well .
We use a simple , but physically-motivated , functional forms for the metallicity of the interstellar medium as a function of radius and time and for the star-formation rate , and a model for the diffusion of stars through phase space to suggest the required functional form of an edf .
We introduce a simple prescription for radial migration that preserves the overall profile of the disc while allowing individual stars to migrate throughout the disc .
Our models explicitly consider the thin and thick discs as two distinct components separated in age .

We show how an edf can be used to incorporate realistic selection functions in models , and to construct mock catalogues of observed samples .
We show that the selection function of the Geneva-Copenhagen Survey ( GCS ) biases in favour of young stars , which have atypically small random velocities .
With the selection function taken into account our models produce good fits of the GCS data in chemo-dynamical space and the Gilmore and Reid ( 1983 ) density data .

From our edf , we predict the structure of the SEGUE G-dwarf sample .
The kinematics are successfully predicted .
The predicted metallicity distribution has too few stars with { [ Fe / H ] } \simeq - 0.5 { dex } and too many metal-rich stars .
A significant problem may be the lack of any chemical-kinematic correlations in our thick disc .
We argue that edf s will prove essential tools for the analysis of both observational data and sophisticated models of Galaxy formation and evolution .