  In this paper we study the realization of lattice models in mixtures of
atomic and dipolar molecular quantum gases. We consider a situation where polar
molecules form a self-assembled dipolar lattice, in which atoms or molecules of
a second species can move and scatter. We describe the system dynamics in a
master equation approach in the Brownian motion limit of slow particles and
fast phonons, which we find appropriate for our system. In a wide regime of
parameters, the reduced dynamics of the particles leads to physical
realizations of extended Hubbard models with tuneable long-range interactions
mediated by crystal phonons. This extends the notion of quantum simulation of
strongly correlated systems with cold atoms and molecules to include
phonon-dynamics, where all coupling parameters can be controlled by external
fields.
