Since vacancy defects are related to boltzmann's equation, I was thinking of simulating the movement of defects in solids using lattice Boltzmann modelling.
Indeed you can. See the attached file. The LB equation, and its numeric counterpart LBM solves the transport of a given scalar field. The Equilibrium distribution characterize the steady state and the physical problem. So in this context, different equilibrium distributions will solve different physics.