  We derive a multi-species BGK model with velocity-dependent collision
frequency for a non-reactive, multi-component gas mixture. The model is derived
by minimizing a weighted entropy under the constraint that the number of
particles of each species, total momentum, and total energy are conserved. We
prove that this minimization problem admits a unique solution for very general
collision frequencies. Moreover, we prove that the model satisfies an H-Theorem
and characterize the form of equilibrium.
