The problem of describing particle transport through a Markovian stochastic mixture of two immiscible materials is generally approximated by the so-called Levermore model, consisting of two coupled transport equations. In this paper, the P2 diffusive equations and the associated boundary conditions for this Levermore model are derived in planar geometry by using a variational principle, and numerical results comparing P2, P1, and S16 (benchmark) calculations are presented. These results demonstrate that the P2 equations are considerably more accurate than the P1 equations away from boundary layers. An asymptotic diffusion approximation to this model is also explored with several different boundary conditions, and the overall conclusion is that the asymptotic diffusion treatment is in general inferior to P2 theory, and its superiority over P1 theory is not overwhelming and not consistent.