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At finite temperatures below the phase transition point, the Bose-Einsteincondensation, the macroscopic occupation of a single quantum state by particlesof integer spin, is not complete. In the language of superfluid helium, thismeans that the superfluid coexists with the normal fluid. Our goal is todescribe this coexistence in trapped, dilute atomic gases with repulsiveinteractions via mean field laws that account for a {\em spatially varying}particle interaction strength. By starting with the $N$-body Hamiltonian, $N\gg1$, we formally derive a system of coupled, nonlinear evolution equations in$3+1$ dimensions for the following quantities: (i) the wave function of themacroscopically occupied state; and (ii) the single-particle wave functions ofthermally excited states. For stationary (bound) states and a scattering lengthwith {\em periodic microstructure} of subscale $\epsilon$, we heuristicallyextract effective equations of motion via periodic homogenization up to secondorder in $\epsilon$.

Bose-Einstein condensation at finite temperatures: Mean field laws with periodic microstructure