Ensemble averaged coherent state path integral for disordered bosons with a repulsive interaction (Derivation of mean field equations)

We consider bosonic atoms with a repulsive contact interaction in a trap potential for a Bose-Einstein condensation and additionally include a random potential. The ensemble averages for two models of static and dynamic disorder are performed and investigated in parallel. The bosonic many body systems of the two disorder models are represented by coherent state path integrals on the Keldysh time contour which allow exact ensemble averages for zero and finite temperatures. These ensemble averages of coherent state path integrals therefore present alternatives to replica field theories or supersymmetric averaging techniques.