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We develop a diagrammatic scattering theory for interacting bosons in athree-dimensional, weakly disordered potential. Based on a microscopic N-bodyscattering theory, we identify the relevant diagrams including elastic andinelastic collision processes that are sufficient to describe diffusive quantumtransport. By taking advantage of the statistical properties of the weakdisorder potential, we demonstrate how the N-body dynamics can be reduced to anonlinear integral equation of Boltzmann type for the single-particle diffusiveflux. Our theory reduces to the Gross-Pitaevskii mean field description in thelimit where only elastic collisions are taken into account. However, even atweak interaction strength, inelastic collisions lead to energy redistributionbetween the bosons - initially prepared all at the same single-particle energy- and thereby induce thermalization of the single-particle current. Inaddition, we include also weak localization effects and determine the coherentcorrections to the incoherent transport in terms of the coherent backscatteringsignal. We find that inelastic collisions lead to an enhancement of thebackscattered cone in a narrow spectral window for increasing interactionstrength.

Microscopic scattering theory for interacting bosons in weak random potentials