  We develop a diagrammatic scattering theory for interacting bosons in a
three-dimensional, weakly disordered potential. Based on a microscopic N-body
scattering theory, we identify the relevant diagrams including elastic and
inelastic collision processes that are sufficient to describe diffusive quantum
transport. By taking advantage of the statistical properties of the weak
disorder potential, we demonstrate how the N-body dynamics can be reduced to a
nonlinear integral equation of Boltzmann type for the single-particle diffusive
flux. Our theory reduces to the Gross-Pitaevskii mean field description in the
limit where only elastic collisions are taken into account. However, even at
weak interaction strength, inelastic collisions lead to energy redistribution
between the bosons - initially prepared all at the same single-particle energy
- and thereby induce thermalization of the single-particle current. In
addition, we include also weak localization effects and determine the coherent
corrections to the incoherent transport in terms of the coherent backscattering
signal. We find that inelastic collisions lead to an enhancement of the
backscattered cone in a narrow spectral window for increasing interaction
strength.

Quantum Physics;Disordered Systems and Neural Networks;Quantum Gases