  The widely used Gross-Pitaevskii equation treats only coherent aspects of the
evolution of a Bose-Einstein condensate. However, inevitably some atoms scatter
out of the condensate. We have developed a method, based on the field theory
formulation, describing the dynamics of incoherent processes which are due to
elastic collisions. We can therefore treat processes of spontaneous emission of
atoms into the empty modes, as opposed to stimulated processes, which require
non-zero initial occupation.
  In this article we study two counter-propagating plane waves of atoms,
calculating the full dynamics of mode occupation, as well as the statistics of
scattered atoms. The more realistic case of Gaussian wavepackets is also
analyzed.
