Improved transport equations including correlations for electron-phonon   systems. Comparison with exact solutions in one dimension
  We study transport equations for quantum many-particle systems in terms of correlations by applying the general formalism developed in an earlier paper to exactly soluble electron-phonon models. The one-dimensional models considered are the polaron model with a linear energy dispersion for the electrons and a finite number of electrons and the same model including a Fermi sea (Tomonaga-Luttinger model). The inclusion of two-particle correlations shows a significant and systematic improvement in comparison with the usual non-Markovian equations in Born approximation. For example, the improved equations take into account the renormalization of the propagation by the self-energies to second order in the coupling. 
J. Fricke, V. Meden, C. Wohler, and K. Schonhammer