We present a new method to derive transport equations for non-relativistic quantum manyparticle systems. This method uses an equation-of-motion technique and is applicable to interacting fermions and (or) bosons in arbitrary time-dependent external fields. Using a cluster expansion of the r-particle density matrices the infinite hierarchy of equations of motion for many-particle expectation values is transposed into an equivalent one in terms of correlations. This new hierarchy permits a systematic breaking of the hierarchy at any order. Diagrams are derived for these transport equations. In a second paper the method is tested for exactly soluble electron-phonon models in one dimension.

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We present a one-dimensional microscopic model of interacting electrons coupled to phonons that allows the exact calculation of the time evolution of the momentum distribution functions starting from a nonequilibrium initial state. The model shows relaxation features known from approximate calculations for more realistic three-dimensional models. The exact results are compared with approximate ones resulting from the use of quantum kinetic equations which avoid the assumption of completed successive scattering processes implicit in simple Boltzmann equations. To renormalize the electronic but not the phonon propagation in the non-Markovian equations yields no improvement uniform in time compared to simpler approximations. Due to the presence of the Fermi sea only the additional renormalization of the phonon propagation leads to a drastic improvement in comparison with the exact results. I. INTROI3UCTIONThe relaxation of hot carriers has traditionally been described using a Boltzmann equation, where the collisi. on term is calculated assuming successive completed electron-electron and electron-phonon collisions with energy conservation in each scattering process. From the standard derivation of Fermi's golden rule in quantum mechanics textbooks, it is obvious that this assumption can certainly not be valid in the short time regime after the nonequilibrium state has been established. The proper inclusion of quantum mechanical coherence leads to quantum kinetic equations of non-Markovian nature.Quantum kinetic equations can be derived by breaking the hierarchy of the equations of motion for the relevant reduced density matrices or by using nonequilibrium Green's functions within the Kelclysh formalism, ' which generalizes equilibrium many-body techniques, like the use of Feynman diagrams, by introducing time ordering on a contour running along the real time axis and backwards. As in equilibrium, an approximate description of the one-particle Green's function involves an approximation for the self-energy. In order to arrive at quantum kinetic equations, unfortunately additional approximations have to be made. To get rid of the two-time electronic nonequilibrium correlation functions G and G the generalized KadanofF-Baym ansatz (GKBA) is usually introduced and the remaining retarded and advanced nonequilibrium Green's functions are replaced by the corresponding equilibrium ones. Recently Schoeller presented an illuminating diagrammatic analysis to clarify the approximation involved in the GKBA. The phonons are usually treated as a thermal bath. 3 While there are very few microscopic models of interacting electrons and phonons that allow an exact calculation of equilibrium properties, the situation concerning exact results for the nonequilibrium behavior of microscopic models is even worse. Due to the uncontrolled approximations involved in the derivation of quantum kinetic equations, which can lead, e.g. , to negative probabilities, it is highly desirable to obtain exact results for models that show relaxation behav...

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 electronphonon 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.1997 Academic Press

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