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The non-Markoffian transport equations for the systems of cold Bose atomsconfined by a external potential both without and with a Bose-Einsteincondensate are derived in the framework of nonequilibrium thermal filed theory(Thermo Field Dynamics). Our key elements are an explicit particlerepresentation and a self-consistent renormalization condition which areessential in thermal field theory. The non-Markoffian transport equation forthe non-condensed system, derived at the two-loop level, is reduced in theMarkoffian limit to the ordinary quantum Boltzmann equation derived in theother methods. For the condensed system, we derive a new transport equationwith an additional collision term which becomes important in the Landauinstability.

Derivation of non-Markoffian transport equations for trapped cold atoms in nonequilibrium thermal field theory