We present a quantum transport equation derived under the simplifying assumption that the inelastic scattering is caused by uncorrelated point scatterers, such as magnetic impurities. While this assumption is not always realistic, we believe that the model can be used to describe much of the essential physics of quantum transport in mesoscopic systems. This assumption allows us to write a quantum transport equation that involves only the diagonal elements of the density matrix which we use to define a distribution function f(r; E). The kernel of this integral equation is calculated from the Schrodinger equation and contains all quantum interference effects. We show that at equilibrium the distribution function relaxes to the Fermi-Dirac function with a constant chemical potential everywhere in the structure. Assuming local thermodynamic equilibrium we then derive a linearized transport equation which has the appearance of a continuous version of the multiprobe Landauer formula. An alternative derivation is provided for the linearized transport equation starting from the multiprobe Landauer formula. Numerical results are presented for the conductivity of a disordered resistor with distributed inelastic scattering. A clear transition is observed from weak to strong localization as the inelastic scattering time is increased. In the present work we restrict ourselves to steady state transport and neglect many-body effects.
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