An Integral Equation for Dissipative Quantum Transport

We present an integral equation derived under the simplifying assumption that the inelastic scattering is caused by uncorrelated point scatterers, such as magnetic impurities or impurities with internal degrees of freedom. 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 transport equation that involves only the electron density and not the spatial correlations of the wave function. The kernel of this integral equation is calculated from the Schrodinger equation and contains all quantum interference effects. We show that at equilibrium the electron density relaxes to the expected equilibrium value with a constant chemical potential everywhere in the structure. Assuming local thermodynamic equilibrium we then derive a linear-response transport equation which resembles the Landauer-Buttiker formula extended to include a continuous distribution of probes. An alternative derivation is provided in the appendix for the kernel of the linear-response transport equation, starting from the Kubo formula for the conductivity. We discuss the conditions under which this transport equation reduces to the well-known drift-diffusion equations describing classical Brownian motion. In the present work we restrict ourselves to steady state transport and neglect many-body effects beyond the Hartree term.