The Quantum Transport Problem

We first recall the notion of classical currents and current densities and then turn to their microscopic definition, in terms of non-equilibrium, quantum mechanical ensemble averages. This will involve the introduction of the quantum statistical operator, enabling us a statistical approach to the transport problem. We then specialize ourselves to open quantum systems. Here small quantum systems are in contact to large reservoirs of particles and heat, the latter being fully characterized by macroscopic quantities like the temperature or the electrochemical potential. This is the case of a nanojunction. By tracing over the many degrees of freedom of the reservoirs, the reduced density operator is obtained. The trace introduces irreversibility in the dynamics of the reduced density operator, leading to equilibrium or steady states at long times. A brief introduction to the full counting statistics shows how the understanding of the transport dynamics of a nanojunction is enriched by considering not only the average current but also higher order cumulants. Finally, we apply these concepts to define currents in nanoscopic set-ups.