So far, we have dealt mostly with the electron system already at an ideal steady state. We never questioned whether this state exists at all, and if so, how a many-body system does actually reach that steady state, whether the steady state we impose via single-particle scattering boundary conditions is actually what the electrons want to realize when they flow across a nanojunction, or even if it is unique.

In addition, we have mostly worked with electrons interacting at a mean- field level (see discussion in Sec. 4.2.4). We discussed in Chapter 4 how, in principle, one can introduce electron-electron interactions beyond mean field using the non-equilibrium Green's function formalism. However, except for simple model systems, it is computationally demanding – and most of the time, outright impossible – to use the interacting version of the NEGF practically in transport calculations (and not only in transport).

A simpler and more efficient way to treat electron-electron interactions in a transport problem would be thus desirable.

In this chapter I will introduce an alternative picture of transport – I will name it micro-canonical (Di Ventra and Todorov, 2004) – that does not rely on the approximations of the Landauer approach. In addition, this formulation does not require partitioning the system into leads and a central region, or assuming the leads contain non-interacting electrons in order to have a closed form for the current.

In fact, within this picture I will prove several theorems of dynamical density-functional theory (Appendices E, F and G) that, in principle, allow us to calculate the exact current – namely with all many-body effects included – within an effective single-particle picture.