What does "heating" mean at the nanoscale?

I've talked before about what physicists mean when they talk about "temperature", and work has me thinking about this a lot these days. Temperature is inherently a statistical concept.  It doesn't really make sense to talk about the temperature of a single electron.  The electron has some momentum (and therefore some kinetic energy), but temperature is not a meaningful concept for a single particle in isolation.  Now, if you have a whole bunch of electrons, you can talk about how many of them have a certain amount of energy.  That distribution of electrons as a function of energy takes on a particular form when the electron system is in thermal equilibrium.  (That is, if the electron system is weakly coupled somehow to an energy reservoir so that energy can be exchanged freely between the reservoir and the electrons.)  When the electron system is in thermal equilibrium with the reservoir, on average no net energy is transferred as a function of time between the electrons and the reservoir; this is what we mean when we say that the electrons and the reservoir have the same temperature.

The situation gets really tricky when a system is driven out of equilibrium.  For example, you can use a battery to drive electrons through some nanoscale system.  When you do that, and you look at different points within the nanoscale system, you will find that, in general, the distribution of the electrons as a function of energy doesn't necessarily look much like the thermal equilibrium case.  So, is there a sensible way to generalize the idea of temperature to quantify how "hot" the electrons are?  The problem is, there are many ways you might want to do this - you are trying to take a potentially very complicated distribution function and essentially summarize it by a single number, some local effective temperature.   A natural direction to go is to consider a thought experiment:  what if you took a reservoir with a well defined equilibrium temperature, and allowed it to exchange energy with the nonequilibrium system at a location of interest.  What reservoir temperature would you have to pick so that there is no net average energy transfer between the system and the reservoir in steady state?   That is one sensible way to go, but in the nano limit the situation can be very tricky, even in the thought experiment.  The details of how the imagined energy exchange takes place can affect the answers you get.  Tough stuff.