  Understanding the interplay between illumination and the electron
distribution in metallic nanostructures is a crucial step towards developing
applications such as plasmonic photo-catalysis for green fuels, nano-scale
photo-detection and more. Elucidating this interplay is challenging, as it
requires taking into account all channels of energy flow in the electronic
system. Here, we develop such a theory, which is based on a coupled
Boltzmann-heat equations and requires only energy conservation and basic
thermodynamics, where the electron distribution, and the electron and phonon
(lattice) temperatures are determined {\em uniquely}. Applying this theory to
realistic illuminated nanoparticle systems, we find that the electron and
phonon temperatures are similar, thus justifying the (classical) single
temperature models. We show that while the fraction of high-energy ``hot''
carriers compared to thermalized carriers grows substantially with illumination
intensity, it remains extremely small (on the order of $10^{-8}$). Importantly,
most of the absorbed illumination power goes into heating rather than
generating hot carriers, thus rendering plasmonic hot carrier generation
extremely inefficient. Our formulation allows for the first time a unique
quantitative comparison of theory and measurements of steady-state electron
distributions in metallic nanostructures.
