Joule heating in Boltzmann theory of metals

Abstract

Boltzmann theory deals with the quasiparticle distributions ${F}_{K}$ for electrons and ${N}_{Q}$ for phonons. It is difficult to reconcile the common situation of ``steady state'' (where distributions are out of equilibrium but carry steady charge current $\stackrel{P\vec}{j}$) and Joule heating $\stackrel{P\vec}{j}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{P\vec}{E}$. This paper offers elementary clarification. The resolution is simplest if a true steady state is abandoned.