  We analyze temperature and thermometry for simple nonequilibrium
heat-conducting models. We show in detail, for both two- and three-dimensional
systems, that the ideal gas thermometer corresponds to the concept of a local
instantaneous mechanical kinetic temperature. For the Phi-4 models investigated
here the mechanical temperature closely approximates the local thermodynamic
equilibrium temperature. There is a significant difference between kinetic
temperature and the nonlocal configurational temperature. Neither obeys the
predictions of extended irreversible thermodynamics. Overall, we find that
kinetic temperature, as modeled and imposed by the Nos\'e-Hoover thermostats
developed in 1984, provides the simplest means for simulating, analyzing, and
understanding nonequilibrium heat flows.
