  Steady-state thermal transport in nanostructures with dimensions comparable
to the phonon mean-free-path is examined. Both the case of contacts at
different temperatures with no internal heat generation and contacts at the
same temperature with internal heat generation are considered. Fourier's Law
results are compared to finite volume method solutions of the phonon Boltzmann
equation in the gray approximation. When the boundary conditions are properly
specified, results obtained using Fourier's Law without modifying the bulk
thermal conductivity are in essentially exact quantitative agreement with the
phonon Boltzmann equation in the ballistic and diffusive limits. The errors
between these two limits are examined in this paper. For the four cases
examined, the error in the apparent thermal conductivity as deduced from a
correct application of Fourier's Law is less than 6%. We also find that the
Fourier's Law results presented here are nearly identical to those obtained
from a widely-used ballistic-diffusive approach, but analytically much simpler.
Although limited to steady-state conditions with spatial variations in one
dimension and to a gray model of phonon transport, the results show that
Fourier's Law can be used for linear transport from the diffusive to the
ballistic limit. The results also contribute to an understanding of how heat
transport at the nanoscale can be understood in terms of the conceptual
framework that has been established for electron transport at the nanoscale.
