  Phonon heat conduction over length scales comparable to their mean free paths
is a topic of considerable interest for basic science and thermal management
technologies. Although the failure of Fourier's law beyond the diffusive regime
is well understood, debate exists over the proper physical description of
thermal transport in the ballistic to diffusive crossover. Here, we derive a
generalized Fourier's law that links the heat flux and temperature fields,
valid from ballistic to diffusive regimes and for general geometries, using the
Peierls-Boltzmann transport equation within the relaxation time approximation.
This generalized Fourier's law predicts that thermal conductivity not only
becomes nonlocal at length scales smaller than phonon mean free paths, but also
requires the inclusion of an inhomogeneous nonlocal source term that has been
previously neglected. We demonstrate the validity of this generalized Fourier's
law through direct comparison with time-domain thermoreflectance (TDTR)
measurements in the nondiffusive regime without adjustable parameters.
Furthermore, we show that interpreting experimental data without this
generalized Fourier's law leads to inaccurate measurement of thermal transport
properties.
