  We study the heat transfer between elastic solids with randomly rough
surfaces. We include both the heat transfer from the area of real contact, and
the heat transfer between the surfaces in the noncontact regions. We apply a
recently developed contact mechanics theory, which accounts for the
hierarchical nature of the contact between solids with roughness on many
different length scales. For elastic contact, at the highest (atomic)
resolution the area of real contact typically consists of atomic (nanometer)
sized regions, and we discuss the implications of this for the heat transfer.
For solids with very smooth surfaces, as is typical in many modern engineering
applications, the interfacial separation in the non-contact regions will be
very small, and for this case we show the importance of the radiative heat
transfer associated with the evanescent electromagnetic waves which exist
outside of all bodies.
