  Quasiparticle theory gives a local relation between heat current and
temperature gradient, provided the quasiparticle mean free path is smaller than
the scale of variation of temperature. When mean free paths are comparable to
sample size, the relation becomes non-local. This non-local relation is
formulated for phonon carriers; an explicit form is found in the approximation
where current relaxation rates are replaced by quasiparticle relaxation rates.
A quasiparticle definition of local temperature is offered. To extract the
spatial variation of the temperature gradient requires inverting the non-local
relation. A variational principle is constructed. The heat current is bounded
above when evaluated for a trial temperature gradient. The true temperature
gradient is the one which minimizes heat current. The simplest variational
approximation (a constant temperature gradient) gives an approximate formula
for the thickness dependence of the heat transport across a film whose
thickness is comparable to mean free paths of phonons. The formula interpolates
between ballistic and diffusive limits.
