Heat Conduction by Phonons across a Film

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.