Temperature in a Peierls-Boltzmann treatment of nonlocal phonon heat transport

In nonmagnetic insulators, phonons are the carriers of heat. If heat enters in a region and temperature is measured at a point within phonon mean free paths of the heated region, ballistic propagation causes a nonlocal relation between local temperature and heat insertion. This paper focuses on the solution of the exact Peierls-Boltzmann equation (PBE), the relaxation time approximation (RTA), and the definition of local temperature needed in both cases. The concept of a nonlocal ``thermal susceptibility'' (analogous to charge susceptibility) is defined. A formal solution is obtained for heating with a single Fourier component P(Pr,t)=P₀exp(iPk·Pr-iωt), where P is the local rate of heating). The results are illustrated by Debye model calculations in RTA for a three-dimensional periodic system where heat is added and removed with P(Pr,t)=P(x) from isolated evenly spaced segments with period L in x. The ratio L/ₘᵢₙ is varied from 6 to ∞, where ₘᵢₙ is the minimum mean free path. The Debye phonons are assumed to scatter anharmonically with mean free paths varying as ₘᵢₙ(qD/q)² where qD is the Debye wave vector. The results illustrate the expected local (diffusive) response for ₘᵢₙL, and a diffusive to ballistic crossover as ₘᵢₙ increases toward the scale L. The results also illustrate the confusing problem of temperature definition. This confusion is not present in the exact treatment but occurs in RTA.