Regime diagram for thermal transport. Three regimes can be distinguished in a diagram reporting the phonon lifetime (τ ) as a function of the phonon frequency (ω). The horizontal black like is the Wigner limit in time, τ = 1/∆ωavg (i.e. the inverse average interband spacing, see Eq. (52), and denotes the center of the non-sharp particle-wave crossover for phonons. The green region well above the line represents phonons that propagate particlelike and mainly contribute to the populations conductivity; the blue region below the line represents phonons that tunnel wave-like and mainly contribute to the coherences conductivity; the red points around the line are phonons that contribute simultaneously and comparably to the particle-like and wave-like conductivities. The purple line τ =1/ω is the center of the non-sharp Ioffe-Regel limit in time [39, 105]. The region above it represents well defined phonons; the Wigner transport equation (37) describes all these phonons (blue, red, green), while the semiclassical Peierls-Boltzmann equation accounts only for phonons that propagate particlelike (green). The yellow region below the Ioffe-Regel limit represents overdamped phonons, which require full spectralfunction approaches [75, 76] to be described correctly.