Phonon-mediated thermal conductance of mesoscopic wires with rough edges

We present an analysis of acoustic phonon propagation through long, free-standing, insulating wires with rough surfaces. Due to a crossover from ballistic propagation of the lowest-frequency phonon mode at $\omega <\omega _{1}=\pi c/W$ to a diffusive (or even localized) behavior upon the increase of phonon frequency, followed by re-entrance into the quasi-ballistic regime, the heat conductance of a wire acquires an intermediate tendency to saturate within the temperature range $T\sim \hbar \omega_{1}/k_{B}$.