Behavior of the phonon gas in restricted geometries at low temperatures

In a previous paper [D. V. Anghel, J. P. Pekola, M. M. Leivo, J. K. Suoknuuti, and M. Manninen, Phys. Rev. Lett. 81, 2958 (1998)], we presented a formalism suitable to describe the thermal properties of the phonon gas in ultrathin membranes. We extend here the formalism to the case of narrow objects, such as wires or ``bridges,'' and to the case of small nanoscopical objects. This will allow us to see crossovers between three-, two-, one- and zero-dimensional distribution of the phonon gas.