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. Suoknuut i, and M.Manninen, Phys. Rev. Lett. 81, 2958 (1998)], we presented a formal ism suitable to describe the thermal properties of the phonon gas in ultrat hin 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-di mensional distribution of the phonon gas. [S0163-1829(99)10915-9].