## Thermodynamics of acoustic phonons in mesoscopic materials

At sub-Kelvin temperatures, the formalism used for describing bulk materials doesn't lead to the correct thermodynamics for mesoscopic materials. One reason is that surface effects do matter. In the case of bulk materials surface effects are weak while volume effects dominate. For this reason alone, one is inclined to use periodic boundary conditions to describe the phonon gas. In the case of mesoscopic systems we tend to encounter size effects which are mostly described by surface effects. To describe the influence of these effects people use stress-free boundary conditions. This type of boundary conditions lead to a quantization of the transversal part of the phononic spectrum.From a thermodynamical point of view this leads to the existence of a temperature cross-over( Tc ). At temperatures well bellow Tc the phononic gas behaves quasi-bidimensional while at temperature well above Tc the phononic gas behaves three dimensional.

In this talk I will mainly review the content of two papers[1,2]. Based on these papers I will explain how one can quantize the modes of an elastic isotropic thin plate and derive the thermodynamics of the associated phonon gas.

References:

1.D. V. Anghel and T. K ̈uhn, J. Phys. A: Math. Theor. 40,10429 (2007)

2.Kuhn, D. V. Anghel, J. P. Pekola, M. Manninen, and Y. M. Galperin, Phys. Rev. 70, 125425 (2004).