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Application of nano-structures requires knowledge of their fundamental physical (mechanical, electro-magnetic, optical, etc.) characteristics. Thermodynamic properties associated with phonon displacements through the nano-samples are particularly interesting. Independent of the type of lattices, the thermodynamics of their subsystems (electrons, excitons, spin waves, etc.) is determined when the subsystem is in thermodynamic equilibrium with phonons. Besides, the acoustical characteristics as well as conductive and superconductive properties etc. could not be realistically explained without phonons. The fact which must be especially pointed out is that the role of phonons in nanostructures is much more impressive than in bulk structures. The main fact concerning phonon properties in nanostructures is the absence of the so-called acoustic phonons: for the exciting of phonons in nanostructures activation energy different from zero is necessary. These unexpected characteristics require revision of all conclusions obtained by bulk theories of phonons. Therefore, the contribution of phonon subsystems to thermodynamic is the first step in a research of nanostructure properties.