We propose a method for the statistical description of nanocrystalline systems. It is demonstrated that a change either in the size or in the surface state of a nanoparticle leads to a corresponding change in the parameters of the statistical distribution of the atomic displacements and, therefore, to changes in the energetic characteristics of the bulk part of the nanocrystal. In particular, for a free particle, size reduction is accompanied by an increase in energy and pressure of the phonon gas. The pressure of the phonon gas depends on temperature and is directed outside the crystal. Since temperature and pressure are independent thermodynamic variables determining the state of the matter, the presence of an excess pressure of the phonon gas leads to a change of the melting temperature and of other thermodynamic properties of the nanocrystal. A consequence of these size-dependent effects in nanocrystals is non-additivity of the energy, which can manifest itself in possible temperature change and heat release due to a tight contact of nanocrystals.
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