Solution of the Functional Differential Equation for the Statistical Equilibrium of a Crystal
Abstract
The sparticle distribution functions (s=1, 2, ...) of classical equilibrium statistical mechanics are determined for a crystal, as power series in the temperature. They are obtained by solving Bogolyubov's functional differential equation. From the distribution functions, the thermodynamic functions of a crystal are computed as power series in the temperature. The leading terms in these series are the usual classical results which are customarily derived by assuming that the potential energy is a quadratic function of the particle displacements. The further terms, which depend upon the nonquadratic or anharmonic terms in the potential, provide corrections to the usual results, which become more important as the temperature increases. If only a few terms in the series are used, the results will be valid at temperatures low compared to some characteristic temperature of the crystal, e.g., the melting temperature. Since they are based on classical mechanics, the results are valid only at temperatures high compared to the Debye temperature. The series expansions of the distribution functions and thermodynamic functions may be viewed as the lowtemperature analogs of the virial expansions, which are lowdensity expansions. As in the case of the virial expansions, all the terms are determined explicitly in analytic form, but their actual evaluation is difficult.
 Publication:

Physical Review
 Pub Date:
 February 1961
 DOI:
 10.1103/PhysRev.121.1022
 Bibcode:
 1961PhRv..121.1022L