The impact of phonon dispersion on thermodynamical properties in   computational models of crystalline solids
  The existing techniques of account for the phonon dispersion are computationally costly, while its impact on a variety of thermodynamic properties appears negligible. We develop a mathematical formalism, which allows for clear understanding of the effect. The theoretical elaborations are then confronted with a widely used phenomenological model of the dispersion relation. The results show that accuracy of the models, which neglect the phonon dispersion, allows for calculation of many thermodynamic quantities up to an admissible precision. Although the context refers mainly to the Helmholtz energy , other properties are concerned as well with less details. Time-efficient schemes of control over the accuracy loss due to the neglected phonon dispersion are discussed. 
Roman Belousov and Mauro Prencipe
cond-mat.stat-mech