Intermediate statistics: Addressing the thermoelectric properties of solids

We study the thermodynamics of a crystalline solid by applying intermediate statistics obtained by deforming known solid state models using the mathematics of q analogs. We apply the resulting q deformation to both the Einstein and Debye models and study the deformed thermal and electrical conductivities and the deformed Debye specific heat. We find that the q deformation acts in two different ways-but not necessarily as independent mechanisms. First, it acts as an effective factor of disorder or impurity, modifying the characteristics of a crystalline structure, which are phenomena described by q bosons. Second, it also manifests intermediate statistics, namely, the B anyons (or B-type systems). For the latter case we have identified the Schottky effect, normally associated with high-T_{c} superconductors in the presence of rare-earth-ion impurities. We also find that it increases the specific heat of the solids beyond the Dulong-Petit limit at high temperature. Such an effect is usually related to anharmonicity of interatomic interactions. Alternatively, since in the q-boson's case the statistics are in principle maintained, the effect of the deformation acts more slowly due to a small change in the crystal lattice. On the other hand, B anyons that belong to modified statistics are more sensitive to the deformation. The results reported here may be verified experimentally, for instance, in experimental samples by inserting impurities, or changes in pressure or temperature if one assumes these tuning quantities are related with the q-deformation parameter.