Some properties of solids through q-Deformed algebras.

Abstract

The motivation of our study lies in the fact that a full understanding of the physical origin of the q-deformation of classical physics is still lacking. It is not clear that there is a standard answer to the q-deformation mechanics inspired by the study of quantum groups. But appeared recently great interest investigating the q-deformed thermodynamic systems at classical level. Deformed theory manages the statistical behavior of complex systems whose underlying dy- namics is calibrated on an area of multi-fractal phase governed by the long-range interaction and the effects of long-term memory [4]. A mechanism capable of generating a deformed version of the classical statistical mechanics is to replace the Boltzmann-Gibbs distribution by its deformed version. In this sense, it is postulated a form of entropy that involves a deformed theory of generalized thermodynamics. Thus, some generalizations of statistical mechanics have been proposed [5]. In this context, it was shown that a natural realization of q-deformed thermodynamics bosons and fermions can be built on the "q-calculus" formalism. In the recent past, some developing in q-deformed quantum group were studied [6]. In a specific case, we focus our attention on the study of the thermal and electrical problem in a solid. Asolid consists of a large number of atoms linked by cohesive forces of various kinds. Atomic motion in a solid is very slight, causing every atom to move only within a small neighborhood, and vibrate around its equilibrium point. In a crystalline solid, the equilibrium points of atomic vibrations form a regular spatial structure, such as a cubic or hexagonal structure. Interaction between atoms allows the propagation of elastic waves in solid media, which can be both horizontal and longitudinal. Although this phenomenon is predominant at room temperature, it is very different at low temperature, where the electronic part take a more important role. Previous studies have analyzed the behavior of the q-deformed phonon contribution [7]. In this report, we are going to add the q-deformed electronic contribution and discuss new properties and obtained parameters. In the first part, we present a brief introduction of the q-deformed quantum algebra, with the creation and annihilation operators. Then in a second part, we set up our fermionic system by calculating the q-deformed Fermi-Dirac statistics. In the third part, we use this statistics to get some q-deformed thermodynamic parameters (the q-deformed total energy, specific heat, Sommerfeld parameter,etc.), where we note that the q-deformation can be a phenomena due to impurities in the material. Finally, we present our conclusions and ongoing research. All calculations and graphs were performed thanks to the software MAPLE 16.