Generalized Kinetic Theory of Electrons and Phonons

A Generalized Kinetic Theory [1] was proposed in order to have the possibility to treat particles which obey a very general statistics. By adopting the same approach, we generalize here the Kinetic Theory of electrons and phonons. Equilibrium solutions and their stability are investigated. 1. Introduction Very recently [1], a Generalized Kinetic Theory (GKT) has been proposed by Rossani and Kaniadakis, in order to have the possibility to treat, at a kinetic level, particles which obey a very general statistics. The quasi–classical Boltzmann equation introduced in ref. [1] is a generalization of the Uehling–Uhlenbeck equation [2], which is intended for bosons and fermions only. In ref. [1] we have shown that our generalized Uehling–Uhlenbeck equation (GUUE) assures particle, momentum and energy conservation. Equilibrium, its uniqueness and stability (via an H–theorem) have been investigated. In ref. [3] Koponen points out that fractal or inverse power law distributions of phonon excitations are of interest in modeling various meaningful situations in solid state physics. Moreover, he feels that until recently there has been little guidance on how to generalize the kinetic theory of electrons and phonons obeying non Gibbsian statistics. Here we propose a Generalized Kinetic Theory for Electrons and Phonons (GKTEP), by following the same ideas which lead to the GUUE. In order to keep our GKTEP as general as possible, we introduce modified collision terms not only for phonons, but also for electrons, so that an application is allowed not only to electrons, but also to other particles (obeying a general statistics) which interact with a crystal lattice. The paper is organized as follows. In section 2 the Bloch–Boltzmann–Peierls [4] equations are briefly recalled, and our generalization is introduced. In section 3 and 4 we study the equilibrium solutions to the GKTEP equations and their stability is investigated, via an H theorem.