Plasma oscillations and spectral index in non-extensive statistics

In this paper, we have discussed the dependency of the plasma oscillations on the numbers of degrees of freedom (dimensionality) involved in the spectral indices (q or κ) of non-extensive statistical mechanics. Generally, the spectral indices have an inherent dimensional dependency, which may be ignored when a reduced velocity distribution function (VDF) has been derived from the higher dimensional ones, e.g., when studying the longitudinal oscillations or the related topics. In such cases, if we ignore the equipartition of energy, the higher dimensional dependency of the original VDF may be still included in the spectral index of our solutions. On the other hand, in many studies of the plasma oscillations in non-extensive statistics, the ordinary formalism of canonical probability distribution (the old version) had been used instead of the escort probability distribution (the modern version). The escort formalism has some advantages, as opposed to the ordinary version, which may provide more appropriate solutions for the plasma oscillations. In this study, we have re-examined this problem by utilizing the modern version of kappa distribution formalism labeled with an invariant kappa index as of zero dimensionality spectral index, κ0, which is independent of the dimensionality, the degrees of freedom, or the numbers of particles. The related dispersion relation and Landau damping rate of plasma oscillations (Langmuir waves) have been analytically derived and studied by using the Vlasov–Poisson equations for the weakly damped modes. The physical and thermodynamics features of the problem have been considered in terms of the variation of the invariant spectral index κ0.