Simplified calculations of plasma oscillations with non-extensive statistics

Abstract

We use the exponential parametrization of the nonextensive distribution to calculate the dielectric constant in an electron gas obeying the nonextensive statistics. As we show, the exponential parametrization allows us to make such calculations in a straightforward way, bypassing the use of intricate formulas obtained from integral tables and/or numerical methods. For illustrative purposes, we apply first the method to the calculation of the permittivity and the corresponding dispersion relation in the ultrarelativistic limit of the electron gas, and verify that it reproduces in a simple way the results that had been obtained previously by other authors using the standard parametrization of the nonextensive distribution. In the same spirit we revisit the calculation of the same quantities for a non-relativistic gas, in the high frequency limit, which has been previously carried out, first by Lima, Silva and Santos, and subsequently revised by Chen and Li. Our own results agree with those obtained by Chen and Li. For completeness, we also apply the method the low frequency limit in the non-relativistic case, which has been previously considered by Dai, Chen and Li in the context of the stream plasma instability. We discuss some features of the results obtained in each case and their interpretation of terms of generalized nonextensive quantities, such as the Debye length $\lambda_{D}^{(q)}$, the plasma frequency $\omega_{p}^{(q)}$ and the ultra-relativistic frequency $\Omega^{(q)}_{e,rel}$. In the limit $q \rightarrow 1$ such quantities reduce to their classical value and the classical result of the dispersion relations are reproduced.