On The Dispersion Relation And Zakharov Equation In Non-extensive Distributed Plasmas

Plasma is a typical electromagnetic medium, which can support a large numberof wave modes. So it is important to study the plasma wave in plasma physics.Dispersion relations are fundamental for studying the wave in the plasma. Accordingto the dispersion relations, we can study the problem of instability, propagation,refraction, absorption and nonlinear effects of the plasma wave.Non-extensive statistical mechanics has been developed recently as a very usefultool to describe the complex systems which present long-range interactions, whoseproperties cannot be exactly described by Boltzmann–Gibbs statistical mechanics.More and more people pay attention to its application in plasma, as a typicallong-range interaction system. Lima et al. have studied the dispersion relation andLandau damping of longitudinal oscillation based on q distribution in anon-relativistic plasma and the result showed that non-extensive formalism presents agood fit to the experimental data, while the standard Maxwellian distribution onlyprovides a crude description. The non-extensive statistical mechanics is successfullyapplied in plasma physics for the first time in this paper. However, the resultsobtained in this paper are not appropriate. So one of the main research content is toderive the correct analytic formulas for both the dispersion relation and Landaudamping of Langmuir wave in the Tsallis formalism based on the plasma kinetictheory. At the same time, we also have studied the dispersion property and Landaudamping of ion acoustic and transverse wave.With the development of nuclear fusion and Astrophysics, especially thedevelopment of laser technology, the electron is relativistic obviously in nuclearfusion by means of laser. so it is necessary to study the dispersion property inrelativistic plasmas with non-extensive distribution. In the second part, we havediscussed the dispersion property for plasma oscillation in an unmagnetized,collisionless, isotropic and relativistic plasma in the context of non-extensivedistribution. The analytical dispersion relation is obtained under the long-wavelength,short-wavelength and near light wave approximation in the ultra-relativistic regime.In the limiting case, the result based on the relativistic Maxwellian distribution is recovered. Because the analytical dispersion curve is discontinuous and restricted toultra-relativistic temperature, the full dispersion curve, ranging from the weak to theultra-relativistic case, is obtained by numerically solving the generalized dispersionequation.All kinds of instability can be developed in the plasma system. There arefrequent interactions between waves and particles because of nonlinear effects. Theinteractions will result in a large amount of energy exchanges among waves, whichwould compose very complex system. In other words, plasma system is in a state ofturbulence. Turbulence research, which is to discuss random interactions of numerousmotions in different scales, is the important branch in plasma physics. People havebeen looking about the nonlinear equations which can depict the turbulence. Theequation, usually called Zakharov equation, has been found by Zakharov in1972. Theevolution process is identical for the same initial conditions based on Zakharovequation. But the experiment results are different under the same macroscopicconditions, which illustrate the complexity of turbulence. Thus we have derived thetwo fluid equations in the Tsallis formalism, and then the non-extensive Zakharovequation is obtained based on two time scale approximation. The non-extensiveZakharov equation may be a physical mechanism of turbulence with the developmentof non-extensive statistics.