Stability analysis of a three-stream quantum-plasma equilibrium

Abstract

We apply the Nyquist method to analyze the stability of small amplitude electrostatic waves in quantum plasmas described by the Wigner-Poisson system. The technique is presented in detail to assess the stability of a threestream equilibrium distribution function. The topology of the Nyquist diagrams is substantially changed with increasing quantum diffraction effects. I IntroductionThe topic of quantum plasmas has received considerable attention in the recent literature [1]- [8]. The main reason for this is the increasing importance of quantum effects in the manufacturing process and performance of today's micro-electronic devices. For instance, no classical transport model can adequately incorporate quantum effects such as resonant tunnelling, which is the basis for the operation of devices like the resonant tunnelling diode. Accordingly, there has been much work dedicated to the development of quantum transport models. One of such models is provided by the Wigner-Poisson system [9,10], which is the main concern of the present paper. We address the question of the stability of small amplitude electrostatic waves described by the Wigner-Poisson system. In the formal classical limit, the Wigner-Poisson system goes into the Vlasov-Poisson system, with linear stability properties much better understood than those of the corresponding quantum model. One natural question in this regard is: how do quantum effects modify the linear stability properties of typical classical plasma equilibria? Here we address this question in detail for the case of a three-stream plasma equilibrium.The subject of quantum plasma stability is a subtle one. In [11] it was shown that two counter-streaming beams generate a linearly unstable mode that is not present in the corresponding classical situation. In the opposite direction, for large wave-numbers quantum effects are responsible for the suppression of all unstable modes. The extra details appearing in the quantum context can be traced back to the subtle nature of quantum wave-particle interactions in a plasma. In [12], the quantum plasma instability question is addressed by means of the Nyquist method. Let ´ µ Ö´ µ · ´ µ ¼ be the dispersion relation for the quantum plasma, where and are the frequency and wave-number of linear waves, and where the dispersion function is decomposed into its real and imaginary parts. In the Nyquist method, the stability of a mode of fixed wave-number is checked by varying (taken as a real variable) from minus to plus infinity. Drawing the corresponding curve in the Ö ¢ plane, we obtain the socalled Nyquist diagrams. The number of unstable modes is equal to the number of times the Nyquist diagram encircles the origin. For Vlasov-Poisson plasmas, the Nyquist method shows that monotonically decreasing equilibrium functions of the energy are stable against small amplitude perturbations. Also, in this classical case the sign of the associated Penrose functional [13,14] determines the linear stability of symmetric equilibria with at most two maxima. For Wigner...

Quantum Zakharov Equations

“…Applied to a two stream system, the multistream quantum plasma model predicts some new instabilities (attenuated by statistics, [2]) and provides an overall stabilization mechanism for large wave numbers. The same dispersive and stabilizing contribution appears for quantum effects in three stream plasmas [3]. Similar behavior also occurs for quantum plasmas described within the framework of the Wigner-Poisson system [4,5].…”

Physical interpretation of the quantum two-stream instability

“…While the mathematical techniques for the quantum plasma problem are the same as for the classical two-stream plasma, the quantum dispersion relation is more subtle as evidenced by Fig. (3). The above analysis can in principle be pursued on similar problems such as the quantum beam-plasma instability and on theories with a discrete structure such as the quantum multi-stream model.…”

“…Later on, a kinetic (Wigner-Poisson) treatment of the problem [2] showed that temperature effects can suppress the quantum instability. The quantum fluid equations have been further applied to several quantum streaming instability problems, like the three-stream quantum plasmas [3], the quantum dusty plasmas [4], the electron-positronion quantum plasmas [5] or the magnetized multi-stream instability [6]. The hydrodynamic formalism has also been applied to the quantum filamentation instability, with [7] or without [8] magnetization.…”

Quantum Weibel instability

“…However, in the intermediate regime where quantum effects are not too intense but are nevertheless not negligible, there are situations in which unexpected quantum instabilities can arise. Examples on this are the quantum two-stream and three-stream instabilities, [32][33][34][35] showing unstable modes of pure quantum nature and no classical counterpart. In addition, unlike classical plasmas, there is no Penrose functional determining the linear stability properties of quantum plasmas.…”