Macroscopic description for the quantum Weibel instability F. Haas * and M. Lazar Institut für Theoretische Physik IV, Ruhr-Universität Bochum, D-44780 Bochum Germany Received 25 January 2008; published 15 April 2008 The Weibel instability in the quantum plasma case is treated by means of a fluidlike moments approach. Quantum modifications to the macroscopic equations are then identified as effects of the first or second kind. Quantum effects of the first kind correspond to a dispersive term, similar to the Bohm potential in the quantum hydrodynamic equations for plasmas. Effects of the second kind are due to the Fermi statistics of the charge carriers and can become the dominant influence for strong degeneracy. The macroscopic dispersion relations are of higher order than those for the classical Weibel instability. This corresponds to the presence of a cutoff wave number even for the strong temperature anisotropy case. DOI: 10.1103/PhysRevE.77.046404 PACS numbers: 52.59.Hq, 71.10.Ca, 52.65.Kj I. INTRODUCTION The field of quantum plasmas has been introduced long ago 1,2 and is presently attracting renewed attention from a variety of viewpoints. It was already confirmed that quantum mechanical effects, e.g., electron tunneling and wave-packet spreading, play a central role in the behavior of metallic or semiconductor nanostructures of the next generation elec- tronic devices 3–5. Some astrophysical compact objects, such as white dwarf or neutron stars, possess very high tem- perature and strong quantum effects due to their large densi- ties 10 6 g / cm 3 6. There have been recent studies in quantum plasmas involving quantum turbulence 7, quan- tum analogs for the Harris sheet 8, quantum models taking into account spin 9,10, stable solitary structures 11, dark soliton and vortices solutions 12, variational structures for the quantum Zakharov system 13, as well as application of quantum hydrodynamic equations for carbon nanotubes 14. The growing interest on quantum plasmas comes in part from the recently introduced hydrodynamic equations 15–17, which are simpler in comparison to the kinetic de- scriptions used in the original developments. However, the Weibel instability 18 is usually treated in terms of kinetic descriptions. The Weibel instability is one of the basic plasma instabilities and is driven by an anisotropic velocity distribution of plasma particles 18,19. The quantum version of the Weibel instability has been recently proposed 20–22 on grounds of the dispersion relation for the Wigner- Maxwell system, which is the quantum counterpart of the Vlasov-Maxwell system. Therefore, the details of the insta- bility are dependent on the precise form of the equilibrium Wigner pseudodistribution function, in a similar way as the traditional Weibel instability is partially dependent on the exact form of the classical equilibrium distribution function. The purpose of this paper is to overcome this condition by means of a moment description for the quantum Weibel in- stability. Recently, the classical Weibel instability was inves- tigated by Basu 23 taking moments of the Vlasov-Poisson system and the present work follows basically the same strat- egy. Here, however, the starting point is the linearized Wigner-Maxwell system. It is also interesting to verify to what extent a fluidlike approach as the moment method is able to capture the essentials of the Weibel instability, in the quantum case. Some peculiar subtleties coming from the quantum nature of the model equations will show up. The transition from a kinetic to a fluidlike approach in a quantum plasma model will be shown to reveal the quantum effects of a different nature according to the density of the system, as explained more thoroughly in the continuation. Classical plasmas frequently have equilibrium distribution functions anisotropic in velocity space 24 –26. In the con- text of quantum plasmas, velocity anisotropy can arise at least for laser plasmas and neutron stars. It is well known 27 that anisotropic heating by resonant absorption can pro- duce a Weibel-like instability in laser plasmas. Also, there is experimental evidence of Weibel instability in laser-solid in- teraction experiments 28. In addition, in tunnel-ionized la- ser plasmas there can be velocity anisotropy driven by a varying laser polarization 29. Quantum effects should be more evident in the next generation of laser-solid interaction experiments, where the densities are very high. For neutron stars, it has been conjectured 30 that anisotropic heating can arise in view of fast rotation, implying a strongly de- formed neutrino sphere and anisotropic neutrino fluxes. There are estimates 31 where the pole-to-equator neutrino flux ratio can assume a value of 2. For these reasons, it is important to have a better understanding of the Weibel insta- bility taking into account quantum effects. As examples of distinct equilibrium Wigner functions for the quantum Weibel instability, one can have Maxwell- Boltzmann or Fermi-Dirac functions, both with anisotropy in velocity space. Using a moment description, there is some lost of information, but more universal statements are made available. As for any moments or fluid modeling, an intrinsic limit of such an approach is in the closure of the equations. Indeed, one is always faced with a system where the equa- tion for the time evolution of the velocity moment of order n depends on the velocity moment of order n +1. In this way 23, it happens that the moment approach is appropriate only for long wavelength and large temperature anisotropy. Moment descriptions have also been applied to cyclotron wave-particle interaction 32. * Also at Universidade do Vale do Rio dos Sinos—UNISINOS, Av. Unisinos 950, 93022–000, São Leopoldo, RS, Brazil. PHYSICAL REVIEW E 77, 046404 2008 1539-3755/2008/774/0464046 ©2008 The American Physical Society 046404-1