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Article Dans Une Revue Journal of Statistical Physics Année : 2021

The Magnetized Vlasov–Ampère System and the Bernstein–Landau Paradox

Résumé

We study the Bernstein-Landau paradox in the collisionless motion of an electrostatic plasma in the presence of aconstant external magnetic field. The Bernstein-Landau paradox consists in that in the presence of the magnetic field,the electric field and the charge density fluctuation have an oscillatory behavior in time. This is radically different fromLandau damping, in the case without magnetic field, where the electric field tends to zero for large times. We considerthis problem from a new point of view. Instead of analyzing the linear magnetized Vlasov-Poisson system, as it isusually done, we study the linear magnetized Vlasov-Amp`ere system. We formulate the magnetized Vlasov-Amp`eresystem as a Schr ̈odinger equation with a selfadjoint magnetized Vlasov-Amp`ere operator in the Hilbert space ofstates with finite energy. The magnetized Vlasov-Amp`ere operator has a complete set of orthonormal eigenfunctions,that include the Bernstein modes. The expansion of the solution of the magnetized Vlasov-Amp`ere system in theeigenfunctions shows the oscillatory behavior in time. We prove the convergence of the expansion under optimalconditions, assuming only that the initial state has finite energy. This solves a problem that was recently posed in theliterature. The Bernstein modes are not complete. To have a complete system it is necessary to add eigenfunctionsthat are associated with eigenvalues at all the integer multiples of the cyclotron frequency. These special plasmaoscillations actually exist on their own, without the excitation of the other modes. In the limit when the magneticfields goes to zero the spectrum of the magnetized Vlasov-Amp`ere operator changes drastically from pure point toabsolutely continuous in the orthogonal complement to its kernel, due to a sharp change on its domain. This explainsthe Bernstein-Landau paradox. Furthermore, we present numerical simulations that illustrate the Bernstein-Landauparadox. In Appendix B we provide exact formulas for a family of time-independent solutions.
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Dates et versions

hal-03542990 , version 1 (28-02-2020)
hal-03542990 , version 2 (15-02-2024)

Identifiants

Citer

Frédérique Charles, Bruno Després, Alexandre Rege, Ricardo Weder. The Magnetized Vlasov–Ampère System and the Bernstein–Landau Paradox. Journal of Statistical Physics, 2021, 183 (2), pp.23. ⟨10.1007/s10955-021-02755-9⟩. ⟨hal-03542990v2⟩
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