Harris sheet solution for magnetized quantum plasmas Fernando Haas Unisinos, Brazil.

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Presentation transcript:

Harris sheet solution for magnetized quantum plasmas Fernando Haas Unisinos, Brazil

Quantum plasmas High density systems (e.g. white dwarfs) Small scale systems (e.g. ultra- small electronic devices) Low temperatures (e.g. ultra-cold dusty plasmas)

Some developments Dawson’s (multistream) model applied to quantum two-stream instabilities [Haas, Manfredi and Feix, PRE 62, 2763 (2000)] Quantum MHD equations [Haas, PoP 12, (2005)] Quantum modulational instabilities (modified Zakharov system) [Garcia, Haas, Oliveira and Goedert, PoP 12, (2005)] Quantum ion-acoustic waves [Haas, Garcia, Oliveira and Goedert, PoP 10, 3858 (2003)]

Modeling quantum plasmas Microscopic models: N-body wave-function  density operator  Wigner function Macroscopic models: hydrodynamic formulation

Wigner-Poisson system

Remarks In the formal classical limit ( ) the Wigner equation goes to the Vlasov equation The Wigner function can attain negative values (a pseudo-probability distribution only) The Wigner function can be used to compute all macroscopic quantities (density, current, energy and so on)

Hydrodynamic variables

Quantum hydrodynamic model (electrostatic plasma)

Bohm’s potential or quantum pressure term:

Application: quantum two-stream instability [Haas et al., PRE (2000)]

The quantum parameter (two-stream instability)

Magnetized quantum plasmas Electromagnetic Wigner equation: [Haas, PoP (2005)] This is an ugly looking equation so I will not try to show it! Sensible simplifications are needed  hydrodynamic models

Quantum hydrodynamics for (non- relativistic) magnetized plasma plus Maxwell’s equations and an equation of state.

Quantum magnetohydrodynamics Highly conducting two-fluid plasma  merging  QMHD [Haas, PoP (2005)] The quantum parameter (QMHD):

One-component magnetized quantum plasma: “1D” equilibrium

Vector potential

A pseudo-potential

Ampere's law  equivalent to a Hamiltonian system

Pressure balance equation It can be shown that

Remarks In general, the balance equation is an ODE for the density n Solving the Hamiltonian system for yields simultaneously and

Rewriting the balance equation

Free ingredients The pressure p = p(n) The pseudo-potential

Harris sheet solution In classical plasmas, the Harris solution more frequently is build using the energy invariant to solves Vlasov In quantum plasmas, in general a function of the energy is not a solution for Wigner This also poses difficulties for quantum BGK modes

Choice for Harris sheet magnetic field

Solving for and then for (using suitable BCs)

Balance equation for quantum Harris sheet solution Using a suitable rescaling:

Quantum parameter (quantum Harris sheet) It increases with 1/m, 1/L, and the ambient density.

Classical limit

Ultra-quantum limit

Numerical simulations (H=3)

Numerical simulations (H=5)

Final remarks In the quantum case, a Harris-type magnetic field (together with ) is associated to an oscillating density The velocity field is also modified (it depends on the density) Stability questions were not addressed - what is the role of quantum correlations?