Harris sheet solution for magnetized quantum plasmas Fernando

Harris sheet solution for magnetized quantum plasmas Fernando Haas ferhaas@unisinos. br Unisinos, Brazil

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

Some developments l l Dawson’s (multistream) model applied to quantum two-stream instabilities [Haas, Manfredi and Feix, PRE 62, 2763 (2000)] Quantum MHD equations [Haas, Po. P 12, 062117 (2005)] Quantum modulational instabilities (modified Zakharov system) [Garcia, Haas, Oliveira and Goedert, Po. P 12, 012302 (2005)] Quantum ion-acoustic waves [Haas, Garcia, Oliveira and Goedert, Po. P 10, 3858 (2003)]

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

Wigner-Poisson system

Remarks l l l 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 l l l Electromagnetic Wigner equation: [Haas, Po. P (2005)] This is an ugly looking equation so I will not try to show it! Sensible simplifications are needed hydrodynamic models

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

Quantum magnetohydrodynamics l l Highly conducting two-fluid plasma merging QMHD [Haas, Po. P (2005)] The quantum parameter (QMHD):

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

Vector potential

A pseudo-potential

Ampere's law equivalent to a Hamiltonian system

Pressure balance equation l It can be shown that

Remarks l l 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 l The pressure p = p(n) l The pseudo-potential

Harris sheet solution l l l 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 l Using a suitable rescaling:

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

Classical limit

Ultra-quantum limit

Numerical simulations (H=3) 1. 2 1 0. 8 0. 6 0. 4 0. 2 -15 -10 -5 5 10 15

Numerical simulations (H=5)

Final remarks l l l 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?