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. 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, 062117 (2005)]

Quantum modulational instabilities (modified Zakharov system) [Garcia, Haas, Oliveira and Goedert, PoP 12, 012302 (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

).(

,),,'(),,'(

00

fdvne

x

E

txvftxvvKx

fv

t

f

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)

0

Hydrodynamic variables

.,

1

,

22

nudvfvmP

dvfvn

u

dvfn

Quantum hydrodynamic model (electrostatic plasma)

).(

),(

,/

2

1

,0)(

00

22

2

2

npp

nne

x

E

n

xn

xmE

m

e

x

p

mnx

uu

t

u

nuxt

n

n

xn

xm

22

2

2 /

2

Bohm’s potential or quantum pressure term:

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

The quantum parameter (two-stream instability)

,20mu

H p

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.

n

n

mBuE

m

ep

mnuu

t

u

unt

n

2

2

2

2)(

1

,0)(

Quantum magnetohydrodynamics

Highly conducting two-fluid plasma merging QMHD [Haas, PoP (2005)]

The quantum parameter (QMHD):

2Aie

i

VmmH

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

)(

,ˆ)(ˆ)(

),(

,0,ˆ)(ˆ)(

npp

zxuyxuu

xnn

EzxByxBB

zy

zy

Vector potential

./,/

,ˆ)(ˆ)(

dxdABdxdAB

zxAyxAA

yzzy

zy

A pseudo-potential

zz

yy

zy

A

V

enu

A

V

enu

AAVV

00

1,

1

),(

Ampere's law equivalent to a Hamiltonian system

.

,

2

2

2

2

x

x

y

y

A

V

dx

Ad

A

V

dx

Ad

Pressure balance equation

It can be shown that

n

dxnd

dx

d

m

nxVnp

dx

d 222

0

/

2)

)()((

0

2

2V

BV

Remarks

In general, the balance equation is an ODE for the density n

Solving the Hamiltonian system for yields simultaneously and

)())(),((~

xVxAxAVV zy

A

B

Rewriting the balance equation

.2

)(

),(4

)(

,0)()(

~

20

22

2

2

3

3

dx

Vdmxg

andn

dpmaaf

xgdx

daaf

dx

ad

dx

da

dx

adana

Free ingredients

The pressure p = p(n)

The pseudo-potential ),( zy AAVV

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

LB

ABV

Tnp

z

B

2exp

2

,2

Solving for and then for (using suitable BCs)

0),/tanh( BBLxBB zy

A

B

BBy /

Lx /

Balance equation for quantum Harris sheet solution

Using a suitable rescaling:

)sec( 222

2

3

32 xha

dx

d

dx

ad

dx

da

dx

adaH

Quantum parameter (quantum Harris sheet)

It increases with 1/m, 1/L, and the ambient density.

LmVH

A

B/1

Classical limit

solution localizedsec

,0

22

xhan

H

Ultra-quantum limit

solution periodiccos

:1)0(,0)0(10For

01

22

2

2

2

2

3

3

xan

xdx

adx

dx

da,)a(x

dx

ad

dx

da

dx

adaH

Numerical simulations (H=3)

n x

n

-15 -10 -5 5 10 15

0.2

0.4

0.6

0.8

1

1.2

n

Numerical simulations (H=5)

-30 -20 -10 10 20 30

1

2

3

4

5

n

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?

Tnp B