Post on 21-Dec-2015
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