Harris sheet solution for magnetized quantum plasmas Fernando Haas [email protected] Unisinos,...
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Transcript of Harris sheet solution for magnetized quantum plasmas Fernando Haas [email protected] Unisinos,...
![Page 1: Harris sheet solution for magnetized quantum plasmas Fernando Haas ferhaas@unisinos.br Unisinos, Brazil.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d755503460f94a56885/html5/thumbnails/1.jpg)
Harris sheet solution for magnetized quantum plasmas
Fernando Haas
Unisinos, Brazil
![Page 2: Harris sheet solution for magnetized quantum plasmas Fernando Haas ferhaas@unisinos.br Unisinos, Brazil.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d755503460f94a56885/html5/thumbnails/2.jpg)
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)
![Page 3: Harris sheet solution for magnetized quantum plasmas Fernando Haas ferhaas@unisinos.br Unisinos, Brazil.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d755503460f94a56885/html5/thumbnails/3.jpg)
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)]
![Page 4: Harris sheet solution for magnetized quantum plasmas Fernando Haas ferhaas@unisinos.br Unisinos, Brazil.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d755503460f94a56885/html5/thumbnails/4.jpg)
Modeling quantum plasmas
Microscopic models:
N-body wave-function density operator Wigner function
Macroscopic models:
hydrodynamic formulation
![Page 5: Harris sheet solution for magnetized quantum plasmas Fernando Haas ferhaas@unisinos.br Unisinos, Brazil.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d755503460f94a56885/html5/thumbnails/5.jpg)
Wigner-Poisson system
).(
,),,'(),,'(
00
fdvne
x
E
txvftxvvKx
fv
t
f
![Page 6: Harris sheet solution for magnetized quantum plasmas Fernando Haas ferhaas@unisinos.br Unisinos, Brazil.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d755503460f94a56885/html5/thumbnails/6.jpg)
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
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Hydrodynamic variables
.,
1
,
22
nudvfvmP
dvfvn
u
dvfn
![Page 8: Harris sheet solution for magnetized quantum plasmas Fernando Haas ferhaas@unisinos.br Unisinos, Brazil.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d755503460f94a56885/html5/thumbnails/8.jpg)
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
![Page 9: Harris sheet solution for magnetized quantum plasmas Fernando Haas ferhaas@unisinos.br Unisinos, Brazil.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d755503460f94a56885/html5/thumbnails/9.jpg)
n
xn
xm
22
2
2 /
2
Bohm’s potential or quantum pressure term:
![Page 10: Harris sheet solution for magnetized quantum plasmas Fernando Haas ferhaas@unisinos.br Unisinos, Brazil.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d755503460f94a56885/html5/thumbnails/10.jpg)
Application: quantum two-stream instability [Haas et al., PRE (2000)]
![Page 11: Harris sheet solution for magnetized quantum plasmas Fernando Haas ferhaas@unisinos.br Unisinos, Brazil.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d755503460f94a56885/html5/thumbnails/11.jpg)
The quantum parameter (two-stream instability)
,20mu
H p
![Page 12: Harris sheet solution for magnetized quantum plasmas Fernando Haas ferhaas@unisinos.br Unisinos, Brazil.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d755503460f94a56885/html5/thumbnails/12.jpg)
![Page 13: Harris sheet solution for magnetized quantum plasmas Fernando Haas ferhaas@unisinos.br Unisinos, Brazil.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d755503460f94a56885/html5/thumbnails/13.jpg)
![Page 14: Harris sheet solution for magnetized quantum plasmas Fernando Haas ferhaas@unisinos.br Unisinos, Brazil.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d755503460f94a56885/html5/thumbnails/14.jpg)
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
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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)(
![Page 16: Harris sheet solution for magnetized quantum plasmas Fernando Haas ferhaas@unisinos.br Unisinos, Brazil.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d755503460f94a56885/html5/thumbnails/16.jpg)
Quantum magnetohydrodynamics
Highly conducting two-fluid plasma merging QMHD [Haas, PoP (2005)]
The quantum parameter (QMHD):
2Aie
i
VmmH
![Page 17: Harris sheet solution for magnetized quantum plasmas Fernando Haas ferhaas@unisinos.br Unisinos, Brazil.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d755503460f94a56885/html5/thumbnails/17.jpg)
One-component magnetized quantum plasma: “1D” equilibrium
)(
,ˆ)(ˆ)(
),(
,0,ˆ)(ˆ)(
npp
zxuyxuu
xnn
EzxByxBB
zy
zy
![Page 18: Harris sheet solution for magnetized quantum plasmas Fernando Haas ferhaas@unisinos.br Unisinos, Brazil.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d755503460f94a56885/html5/thumbnails/18.jpg)
Vector potential
./,/
,ˆ)(ˆ)(
dxdABdxdAB
zxAyxAA
yzzy
zy
![Page 19: Harris sheet solution for magnetized quantum plasmas Fernando Haas ferhaas@unisinos.br Unisinos, Brazil.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d755503460f94a56885/html5/thumbnails/19.jpg)
A pseudo-potential
zz
yy
zy
A
V
enu
A
V
enu
AAVV
00
1,
1
),(
![Page 20: Harris sheet solution for magnetized quantum plasmas Fernando Haas ferhaas@unisinos.br Unisinos, Brazil.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d755503460f94a56885/html5/thumbnails/20.jpg)
Ampere's law equivalent to a Hamiltonian system
.
,
2
2
2
2
x
x
y
y
A
V
dx
Ad
A
V
dx
Ad
![Page 21: Harris sheet solution for magnetized quantum plasmas Fernando Haas ferhaas@unisinos.br Unisinos, Brazil.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d755503460f94a56885/html5/thumbnails/21.jpg)
Pressure balance equation
It can be shown that
n
dxnd
dx
d
m
nxVnp
dx
d 222
0
/
2)
)()((
0
2
2V
BV
![Page 22: Harris sheet solution for magnetized quantum plasmas Fernando Haas ferhaas@unisinos.br Unisinos, Brazil.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d755503460f94a56885/html5/thumbnails/22.jpg)
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
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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
![Page 24: Harris sheet solution for magnetized quantum plasmas Fernando Haas ferhaas@unisinos.br Unisinos, Brazil.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d755503460f94a56885/html5/thumbnails/24.jpg)
Free ingredients
The pressure p = p(n)
The pseudo-potential ),( zy AAVV
![Page 25: Harris sheet solution for magnetized quantum plasmas Fernando Haas ferhaas@unisinos.br Unisinos, Brazil.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d755503460f94a56885/html5/thumbnails/25.jpg)
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
![Page 26: Harris sheet solution for magnetized quantum plasmas Fernando Haas ferhaas@unisinos.br Unisinos, Brazil.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d755503460f94a56885/html5/thumbnails/26.jpg)
Choice for Harris sheet magnetic field
LB
ABV
Tnp
z
B
2exp
2
,2
![Page 27: Harris sheet solution for magnetized quantum plasmas Fernando Haas ferhaas@unisinos.br Unisinos, Brazil.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d755503460f94a56885/html5/thumbnails/27.jpg)
Solving for and then for (using suitable BCs)
0),/tanh( BBLxBB zy
A
B
BBy /
Lx /
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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
![Page 29: Harris sheet solution for magnetized quantum plasmas Fernando Haas ferhaas@unisinos.br Unisinos, Brazil.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d755503460f94a56885/html5/thumbnails/29.jpg)
Quantum parameter (quantum Harris sheet)
It increases with 1/m, 1/L, and the ambient density.
LmVH
A
B/1
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Classical limit
solution localizedsec
,0
22
xhan
H
![Page 31: Harris sheet solution for magnetized quantum plasmas Fernando Haas ferhaas@unisinos.br Unisinos, Brazil.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d755503460f94a56885/html5/thumbnails/31.jpg)
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
![Page 32: Harris sheet solution for magnetized quantum plasmas Fernando Haas ferhaas@unisinos.br Unisinos, Brazil.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d755503460f94a56885/html5/thumbnails/32.jpg)
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
![Page 33: Harris sheet solution for magnetized quantum plasmas Fernando Haas ferhaas@unisinos.br Unisinos, Brazil.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d755503460f94a56885/html5/thumbnails/33.jpg)
Numerical simulations (H=5)
-30 -20 -10 10 20 30
1
2
3
4
5
n
![Page 34: Harris sheet solution for magnetized quantum plasmas Fernando Haas ferhaas@unisinos.br Unisinos, Brazil.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d755503460f94a56885/html5/thumbnails/34.jpg)
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