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




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
f
f
v
  K (v'v, x, t ) f (v' , x, t ),
t
x
E
e

(n0   fdv).
x  0
Remarks



In the formal classical limit (   0 ) 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
n

1
u
n
f dv ,

Pm
fv dv ,
 fv
2

dv  nu2 .
Quantum hydrodynamic model
(electrostatic plasma)
n


( nu)  0,
t x
u
u
1 p e
 2    2 n / x 2

u

 E
2
t
x
m n x m
2m x 
n
E
e

( n0  n),
x
0
p  p ( n).

,


Bohm’s potential or quantum pressure
term:
2
2

   n / x 


2

2m x 
n

2
Application: quantum two-stream
instability [Haas et al., PRE (2000)]
The quantum parameter (two-stream
instability)
H 
 p
2
0
mu
,
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 (nonrelativistic) magnetized plasma

n
   (nu )  0,
t


u 
1
e   
2  2 n 

 u  u  
p  ( E  u  B) 

2
t
mn
m
2m 
n 
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):
H 
i
2
me mi VA
One-component magnetized quantum
plasma: “1D” equilibrium

B  B y ( x) yˆ  Bz ( x) zˆ,
n  n( x),

u  u y ( x) yˆ  u z ( x) zˆ,
p  p ( n)
 
E  0,
Vector potential

A  Ay ( x) yˆ  Az ( x) zˆ,
B y  dAz / dx,
Bz  dAy / dx.
A pseudo-potential
V  V ( Ay , Az )

1 V
nuy  
,
e 0 Ay
1 V
nuz  
e 0 Az
Ampere's law  equivalent to a
Hamiltonian system
2
d Ay
2
dx
V

,
Ay
d Ax
V

.
2
dx
Ax
2
Pressure balance equation
d
V ( x)
 2 n d  d 2 n / dx2

( p ( n) 
)
dx
0
2m dx 
n

It can be shown that
B2
V 
 V0
2




Remarks


In general, the balance equation is an ODE for the
density n

A
Solving the Hamiltonian
system
for
yields

simultaneously Band
~
V  V ( Ay ( x), Az ( x))  V ( x)
Rewriting the balance equation
d 3 a da d 2 a
da
a na 3 
 f (a)
 g ( x)  0,
2
dx
dx dx
dx
4m a dp
f (a)   2
(n  a 2 ),
 dn
~
2m d V
g ( x) 
.
2
 0  dx
Free ingredients

The pressure p = p(n)

The pseudo-potential
V  V ( Ay , Az )
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
p  n B T ,
 2 Az 

V 
ex p
B L
2
  
2
B


Solving for A and then for B
(using suitable BCs)
By  B tanh(x / L),
Bz  B0
By / B
x/L
Balance equation for quantum Harris
sheet solution

Using a suitable
rescaling:
3
2

d
a
da
d
a d
2
2
2

H  a 3 

(
a

sec
h
x)
2 
dx dx  dx
 dx
Quantum parameter (quantum Harris
sheet)

H
m VA L
It increases with 1/m, 1/L, 1 / B and the
ambient density.
Classical limit
H  0,
n  a 2  sec h 2 x

localizedsolut ion
Ultra-quantum limit
d 3 a da d 2 a
H  1  a 3 
0
2
dx
dx dx
For a(x  0 )  1,

da
( x  0)  0,
dx
n  a 2  cos2 x

d 2a
( x  0)  1 :
2
dx
periodicsolut ion
Numerical simulations (H=3)
n
1.2
n
x
1
0.8
0.6
0.4
n
0.2
-15
-10
-5
5
10
15
Numerical simulations (H=5)
n
5
4
3
2
1
-30
-20
-10
10
20
30
Final remarks



In the quantum case, a Harris-type magnetic field
(together with p  n BT ) 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?