BGK Modes
Lucio Demeio
Dipartimentodi MatematicaV.Volterra
Universita'degli Studidi Ancona
Via BrecceBianche1 ,60131Ancona- Italy
E-mail: demeio@popcsi.unian.it
Abstract
First-order quantumcorrectionstotheclassicalBernstein-Greene-
Kruskalequilibriawithzerophasevelo cityareconsidered. TheWigner-
Poisson systemwith p erio dic b oundary conditions is solved byusing
a p erturbative approachab out the classical solution. The numerical
results indicate that themost imp ortant quantum corrections to the
classicalBGKsolutiono ccurnearthephase-spaceseparatricesb etween
classically trapp edanduntrapp edparticles.
1 Classical and quantum BGK modes
In this contribution, westudy the quantumanalogue of the classicalsteady-
state solutions of the Vlasov-Poisson system (BGK mo des), intro duced by
Bernstein, Greene and Kruskal in 1957 [1] and describing the spatially in-
homogeneous equilibria of a collisionless unmagnetized plasma with immo-
bile ions. The classical system is characterized by the electron distribu-
tion function f(x;p) and by the self-consistent electric eld E(x),which are
the unknown quantitiesand ob ey the nonlinear,stationary, one-dimensional
Vlasov-Poisson system[2]
p
@f
@x E
@f
@p
=0 (1)
@E
@x
=1 Z
fdp; (2)
wehaveused themomentumvariablepinsteadofthe usualvelo cityvariable
v, since the quantumformulation uses p as a phase-space variable together
with x. Intheclassicalcase, wesimplyhavep=mv. Here,allquantitiesare
dimensionless;xismeasuredinunitsoftheDebyelength
D
= q
T=(4ne 2
),
pinunitsofmv
th
,withv
th
= q
T=mthethermalsp eed,f inunitsofn=(mv
th )
and E in units of 4ne
D
(m is the electron mass, e the electron charge,
T the plasma temp erature and n the plasma density). Furthermore, we
imp osep erio dic b oundary conditions inspace, of the typ e f(0;p) =f(L;p).
However,instead of prescribing the p erio d L at the outset, the distribution
functionf(x;p)isassignedatx=0andthep erio dListhendeterminedfrom
theequations. Werecallthat,duetotheGalileaninvarianceoftheequations,
any solution (f(x;v);E(x)) of (1)-(2) can b e transformed into a solution of
the time-dep endentVlasov-Poisson systemwith an arbitrary phase velo city
V. These are BGK mo des with non-zero phase velo city. Here, however,
weconsideronly steady-statesolutions, thatis BGKmo deswith zero phase
sp eed. The solutions of (1)-(2) can b echaracterizedin the followingway.
First, note that the single-particle energy E p 2
=2 (x) is a constant
of motion, thereforef(x;p) =F(E) satis es the Vlasov equation identically.
The electrostaticp otential (x) partitionsthe electronp opulationintothree
groups of particles: trapp edparticles, untrapp edparticleswith p ositive mo-
mentumand untrapp edparticleswithnegativemomentum. The curvesthat
divide the trapp ed and untrapp ed p opulations of the phase space are sepa-
ratrices of a given energy
s
. Then, one can think of a problem with four
comp onents: the three groups of particles and the electrostatic p otential.
In general, three of these comp onents can b e xed with some arbitrariness
and the fourth one isthen found by solving the equations. In our approach,
we assign the distribution function over the entire phase space; Poisson's
equation then b ecomes a dierential equation for the electrostatic p oten-
tial (x). The distribution function is assigned at one b oundary, x = 0,
by f(0;p) = m(p)(1 + ), for a given function m(p) and arbitrary. In
general,
s
and are small p ositive numb ers. As they vary, a family of
BGK equilibria is obtained, which we identify as BGK equilibria generated
bythe function m(p). The electrostaticp otentialissubjectto the b oundary
conditions (0) =
s
and 0
(0) = 0. These b oundary conditions and the
numerical generation of BGK mo des are discussed in more detail in [3]. If
(x) istheunknown electrostaticp otential,the equationsof theseparatrices
are givenby
p(x) = q
2( (x)+
s
)=p
0 (x):
Then for p (x) < p < p (x) (trapp ed particles) the distribution is left
t 0 0
particles) we take f(x;p) = f(0;p ), where pis the momentumobtained by
considering the constant energy curve passing through the p oint (x;p) of
the phase space and tracing it back to x = 0. The value of pis given by
p 2
=2 (x)=p 2
=2+
s
. Summarizing,f(x;p) =F(E) where
F(E)= 8
>
>
<
>
>
:
F (E)=f
0;
q
2(E
s )
; E >
s
andp< p
0 (x)
F
t
(E);
MAX
<E <
s
F
+
(E)=f
x;
q
2(E
s )
; E >
s
andp>p
0 (x)
(3)
where
MAX
is the maximumvalue of the p otential. The distribution func-
tion thus constructed is a functional of the electrostatic p otential (x).
Therefore, Poisson's equation can b e writtenas:
d 2
dx 2
=1 R( ) (4)
with R( ) = R
f(x;p)dp = R
jjJjj
F(E)dE, where jjJjj
is the Jacobian of
the transformation from the momentumvariable to the energy variable for
xed x. For an arbitrary function w (x;p) =W(E), we have
Z
+1
1
w (x;p)dp = Z
jjJjj
W(E)dE = Z
1
s
(W (E)+W
+
(E))dE
q
2(E + (x)) +
+ 2 Z
s
(x) W
t (E)dE
q
2(E + (x))
(5)
where W
+
, W and W
t
corresp ond to the untrapp ed (withp>0and p<0)
and trapp edparticles,resp ectively. Then,
R( )= Z
1
s
(F (E)+F
+
(E))dE
q
2(E )
+2 Z
s
(x) F
t (E)dE
q
2(E )
: (6)
With this expression, eq. (4) b ecomes a second order dierential equation
for the p otential (x), equipp ed with the b oundary conditions (0) =
s
and 0
(0)=0. In our numericalapproach,the integralsapp earing in (6) are
calculated by standard Laguerre and Chebychev collo cation rules and the
dierentialequationfor thep otentialissolvedby usingastandardmarching
techniquefrom the b oundary at x=0. The p erio dicity length L isthen ex-
tracted from the solution of the dierentialequationby using the condition
(L)= (0). InourformercalculationsofclassicalBGKmo des[3],thecom-
p otentialis governed by the parameter . For =0 wehavef(0;p) =m(p)
andtheonlysolutionistheonewithzero eldandauniformdistribution. As
b ecomesp ositive,aBGKmo dewith nonzero eldamplitudeisgenerated.
ThequantumversionoftheVlasov-PoissonsystemistheWigner-Poisson
system[5,6],inwhichtheWignerdistributionf
W
takestheplaceoftheclas-
sical distribution function and ob eys an evolutionequation where the accel-
eration termis given by a pseudo-dierentialop erator, in place of the usual
dierential op erator of the Boltzmann equation. The steady-state Wigner-
Poisson systemis:
p
@f
W
@x i
~
(f
W
)=0 (7)
@E
@x
=1 Z
f
W
dp; (8)
where f
W
(x;p) is the Wigner function, ~ = h =(mv
th
D
) and the pseudo dif-
ferentialop erator isde ned by
(f
W
)(x;p)= Z
dÆ (x;) b
f
W
(x;)e ip =~
; (9)
with
b
f
W
(x;)= 1
2~ Z
dpf
W
(x;p)e ip =~
(10)
the Fourier transform of the Wigner function and Æ (x;) = (x+=2)
(x =2)thesymb olofthepseudo dierentialop erator. Onthe system(7)-
(8) we imp ose the same b oundary conditions used for the classical Vlasov-
Poisson system (1)-(2), that is f
0
(0;p) = m(p)(1+ ),
0
(0) =
s and
0
(0) =0. The steady-state Wigner-Poisson system(7)-(8) admits spatially
p erio dic solutions, as was proved by Lange, To omire and Zweifel [7]. Fur-
thermore, it is well known that the Wigner-Poisson system reduces to the
classical Vlasov-Poisson system as ~ ! 0 [6, 8]. It is therefore natural to
lo ok for smallquantumcorrections to the classical BGK equilibriaby using
p erturbation metho ds. In this work, weshall develop only a rst-order the-
ory and pro duce numerical results showing the quantum corrections to the
classical BGKequilibria generated bya two-streamdistribution.
tum correction
Asa rststep,weconsidertheasymptoticexpansionofthepseudo dierential
op erator in p owersof ~. It can b eshown that
i
~
(f
W
)(x;p) = E(x)
@f
W
@p +
~
2
24
@ 2
E
@x 2
@ 3
f
W
@p 3
+O (~ 4
): (11)
By setting = ~ 2
=24, the steady-state Wigner-Poisson system (7)-(8) b e-
comes:
p
@f
W
@x
E(x)
@f
W
@p +
@ 2
E
@x 2
@ 3
f
W
@p 3
+O (
2
)=0 (12)
@E
@x
=1 Z
f
W
dp: (13)
We nowintro duce a p erturbative expansionof f
W
and inp owersof :
f
W
(x;p)=f
0
(x;p)+f
1
(x;p)+O (
2
) (14)
(x)=
0
(x)+
1
(x)+O (
2
): (15)
Theexpansionofthep otential(15)inducesananalogousexpansionofE(x)=
0
(x): E(x)=E
0
(x)+E
1
(x)+O (
2
),whereE
n
(x)= 0
n
(x),n =0;1;:::.
Toorder 0
we have:
p
@f
0
@x E
0
@f
0
@p
=0 (16)
@E
0
@x
=1 Z
f
0
dp (17)
which is the classical steady-state Vlasov-Poisson system. If we imp ose the
b oundary conditions f
0
(0;p) =m(p)(1+ ),
0
(0) =
s and
0
0
(0) =0, the
solution toorder 0
issimplythefamilyofclassicalBGKequilibriagenerated
by m(p), that is f
0
(x;p) =F(E), with E p 2
=2
0
(x)and F(E)given by
(3). Toorder we have:
p
@f
1
@x E
0
@f
1
@p E
1
@f
0
@p +
@ 2
E
0
@x 2
@ 3
f
0
@p 3
=0 (18)
@E
1
@x
= Z
f
1
dp; (19)
subjectto theb oundary conditionsf
1
(0;p)=0,
1
(0)=0and 0
1
(0)=0. In
order to solveequations (18)-(19), it is convenientto adopt the transforma-
0
the rstordercorrectiongivesthe variationofthe Wignerfunctionalongthe
curves of constant E, and so f
1
(x;p) = F
1
(x;E). The partial derivatives of
f
0 and f
1
inthe new variablesx and E then b ecome:
@f
0
@p
=p(x)F 0
(E)
@ 3
f
0
@p 3
=p(x) h
p(x) 2
F 000
(E)+3F 00
(E) i
@f
1
@x
=
@F
1
@x
0 (x)
@F
1
@E
@f
1
@p
=p(x)
@F
1
@E
;
where p(x) = q
2[E +
0
(x)]. The dierential equation (18) for f
1 then
b ecomes
p(x)
"
@F
1
@x E
1 F
0
(E)+
@ 2
E
0
@x 2
p(x) 2
F 000
(E)+3F 00
(E)
#
=0; (20)
whose solution is
F
1
(x;E)=C(E) F 0
(E)
1
(x)+3
00
0 (x)F
00
(E)+
h
0
0 (x)
2
2
00
0
(x)(E+
0 (x))
i
F 000
(E):
The integration constant C(E) can b edeterminedfromthe b oundarycondi-
tions. Sincef
1
(0;p) =0wehaveF
1
(0;E)=0,and by usingthat 00
0
(0)=
and
1
(0)=0,weobtain
C(E)= [3F 00
(E)+2(E
s )F
000
(E)]
and thus
F
1
(x;E)= F 0
(E)
1
(x)+[3(
00
0
(x) )]F 00
(E)+
+2
"
E(
00
0
(x) )+ 00
0 (x)
0
(x)+
s
0
0 (x)
2
2
#
F 000
(E):(21)
The function
1
is stillunknown and must b e obtained from equation (19),
which involvesan integrationof f
1
overthe momentumspace. By using (5)
to p erformthe integrationin the variable E, equation (19) b ecomes:
d 2
1
2
(x)+ 2
(x)
1
(x)=g(x); (22)
zero ordersolution) and are given by
2
(x)= Z
jjJjj
0 F
0
(E)dE
g(x)=[
00
0
(x) ][3Æ(x)+2 (x)]+2
"
00
0 (x)
0
(x)+
s
0
0 (x)
2
2
#
(x);
where
Æ(x)= Z
jjJjj
0 F
00
(E)dE
(x)= Z
jjJjj
0 EF
000
(E)dE
(x)= Z
jjJjj
0 F
000
(E)dE
are p erio dicfunctionsofp erio dL. Weshallcho osethe b oundaryconditions,
and thus the zero order BGK equilibrium,so that 2
(x) > 0 and therefore
(x)isreal (andchosenp ositive). Equation(22)for
1
has to b esolvedsub-
ject to the b oundary conditions
1
(0) = 0
1
(0) =0. Toobtain the numerical
solution of equation (22), we use the same techniques used for the classical
BGKmo des. Substitutingfor
1
in(21)givesthe rst-ordercorrection f
1 to
the Wigner function.
3 Numerical results
In this Section,we consider BGK equilibria generated by a two-stream dis-
tribution,
m(p)= 1
2 p
2
h
e (p ps)
2
=2
+e (p+ps)
2
=2 i
(23)
with p
s
= 2. The trapp ed particle distribution must b e chosen so that the
existence conditions for the classical BGK equilibriumthat gives the zero-
order solution are satis ed [4]. Following our previous calculations of BGK
mo des[3],thetrapp edparticledistributionofequation(3)istakenconstant,
that is F
t
(E) = f(0;0). Of course, other choices are p ossible, but then the
existence of the zero-order solution would also haveto b e addressed.
In Figure 1 we show the classical BGK equilibrium, generated by the
function (23) with
s
=0:5 and =0:05, in the phase-space region around
the separatrices. The p erio d of this solution is L 17. Figures 2, 3 and 4
showthe rst-orderquantumsolutionfor=0:004. Figure2shows theBGK
quantumequilibriumWigner distribution in the phase-space region around
1
self-consistent p otential over the interval 0 x 2L. Figure 4 shows the
self-consistentQuantumBGKp otential
0
(x)+
1
(x)(dashedline)together
with the classical BGK p otential (solid line), overthe interval0 x 2L.
We rst note that
1
isnot p erio dic;if we lo ok at
1
overseveral p erio ds of
length L, it shows secularb ehaviour. Therefore, the straightforward expan-
sion intro duced in (14)-(15) is nonuniform,and it is meaningful onlyovera
spatialscaleofone p erio d. Thissuggests thattheproblemshouldb estudied
by other p erturbative techniques, such as the multiple-scaleanalysis, which
areb etterdesignedtoapproachthesecularb ehaviour. Fromtheseresults,we
seethatthequantumcorrectionto theBGKp otentialamountsto adecrease
of the amplitudeand a shortening of the p erio d. We also seethat, for small
, the dierences b etweenthe classical distributionfunction and the Wigner
function as determined by our straightforward p erturbative expansion, are
most evidentnear the separatrices.
x
p
f
Figure 1: ClassicalBGKequilibrium distribution f(x;p)asa functionof xand p
around thetrapp edparticle region. Here,0xLand 3p3.
x
p
f
Figure 2: QuantumBGK equilibrium Wigner distribution f
W
(x;p) f
0
(x;p)+
f
1
(x;p) as a function of x and p around the trapp ed particle region. Here, 0
xL and 3p3.
10 20 30
x
-20 -10 0 10
Figure3: Quantumcorrection,
1
(x),totheclassicalBGKp otentialasafunction
of x,0x2L.
10 20 30
x
-0.8 -0.6 -0.4 -0.2 0 0.2
Figure 4: ClassicalBGK p otential (x) (solidline) and quantumBGK p otantial
0
(x)+
1
(x)(dashed line) asa functionof x,0x2L.
References
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[3] L. Demeio,A numericalstudy of linearand nonlinear sup erp ositions of
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[4] M.Buchanan,J.J.Dorning,NonlinearElectrostaticWavesinCollision-
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[5] A.Arnold,P.Markowich,Thep erio dicquantumLiouville-Poissonprob-
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