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On the modeling of quantum and complex plasmas
Hugo Fernando Santos Terças
Dissertação para obtenção do Grau de Mestre em
Engenharia Física Tecnológica
Júri
Presidente: Prof. João Seixas
Orientador: Prof. José Tito Mendonça
Vogais: Prof. Vítor Rocha Vieira
Prof. Jorge Loureiro
Junho 2007
Agradecimentos
Em primeiro lugar e de forma especial, o meu sincero e profundo agradecimento ao meu pro-
fessor e orientador José Tito Mendonça por ter sempre acredidato em mim, por me ter colocado
completamente à vontade aquando da escolha do tema desta tese, pels momentos de diálogo e
discussão, que embora fugazes, foram também momentos de descontracção, mas acima de tudo,
por me ter dado a esperança de continuar a acreditar na ciência e em particular na minha carreira
científica. Ao professor Vítor Vieira, que sempre tão gentilmente se disponibilizou a tirar dúvidas,
dentro e fora da sala de aula, e cuja inteligência e perspicácia eu admiro profundamente. Ao pro-
fessor Pedro Sacramento, pela sua indiscutível responsabilidade, pelo seu profissionalismo e pela
sua gentileza, que sem dúvida despertaram o meu interesse crescente pela mecânica quântica.
A ocasião é propícia para agradecer também aos meus colegas de curso, que de uma forma ou
de outra sempre estiveram solidários comigo, que acompanharam as minhas dificuldades e muitas
vezes deram óptimas sugestões. Aqui deixo uma saudação especial à Mariana Cardoso pela sua
humanidade e positivismo inspiradores e ao Marco Mercier, pela sua alegria, pela sua criatividade,
audácia e pragmatismo.
Aos meus amigos, por quem nutro a mais profunda e genuína gratidão, que acompanharam de
perto as minhas alegrias, as minhas tristezas, as preocupações e euforias. Um grande abraço e
votos de sucesso aos meus amigos Saulo Rego, Bruno Coelho, Alexandre Paulo e Ricardo Nunes
pelo especial impacto que tiveram durante este cinco anos da minha vida. Um muito obrigado pelos
momentos de apoio, carinho e amizade que especialmente me proporcionaram.
Finalmente, e de forma muito terna e especial, um muito obrigado à minha família. À minha
mãe, Maria Helena, pelo seu espírito guerreiro, pela sua força e vitalidade, pela sua capacidade
de sacrifício, pela sua honestidade e pela sua ternura que, mesmo após anos de distância, nunca
deixaram de me comover. Um muito obrigado e um saudoso beijo à mulher mais fantástica da
minha vida. À minha irmã, Carina Terças, pelo seu exemplo de força, vigor e carácter que tanto
têm contribuído para a ordem no lar. Ao meu irmão, Eduardo Terças, pela sua ternura, juventude,
bondade e justiça, qualidades em que tanto me revejo e dais quais tanto me orgulho. Aos meus
avós, Palmira Sá e José Santos, pelos seus testemunhos de vida e palavras sábias que tanto me
instruiram; pelos seus valores de humildade, pela sua honra e espirito de sacrifício, sobre os quais
me continuo a guiar após estes anos. Aos meus tios, José Maria Santos e Manuel Santos por terem
sempre depositado tanta confiança em mim e por me transmitirem sempre tanta força, coragem e
segurança.
A todos os demais que passaram pela minha vida durante estes cinco anos, e que por alguma
razão não foram mencionados, os meus sinceros agradecimentos.
i
Resumo
A teoria clássica dos plasmas tem-se dedicado essencialmente a regimes de altas temperaturas
e baixas densidades, onde os efeitos quânticos são irrelevantes. Contudo, avanços recentes na
tecnologia, nomeadamente a miniaturização de dispositivos semicondutores e o desenvolvimento
de objectos à nanoescala, têm viabilizado a aplicação da física dos plasmas em regimes onde
os efeitos quânticos são consideráveis. Existem também as experiências realizadas em regimes
especiais, tais como microgravidade e armadilhamento óptico (optical trapping), que apesar de
menos recentes, continuam a merecer a atenção da comunidade científica e reúnem as condições
necessárias para o surgimento de fenómenos complexos em plasmas. Numa primeira fase desta
tese, apresento os principais ingredientes que estão na base da modulação dos plasmas quânti-
cos, em regime não-colisional, de acordo com os resultados presentes nas diversas publicações
científicas. Os modelos aqui apresentados têm por base o formalismo de Wigner-Moyal, que é for-
malmente equivalente aos modelos de Shcrödinger não-linear e de fluido. Por fim, apresento um
estudo, ainda em curso, sobre as oscilações colectivas em plasmas complexos, através da definição
de um factor de forma de plasma que introduz correcções aos modos próprios de plasmão.
Palavras-chave:Função de Wigner, hidrodinâmica quântica, teoria cinética, factor de forma.
ii
Abstract
Classical plasma physics has mainly focused on regimes of high temperatures and low desities,
in which quantum mechanical effects play no role. Nevertheless, recent techonologial advances,
such as the miniaturization of semiconductor devices and nanoscale objects, have made possible
to preview the application of plasmas physics where the quantum effects may show up. Also, not
so recent but still on research experiments under microgravity and optical trapping provide special
conditions for plasmas to show complex behaviour. Dusty and complex plasmas have captured
the attention of plasma physicists, specially in what concerns to the envisage of technologial ap-
plications. In this thesis, I present the principal features in the modulation of quantum collisionless
plasmas and an original attempt on the modulation of some special oscillations in two-dimensionall
dusty structures, recently reported in experimental and theoretical review papers. The description
of quantum plasmas is made via Wigner-Moyal formalism and equivalentely via Schrödinger-like
equations, which can both be simplified to reduced fluid equations. The study of the oscillations in
complex plasmas is suggested by defining a plasma form factor for the single plasmon modes.
Key-words:Wigner Function, quantum hydrodynamics, kinetic theory, form factor.
iii
Contents
1 Introduction 2
2 From classical to quantum plasmas 42.1 Classical plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Quantum plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Plasma regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Wave and kinetic descriptions of quantum plasmas 93.1 Wave description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.1.1 Fluid model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.1.2 Nonlinear Schrödinger-Poisson system . . . . . . . . . . . . . . . . . . . . . . 12
3.2 Kinetic description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 Magnetized quantum plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4 Waves and instabilities in quantum plasmas 184.1 Electrostatic waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.1.1 Quantum Langmuir waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.1.2 Quantum ion-acoustic waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.1.3 Quantum Alfvén waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.2 Kinetic effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2.1 Zero-temperature Fermi gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2.2 One-stream instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2.3 Two-stream instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5 Nonlinear phenomena in quantum plasmas 285.1 Quantum Zakharov equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.2 Numerical solutions to the nonlinear Schrödinger-Poisson system . . . . . . . . . . . . 31
5.2.1 Time dependent solution of the Wigner function . . . . . . . . . . . . . . . . . . 31
5.3 Solitons in two-dimensional electron plasmas . . . . . . . . . . . . . . . . . . . . . . . 31
6 Complex plasmas 346.1 Plasma form factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
6.1.1 Integral formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
6.1.2 Mie resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
6.1.3 Dust Balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
6.1.4 Two-dimensional Yukawa liquids . . . . . . . . . . . . . . . . . . . . . . . . . . 37
iv
7 Conclusion 38
Appendices 40
A 40A.1 Fermi gas in d dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
A.2 Weyl transformation and the Liouville operator . . . . . . . . . . . . . . . . . . . . . . 41
A.2.1 The Wigner function and Weyl’s transformation . . . . . . . . . . . . . . . . . . 41
A.2.2 The Liouville operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
B 44B.1 Crank-Nicholson implicit scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
B.2 Alternating Direction Implicit scheme (ADI) . . . . . . . . . . . . . . . . . . . . . . . . 45
v
List of Figures
2.1 Plasma diagram in the log T − log n0 plane, separating the quantum and classical
regimes. METAL: electrons in a metal; IONO: ionospheric plasma; TOK: plasma in
the typical tokamak experiments for nuclear fusion; ICF: inertial confinement fusion;
SPACE: interstellar plasma; DWARF: white dwarf star. . . . . . . . . . . . . . . . . . . 7
4.1 Two stream-instability: (a): Plot of H = 1/K, H = 2/K and H2 = 2√K2 − 1/K2.
The thin line corresponds to the one-stream instability condition and the dashed area
represents the instability zone; (b): Square of the growth rate γ2. dashed line, H = 0;
thin line, H = 0.5; bold line, H = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.1 Ploting the numerical Wigner function for t = 0, t = 0.2, t = 0.7, and t = 1 (from top to
bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.2 Dispersion of the wave packet along the simulation box. Screen shots at t = 0, t = 0.1
and t = 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.3 Soliton in a two-dimensional plasma. ΓQ = 2. The time steps presented go from t = 0
until t = 0.9, in time steps of ∆t = 0.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1
Chapter 1
Introduction
The study of N -body systems in charged media is ubiquituos in several areas of physics, spe-
cially in plasma physics. Traditionally, the plasma state, actually referred as the fourth state of matter,
corresponds to an ionized gas with many species, where collective effects take place (waves, os-
cillations and instabilities). Technically, a plasma is said to be a quasineutral medium, which obeys
Laplace equation, but the concept is extended to non-neutral media, where one replaces Laplace
equation for the Poisson equation. Because of the complexity of such systems, plasma state is a
nonlinear and complex medium. The usual description of plasmas is based on magnetohydrody-
namics and kinetic theory, and is quite enough to characterize the dynamics of classical plasmas,
where the species are understood as material particles.
Traditional plasma physics has mainly focused on high-temperature and low-density regimes,
where quantum-mechanical effects play no role. Such regimes can be found, for instance, in fusion
and space plasmas. Recent technological advances, such as miniaturization of electronic devices
and nanoscale objects have made it possible to envisage applications at the nanoscale, where
quantum mechanics plays a very important role. Quantum effects become important when the
particle approximation is not possible, i.e, whenever the de Broglie length does not vanish. A good
example of such systems is a Fermi gas in an ordinary metal. It is quite obvious that different
combinations of the plasma parameters may result on different regimes of the dynamics. One may
expect high-density low-temperature plasmas and high-temperature low-density plasmas as good
candidates to quantum plasmas.
A possible application of quantum plasmas arises from semiconductor physics. Although the
density ot the charger carriers is much lower than in metals, the question of miniaturization intro-
duces characteristical lengths of the order of the de Broglie length λB , and so, quantum effects such
as tunneling and interference may occur. Also the manipulation of metallic nanostructures composed
by few atoms (clusters, thin metal films [1]) represents a feature of such phenomena [2]. Another
possible application of quantum plasma models is concerned with the study of astrophysical plasma
in extreme temperature and density conditions, such as white dwarfs, where the electron density is
much higher than ordinary metallic densities. Under these conditios, a white dwarf can behave as a
quantum fusion plasma.
In this context, we must understand how the correspondence between classical and quantum
plasmas should be established and which parameters are relevant to define. Let us motivate this job
with the quasineutrality condition. Technically, for classical plasmas, the quasineutrality is assured
if the charge separation occur only for a short distance, which is given by the Debye length λD. On
2
distances larger than the Debye length, the medium is basically neutral, and the so called plasma
approximation condition is satisfied. We must agree that a definition of these conditions in terms of
typical quantum parameters is very useful. As we shall see, there are a number of few parameters
that can be defined in order to establish the regime of a given plasma system.
In the next few chapters of this thesis we present a brief overview of quantum plasmas and how
to establish the paralell between classical and quantum plasmas, trying to report the main results
on quantum plasma modulation in a simple and pedagocial way. Then, we present an alternative
and original way of exploring quantum effects in few simple charged systems. We start the descrip-
tion of these quasi plasma systems with the classical equations and perform a quantization of the
oscillations, which result, by they own, on a quantum effect in charged systems. In other words, we
are interested in the study of the collective and individual plasmon modes, which, eventually, can be
quantized.
3
Chapter 2
From classical to quantum plasmas
2.1 Classical plasmas
Let us consider a classical plasma of electron density ne = n0 + n, where n0 is the density
and n represents its fluctuation. Neglecting the effects of temperature, and taking the ideal gas
approximation the pressure P vanishes as well. In this case, the system can be described by the
following set of linear equations
∂v∂t
= − e
mE
∂n
∂t+∇ · (n0v) = 0
∇ · E = − e
ε0n.
(2.1)
Here we assume that the ions are imobilized and their density equals the equilibrium density, ni ≈n0. After some simple calculations, we can derive the following equation for the density fluctuation
∂2n
∂t2+ ω2
pn = 0, (2.2)
where ωp =√n0e2/ε0m represents the plasma frequency, and appears as our first plasma param-
eter. This frequency represents the natural oscillation of electron inserted in the neutralizing ionic
background. Due to their inertia, ions can be assumed to be imobilized (me/mi ≈ 10−4). Although
this approximation is not completely true in the case of complex plasmas, where one can find species
as massive as ions, the frequency ωp as a typical plasma parameter can still be defined, replacing
the electron charge e for the dusty charge Q. The ideia remains the same: describing the oscillations
of negative charged particles in a neutralizing background. There are other corrections applied to
the charged media, in order to generalize a plasma frequency ωp, namely charge density distribution
and geometrical effects. In chapter (6) we discuss these generalizations.
As refered above, the plasma frequency does not take into account the effect of temperature. To
do that, we recall the energy equipartition theorem
K =12νkBT, (2.3)
4
where ν represents the number of freedom degrees, and define a typical plasma velocity
vth =
√kBT
m, (2.4)
where kB represents Boltzmann’s constant. vth is the so called thermal velocity of electrons. Using
the expressions for ωp and vth, one can define the Debye length
λD =√ε0kBT
n0e2, (2.5)
which represents a typical lenght in a plasma. To understand the physical meaning of this quantity,
let us consider a positive charged particle located at the origin immersed in a free electron sea. The
potential perturbation created by this charge verifies the Poisson equation
∇2δφ =e
ε0(ni − ne + δρi) . (2.6)
Expecting the ions to be uniformly distributed, ni ≈ n0 and electrons to follow a Maxwell-Boltzmann
distribution, in thermal equilibrium
ne = n0eeδφ
kBT ≈ n0
(1 +
eδφ
kBT
), (2.7)
and assuming a localized perturbation, δρi(r) = δ(r), eq. (2.6) leads to(∇2 − 1
λ2D
)δφ(r) =
e
ε0δ(r). (2.8)
Taking a Fourier transformation, we can easily integrate (2.8) and get
δφ(r) =e
4πε0re−r/λD , (2.9)
which represents the Thomas-Fermi screening. The physical meaning of λD becomes clear: the
Debye length describes the partial screening range of a positive charge potential due to electrons.
Also, it measures the distance beyond which the plasma approximation is valid.
The mean distance between the species in a plasma is given by 〈d〉 = n−1/30 . This allows us
to define an interaction (electrostatic) energy between them, Eint = e2n1/30 /ε0. Let us define a
dimensionless parameter with this interaction energy Eint and the kinetic energy K = kBT (ν = 1),
ΓC =Eint
K=e2n1/3
ε0kBT. (2.10)
This is known as the classical coupling parameter and is usefull to distiguish the several regimes
of a (classical) plasma. For small values of ΓC , the plasma is dominated for thermal effects and
electrostatic interactions remain weak. In this regime, the plasma is said to be collisionless. For
large values of ΓC , Coulomb interactions must be taken into account and the plasma is said to be
collisional or strongly coupled. This regime can be found in two dimensional Yukawa liquids, where
the dusty particles are confined by an harmonic potential, due to the balance between gravity and
electrostatic interactions between them [22].
Up to now, we have not seen how the quantities defined above are related to quantum plasmas.
In fact, they are all classical. As we could see, these quantities contain many general and important
informations about a classical plasma and its regimes. The idea is then to build a mapping between
these parameters and their quantum analogues, allowing us to establish a direct correspondence
between classical and quantum plasmas.
5
2.2 Quantum plasmas
Quantum mechanical effects start playing role whenever the average distance is comparable to
the de Broglie length
λB =~
mvth, (2.11)
i.e., whenever the approximation n−1/30 λB ≥ 1 is valid. In classical regimes, ~ → 0 and particles
can be considered pointlike and no quantum interference shows up. Thus, classical and quantum
regimes may not occur at the same time. However, recent studies have revealed that it is possible
to observe a phase transition between these two regimes [3]. To introduce the temperature to
quantum effects, let us recall a typical temperature. From solid state physics, we know that this
typical temperature is the Fermi temperature
TF =~2(3π2n0)2/3
2mkB. (2.12)
When T approaches TF , the relevant statistical distribution changes from Maxwell-Boltzmann to
Fermi-Dirac. The thermal transition between these two regimes are related with the scale parameter
n−1/30 λB :
χ =TF
T=
12(3π)2/3(n0λ
3B)2/3. (2.13)
Thus, quantum effects become important when χ ≥ 1. In order to establish a relevant set of typical
scales (time, space and velocity), we must stress that simple expressions can only be found in the
limiting cases χ 1 and χ 1, corresponding to pure classical and quantum regimes, respectively.
It is important to keep in mind the smooth transition between these two limits, which is, however,
hardly treatable in terms of dimensional analysis. In this transition regimes, quantum-mechanical
models based on both fluid and kinetic theories are used.
The typical time scale for collective behaviour in quantum plasma is still given by the inverse
of plasma frequency τQ = 1/ωp. However, in the deeply quantum regime χ 1, equation (2.4)
becomes meaningless and should be replaced by the typical velocity
vF =(
2EF
m
)1/2
=~m
(3π2n0)1/3, (2.14)
which is the well known Fermi velocity. Using the later quantity and the plasma frequency it is
possible to define the typical lenght scale
λF =vF
ωp, (2.15)
which is the quantum analog of the Debye lenght in (2.5), since it defines the length above which
a positive charge is completely screened and thus the plasma approximation is valid. This fact will
become clear in the following discussion.
The quantum coupling parameter can be obtained by generalizing (2.10). Replacing T for TF in
the expression of the kinetic energy, it is possible to write
ΓQ =Eint
EF=
2me2n1/30
ε0~2(3π2n0)2/3= 6π2
(~ωp
EF
)2
. (2.16)
Making use of (2.14) and (2.15), one can rewrite the later expression to obtain
ΓQ =(
1n0λ3
F
)2/3
. (2.17)
6
Figure 2.1: Plasma diagram in the log T − log n0 plane, separating the quantum and classical
regimes. METAL: electrons in a metal; IONO: ionospheric plasma; TOK: plasma in the typical toka-
mak experiments for nuclear fusion; ICF: inertial confinement fusion; SPACE: interstellar plasma;
DWARF: white dwarf star.
The advantage of using the later definition is obvious: by rewriting (2.10) in terms of λD, we can
redefine the classical coupling parameter as
ΓC =(
1n0λ3
D
)2/3
, (2.18)
which shows that λF is completely analogous to λD. Equations (2.16) and (2.17) are equivalent, but
the first has no classical equivalence and describes the quantum coupling parameter as the ratio
between the energy of an elementary excitation and the Fermi energy.
2.3 Plasma regimes
In the last section, we defined the coupling parameters ΓC and ΓQ, and the dimensionless tem-
perature χ. At this point, we should be able to define all plasma regimes in function of these di-
mensionaless quantities. While the later separates the plasma between its quantum and classical
regimes, ΓC and ΓQ provides the distinction between collisional and colisionless regimes, in both
quantum and classical regimes. In figure (2.1), we divide the temperature-density plane (in log-
scale) by setting ΓC , ΓQ, and χ equal to 1. The log T − log n0 plane is divided into four different
regions, two of which are classical (above T = TF ) and two quantum. Each classical/quantum re-
gion is divided into two collisional/collisionless subregions, identified by their characteristic transport
equations. As we shall see in the next chapter, Vlasov and Wigner equations describe the collision-
less regimes of classical and quantum plasmas, respectively. Boltzmann equation and a generalized
7
Winger equation are their collisional couterparts. We must though stress out that collisional effects
in the quantum regime are much hard to deal with and the present models are still controversial.
Also in fig. (2.1), we give examples of some natural and lab plasmas which illustrate both classical
and quantum, collisional and collisionless, regimes [29].
In order to conclude the discussion, we must remark that all previous considerations and defini-
tions have implicitly assumed thermal equilibrium condition. Out-of-equilibrium conditions involve a
kinetic description and the above results may not be entirely correct. For example, as we shall see
in the incoming chapters, if an electrom-beam is injected in the plasma, one must include its velocity
v0 when computing the de Broglie wavelenght λD, which will be smaller, since usually v0 > vth. For
this reason, systems that are far from equilibrium may be sometimes treated semiclassically, even
though the equilibrium regime is fully quantum.
8
Chapter 3
Wave and kinetic descriptions ofquantum plasmas
Quantum mechanics describes the N -body system exactly if we can solve Schrödinger’s equa-
tion for the N -particle wave function
Ψ(r1, r2, · · · , rN; t). (3.1)
This task is obviously impossible, both for analytical and numerical approaches. In fact, the drastic
but useful alternative is to neglect the correlation between the particles for every order and describe
the full wave function as the product of single particle wave functions
Ψ(r1, r2, · · · , rN; t) =N∏
i=1
ψi(ri, t). (3.2)
A first consequence of the single-particle decomposition is related to the Pauli’s exclusion principle,
since (3.1) must satisfies antisymmetry, for a fermionic system. Fortunatelly, this property can be
recovered if we replace the right-hand side of (3.2) by the associated Slater determinant, which pro-
vides a full antisymmetrization of the wave function. Physically, this weak version of Pauli’s principle
is satisfied when the coupling parameter ΓQ is small, and so, we may have a first guess for which
regimes (3.2) is valid. In the next sections, we present two different descriptions of quantum plasmas
(in fact, general enough to be extended to non-plasma systems) consisting in the wave description
(Schrödinger-Poisson (SP)) and in the phase space or kinetic description (Wigner-Poisson (WP)).
At the end of this chapter, we generalize the main results for a magnetized quantum plasma.
3.1 Wave description
3.1.1 Fluid model
Once discussed the validity of (3.2) in the context of quantum plasmas, we should understand
how does the single-particle wave function ψj behave. However, the N−body question must not be
neglected: how does the particle j “feel” the influence of the sorrounding ones? An approximate
solution to this question can be given in the context of mean field theory, which is know as the
Hartree approximation. The Hartree equation can be written in the form
9
(− ~2
2m∇2 + φion
)ψj +
e2
4πε0
∑k
∫dr′|ψk(r′)|2
|r− r′|ψj = εjψj , (3.3)
where φion represents the ionic potential. Making the correspondence between charge density and
probability in a quantum mixed state
ρj =∑
j
pj |ψj |2, (3.4)
where pj is the weigth of the quantum state, and rewriting the Hartree potential in terms of an
electronic self-consistent potential, leads
φe(r) =e
4πε0
∑k
∫dr′|ψk(r′)|2
|r− r′|. (3.5)
Putting (3.3), (3.4) and (3.5) together, we can write the following system for the specie j in a quantum
plasma
i~∂ψj
∂t= − ~2
2m∇2ψj + qjφψj
∇2φ =e
ε0
∑j
pj |ψj |2 − n0
,(3.6)
where qj denotes the charge of the specie j. The problem is still hard to deal with, if we do not
go further in assuming anything else. To contour this question, we should treat the set (3.6) semi-
classically. According to recent works, it is useful to think of the Schrödinger-Poisson system as the
quantum analog of Dawson’s multistream model [34]. In this model, the fluid in the phase space
is supposed to be a weighted sum over single-stream fluids. Therefore, the classical distribution or
phase space function can be writen as
f(r,v, t) =N∑
i=1
pjnj(r, t)δ(v − uj(r, t)) (3.7)
and must satisfy the Vlasov equation for an unmagnetized plasma [9].
∂f
∂t+ v · ∇f +
qjmj∇φ · ∇vf = 0, (3.8)
where qj denotes the charge of the specie j. Equation (3.8) is obtained from the Boltzmann equation
when the collisional term vanishes and the ponderomotive force is purely due to the self-consistent
potential. Inserting (3.7) in (3.8) and computing the momenta,
µs(r, t) =∫f(r,v, t)|v|s dv, (3.9)
where the zeroth (s = 0) order moment is the density and the first moment (s = 1) is the velocity, we
can derive the conservation equations for a single-stream j:
∂uj
∂t+ (uj · ∇)uj = ∇φ, (3.10)
∂nj
∂t+∇ · (njuj) = 0, (3.11)
10
∇2φ =e
ε0
∑j
pjnj − n0
. (3.12)
We must stress out that the vanishing collisional η∇2uj, compressive ξ∇∇ · uj and pressure ∇Pterms are consistent with the non-collisional and single-stream limits. To progress in this semiclassi-
cal approach, let us introduce a real phase αj(r, t) and a real amplitude Aj =√nj(r, t) and define
a parametric wave function
ψj = Ajeiαj/~. (3.13)
By inserting (3.13) in the system (3.6) and separating it into real and imaginary parts, we can find
∂A2j
∂t+m∇ · (A2
j∇αj) = 0 (3.14)
m∂∇αj
∂t+m2 (∇αj · ∇)∇αj =
qjm∇φ+
~2
2m2∇(∇2Aj
Aj
). (3.15)
Identifing the single-stream velocity with the phase in (3.13)
uj =1m∇αj , (3.16)
we can rewrite (3.14) and (3.15) and get
∂nj
∂t+∇ · (njuj) = 0, (3.17)
∂uj
∂t+ (uj · ∇)uj =
qjm∇φ+
~2
2m2∇
(∇2√nj√nj
). (3.18)
The latest two equations appears as a first fluid model for unmagnetized quantum plasmas. As
we can see, quantum effects are described by a pressurelike term in the momentum conservation
equation, also called the Bohm potential. It becomes clear that the classical fluid model (3.10)
and(3.11) is obtained in the limite ~→ 0.
Proceeding with the intention of simplifying (3.6), let us pull the physical frame back and de-
scribe the system, instead of its multistream representation. To do that, let us define some physical
quantities of the full system in terms of the single-stream quantities:
n =N∑
j=1
pjnj , (3.19)
v =1n
N∑j=1
pjnjuj, (3.20)
where the mean value of the velocity in (3.20) is computed for each component uα. By multiplying
(3.17) by pj and summing over the streams j
N∑j=1
∂
∂tpjnj +
N∑j=1
∇ · (pjnjuj) = 0, (3.21)
and using the definitions (3.19) and (3.20), one recovers the usual continuity equation.
∂n
∂t+∇ · (nv) = 0. (3.22)
11
Repeating the procedure for the single-stream momentum equation (3.18), one obtains
∂v∂t
+ (v · ∇)v =e
m∇φ− ∇P
mn+
~2
2m2∇
N∑j=1
pj
(∇2√nj√nj
), (3.23)
where the pressure P is defined in terms of the mean velocities
P = mn
N∑j=1
pjnju2j
n−
N∑j=1
pjnjuj
n
2 = mn
(〈u2
j 〉 − 〈uj〉2). (3.24)
However, a summation over all j streams still remains in the right-hand side of (3.23). In order
to obtain a closed system of two equations for the global averaged quantities n and v, two more
approximations are needed: First, we postulate the existance of a state equation in the form P =
P (n); secondly, we assume that we can replace the summation term
N∑j=1
pj
(∇2√nj√nj
)≈ ∇
2√n√n
. (3.25)
An interesting and pratical result is concerned with the fact that (3.25) is true for length scales larger
than the Fermi length λF . So, it means that our fluid model is valid since the quasineutrality condition
is satisfied. We should remark that the initial 2N system of equations is about to be replaced by the
fluid system of equations∂n
∂t+∇ · (nv) = 0. (3.26)
∂v∂t
+ (v · ∇)v =e
m∇φ− ∇P
mn+
~2
2m2∇(∇2√n√n
), (3.27)
where the self-consistent potential φ is given by the Poisson equation.
3.1.2 Nonlinear Schrödinger-Poisson system
Once obtained (3.26) and (3.27), we should reverse the problem and rewrite Schrödinger’s equa-
tion in through these global quantities. Let us define a global wave function for the unmagnetized
quantum plasma:
Ψ(r, t) = Aeiα/~. (3.28)
The relations v = ∇α/m and A =√n are the straight-forwarded generalization of the previous
definitions for the single-stream. By inserting (3.28) in (3.26) and (3.27) and after some algebra, we
can obtain the following equivalent Schrödinger-Poisson system of equationsi~∂Ψ∂t
= − ~2
2m∇2Ψ + qφΨ + U(|Ψ|2)Ψ
∇2φ =q
ε0
(|Ψ|2 − n0
),
(3.29)
where U = U(n) represents an effective potential given by
U(n) =∫ n
0
|∇P (ξ)|ξ
dξ. (3.30)
12
The SP system (3.29) can describe both electron and ions, or eventually dusty particles, in an
unmangetized quantum plasma. It is also used to model semiconductors [4], [5] and nonlinear
optics.
The nonlinear term (3.30) depends on the dimension d of the system. This dependence is very
important, since nonlinearity is the cause of very interesting phenomena in many physical fields,
such as solitons and vortices [14]. We may choose the plasma to be a polytropic fluid with the
following equation of state:
P (n) = Cnβ , (3.31)
where β is the polytropic exponent. In addition, we may assume that the plasma undergoes only
adiabatic processes, and thus set β = γ, with
γ =Cp
Cv=
2 + d
d, (3.32)
if we make use of the Dulong-Petit models for specific heat. Putting (3.32) and (3.30) together, the
nonlinear term reads
U =Cγ
γ − 1nγ−1. (3.33)
The constant C is set by concretizing the state equation for a d−dimensional Fermi gas at T = 0.
The Fermi energy is given by (Appendix A.1)
EF =~22πm
(Γ(d/2 + 1)d− 1
n
)2/d
, (3.34)
and the average energy E0 values
E0
N=∫ EF
0
D(ε)ε dε =d
d+ 2EF , (3.35)
where D(ε) represents the density of states. Therefore, the pressure is computed from the usual
thermodynamical relation
P = −∂E0
∂V=
2dE0n =
4d+ 2
~2π
m
(Γ(d/2 + 1)d− 1
)2/d
nγ , (3.36)
which means that the value of C is
C =4πd+ 2
~2
m
(Γ(d/2 + 1)d− 1
)2/d
. (3.37)
Finally, by making the identification between n and |Ψ|2, we can write the nonlinear wave equation,
exactly equivalent to the reduced fluid model:
i~∂Ψ∂t
= − ~2
2m∇2Ψ + qφΨ + C
2 + d
d|Ψ|4/dΨ. (3.38)
Together with the Poisson equation, the later compose the so-called nonlinear Schrödinger-Poisson
system, in which many interesting phenomena undergo. This equation is formally equivalent to
Gross-Pitaevskii equation, used extensively in the study of the dynamics of Bose-Einstein conden-
sates. The difference remains in the physical meaning of the potential φ, which in the later corre-
sponds to the external confinement field.
13
The treatment taken in this section allowed us to reduce significantly the complexity of the system
passing through many interesting aspects, in a very pedagogical way. In fact, the fluid model is an
useful approximation for a large set of dynamical applications, since we keep in mind the limitation
imposed by λF itself. In this approach, the typical kinetic effects can not be described. In particular,
collisionless Landau damping can not be reproduced by the set of equations (3.29). To do that, we
must use a kinetic description, based on the Wigner formalism.
3.2 Kinetic description
The kinetic theory approach in classical mechanics has been one of the most important tools to
describe complex phenomena, specially in plasma physics. The aim of this chapter is to present this
formalism in the context of quantum mechanics. Let us assume that the quantum N -body problem
can be represented by the wave function (3.2) and define the density operator for the mixture state
ρ(r, s, t) =N∑j
pjψ∗j (s, t)ψ(r, t). (3.39)
Let us define the usual phase space canonical variables
q = r, p = mv. (3.40)
In this case, the density operator ρ(q, p; t) follows the Liouville equation
i~∂ρ
∂t= −[ρ,H], (3.41)
The Wigner quasi-probability function is defined as the Weyl transformation of the density operator
(appendix A.2)
f(q, p; t) =∫〈q − s
2|ρ|q +
s
2〉eips/~ds, (3.42)
where 〈x|ψ〉 = ψ(x) denotes the usual Dirac notation. The reason why f is not a true probability
function is related to the fact that it is not positive defined, but it can still be used to compute averages
just like in classical statistical mechanics. Inserting (3.42) in (3.41), reads
∂
∂tf(q, p; t) = −iL(q, p)f(q, p; t), (3.43)
where L(q, p) is the quantum Liouville operator (appendix A.2)
L(q, p) = 2i sin[
~2
(∂H∂q
∂f
∂p− ∂H∂p
∂f
∂q
)]. (3.44)
This far, we can already notice quantum contributions for both f and L, which are related with ~.
Thus, equation (3.43) is the formal time evolution equation for the Wigner function defined above.
To derive an explicit evolution equation of f we must assume that the most general Hamiltonean of
a certain system can be writen as
H(p, q) =p2
2m+ V (q), (3.45)
which is valid if one does not take in account virtual potentials. By taking the Taylor expansion of the
Liouville operator in (3.44), and using the later definition, we have, after some algebra,(∂
∂t+
p
m
∂
∂q− ∂V
∂q
∂
∂p
)f =
∞∑k=1
(−1)k
(2k + 1)!
(~2
)2k∂2k+1
∂q2k+1V∂2k+1
∂p2k+1f. (3.46)
14
Keeping terms in (3.46) up to O(~2), and inverting the relations (3.40), we obtain the following kinetic
equation for a quantum plasma
∂f
∂t+ v · ∇f − q
m∇φ · ∇vf =
q~2
24m3∇3φ∇3
vf. (3.47)
The Vlasov equation is recovered in the formal semiclassical limit ~→ 0. The complete description of
a quantum plasma is obtained remembering that the potential V = qφ is a self-consistent potential,
and thus, the Wigner equation (3.47) must be coupled to the Poisson equation
∇2φ =q
ε0
(∫fdv − n0
), (3.48)
if we assume ions to be montionless with uniform density n0. The set of equations (3.47) and (3.48)
is the so-called Wigner-Poisson system and has been extensively used in the study of quantum
transport in semiconductors and thin films [18].
The equivalence between Wigner-Poisson and Schrödinger-Poisson systems can be clarified
trying to obtain (3.47) through equation (3.29). The Wigner function (3.42) can be rewriten in the
form
f(q, p; t) =∫dq′dp′
2π~e−i/~(q′p−p′q)Tre−i/~(qp′−px′)ρ. (3.49)
Since the commutation relations [q, p] = i~ and [q, [q, p]] = [p, [q, p]] = 0 are valid, we have
ei/~(qp−pq) = ei/~qpei/~pqe−qp/2. (3.50)
Writing (3.39) for a pure state, ρ = |ψ〉〈ψ|, and inserting (3.50) in (3.49), reads
f(q, p; t) =∫dq′dp′
2π~e−iq′p′/2~Treipq′/~|ψ〉〈ψ|e−iqp′/~
=∫dp′dq′
2π~e−iq′p′/2~
∫dy〈y|eipq′/~|ψ〉〈ψ|e−iqp′/~|y〉
=∫dp′dq′
2π~e−iq′p′/2~
∫dy(eq′∂/∂yψ(y)
)ψ∗(y)e−iyp′/~
=∫dp′dq′
2π~e−i(q′p−p′q)~
∫dyψ(y + q′)ψ∗(y)e−iyp′/~e−iq′p′/2~
=1
2π~
∫dq′δ (q − q′/2− y) e−iq′p
∫dyψ(y + q′)ψ∗(y)
=1
2π~
∫dyeiyp/~ψ(q − y/2)ψ∗(q + y/2), (3.51)
where we made use of the relation between canonical variables
p =~i
∂
∂qq = −~
i
∂
∂p(3.52)
Writing the later result for the mixture case, and using (3.40), it is possible to write the Wigner
function in terms of the wave functions
f(r,v; t) =1
(2π)d~∑
j
mjpj
∫dsψj (r− s/2; t)ψ∗j (r + s/2; t) eims·v/~ (3.53)
Although the later definition is not necessarily positive, it must reproduce the correct quantum-
mechanical marginal distributions, such as the (electronic) density
n(r, t) =∫f(r,v; t)dv =
N∑j=1
pj|ψj |2, (3.54)
15
and must also reproduce the correct averages for generic quantities
〈A〉 =1NTr ρA =
1N
∫ ∫drdvfA, (3.55)
where the normalization N is introduce in the sake of generality.
Using both (3.53) and (3.29), it possible to derive the following evolution equation:
∂f
∂t+ v · ∇f =
∫du K(u− v, r; t)f(r,u; t), (3.56)
where
K(u− v, r; t) = − qm
2iπ~2
∫ds exp (ims · (v − u)/~) [φ(r + s/2)− φ(r− s/2)] (3.57)
Performing in the same way of (3.51), one may show that in the formal limit ~→ 0, the later equation
can lead to same result of (3.46) (keeping the same order of approximation), which shows the
equivalence between wave and kinetic descriptions. The exact correspondence for higher orders
in ~ is hard to obtain, because it involves weak convergence. For the sake of simplicity, equation
(3.46) may produce the majority of the analytical results. On the other hand, equation (3.56) is
exact but highly nonlinear, and should be used specially for computational analysis. In the next
chapters we will report some analytical and numerical results on quantum plasmas, based on both
descriptions.
3.3 Magnetized quantum plasma
In the previous sections of this chapter, we have neglected the effect of an exterior magnetic field.
In fact, it is not essential for the physics derivation of the models described above and the inclusion
of an external magnetic field is quite straightforward. Following the strategy of Haas [32], the effect
of a magnetic field can be introduced through the minimal coupling, yielding
12m
(−i~∇− qA)2 ψj + qφψj = i~∂ψj
∂t, (3.58)
where A represents the potential vector. Again, using the definition in (3.53) for the Wigner function
and using the Coulomb gauge ∇ ·A = 0, it is possible to derive, after some tedious algebra [32]
∂f
∂t+ v · ∇f =
iqm
~(2π~)d
∫ ∫dsduei(v−u)·s/~ [φ (r + s/2)− φ (r− s/2)] f(r,u; t)
+iq2
2~(2π~)d
∫ ∫dsduei(v−u)·s/~ [A2 (r + s/2)−A2 (r− s/2)
]f(r,u; t)
+q
2~(2π~)d∇ ·∫ ∫
dsduei(v−u)·s/~ [A (r + s/2)−A (r− s/2)] f(r,u; t)
− iq
2~(2π~)dv ·∫ ∫
dsduei(v−u)·s/~ [A (r + s/2)−A (r− s/2)] f(r,u; t) (3.59)
By using the the relations v = (−i~∇ − qA)/m, E = −∇φ − ∂tA and B = ∇ ×A, it is possible to
approximate (3.59) up to O(~2) to obtain
∂f
∂t+ v · ∇f +
q
m(E + v ×B) · ∇vf =
q~2
24m3∇2E · ∇v∇2
vf, (3.60)
16
which provides an approximate quantum correction to the Vlasov equation for magnetized plasma.
Similary, the reduced fluid equations result naturally by applying the minimal coupling relation to
equations (3.24) and (3.23). However, we must remark that strong magnetic field must be treated
carefully, since they are associated with anisotropic pressure dyads. Fortunatelly, we are chiefly
interested in the role of quantum effects, which allows us to disregard such possibility. Therefore,
the reduced fluid model for a magnetoquantum plasma reads
∂v∂t
+ u · ∇u = −∇Pmn
+q
m(E + v ×B) +
~2
2m2∇(∇2√n√n
). (3.61)
The continuity and Poisson equations remain unchanged and no further equations are needed to
describe the magnetized plasmas, since we disregard source terms for the magnetic field and thus
the Coulomb gauge holds.
In what concerns to the nonlinear Schrödinger equation, we must stress out that generalizations
are not that straightforward. Since spin effects have not been included in these derivations, no terms
depending of interaction between spin and magnetic field appear. We know, however, that the case
of an ultra cold plasma, namely, a Fermi gas, the effect of spin is relevant and the dynamics is
described by Pauli equation.
17
Chapter 4
Waves and instabilities in quantumplasmas
It is a widely understood fact that perturbations to the equilibrium state of charged systems
are followed by oscillations, which can be propagated through the medium, giving origin to waves.
Depending both on the nature of the interaction and on the amplitude of the perturbation, these
waves and oscillations can lead to linear and nonlinear phenomena. In order to present the principal
ones, we make use of both fluid and kinetic models. In this chapter, we present the principal linear
quantum corrections to the electrostatic waves, in the absence and presence of magnetic field. Also,
we present some phenomena which can be obtained only if one uses kinetic theory. As well as
possible, the results should be compared to those which have been reported in several refereed
review papers.
4.1 Electrostatic waves
4.1.1 Quantum Langmuir waves
Electron streamings into adjacent layers of plasma with nonzero thermal velocities will carry
information about the perturbation. This effect can be fully treated by adding a term −∇Pe and
the quantum correction introduced by the Bohm potential. Common choices for Pe are the isobaric
(Pe = const), isothermal (Pe ∝ ne) and isentropic (Pe ∝ nγe ) laws. In this case, (2.1) should be
replaced by equation (3.23). Let us assume ions to be motionless with equilibrium density profile,
ni ≈ n0. Taking the usual procedure of linear perturbation, where a certain quantity is given by
X = X0 + X, the fluid description of a one-dimension plasma reads
∂v
∂t= − e
mE − 1
mn0
∂P
∂x+
~2
2m2
∂
∂x
(1√n0
∂2√n
∂x2
)
∂n
∂t+
∂
∂x(n0v) = 0
∂E
∂x= − e
ε0n.
(4.1)
18
Here, we dropped the subscript e for sake of simplicity. Using the equation of state of a perfect fluid
in isothermal compression, P = γnkBT , where γ = 1, and inserting in (4.1), it is possible to write
the following PDE
∂2n
∂t2+ ω2
pn−kBT
m
∂2n
∂x2+
~2
4m2
∂4n
∂x4= 0. (4.2)
Making use of definition (2.4) and taking the Fourier transformation of the later equation, we can
obtain the following dispersion relation
ω2 = ω2p + v2
thk2 +
~2
4m2k4. (4.3)
The later is the quantum version of the Langmuir dispersion relation. Equation (4.3) describes the
wave created by electrons in the presence of a neuralizing ionic backgroung and is in agreement
with previous works [6]. In oposition to classical theory, where the frequency electrons reduces to ωp
in the zero-temperature limit, the quantum correction introduced by the Bohm potential makes these
waves to be dispersive in this limit. For ultracold plasmas, the quantum corrected Langmuir waves
obey to the following dispersion relation
ωQ =(ω2
p +~2
4m2k4
)1/2
. (4.4)
4.1.2 Quantum ion-acoustic waves
Ordinary sound waves may not occur in colisionless media. However, since ions are charged
particle, the vibrations can be propragated and acoustic waves can occur through the intermediary
of an electric field. Since the motion of massive ions will be involved, these will be low-frequency
oscillations. In such case, the plasma approximation is still valid and one should make use of Laplace
equation,∇·E = 0. In the classical hydrodynamical frame, the ion-acoustic waves dispersion relation
reads
ω =(kBTe + γikBTi
mi
)1/2
k ≡ vsk, (4.5)
where vs is the sound speed in a plasma. Because of the electrons are inertialess compared to the
ions, they can be assumed to reach thermal equilibrium everywhere, and thus γe = 1. The value
of γi sould be set in agreement with the discussion presented above for the electronic equation of
state. The interest of relation (4.5) remains in the fact that it is derived in the linear theory frame, and
that, generaly, an ion-acoustic wave is a two-species effect. Usual derivations of weakly nonlinear
properties (involving singular expansion in power series for the perturbed quantities) of this acoustic
waves lead to the Korteweg-de Vries (KdV) equation [7].
The quantum version of the ion-acoustic waves should be obtained if we proceed in a similar way
toP the preceding section. Assuming that the equilibrium densities ne0 = ni0 = n0, let us set the
quantum hydrodinamical model (QHD) for the one-dimensional electron-ion plasma:
19
∂ve
∂t+ ve
∂ve
∂x= − e
meE − 1
men0
∂Pe
∂x+
~2
2m2e
∂
∂x
(1√n0
∂2√ne
∂x2
)
∂vi
∂t+ vi
∂vi
∂x=
e
miE +
~2
2m2i
∂
∂x
(1√n0
∂2√ni
∂x2
)∂ne
∂t+
∂
∂x(n0ve) = 0
∂ni
∂t+
∂
∂x(n0vi) = 0
∂E
∂x=
e
ε0(ni − ne)
(4.6)
The pressure Pi vanishes because of the ionic intertia. Notice that the convective terms are taken
into account. The reason for this is related with the fact that, experimentally, ion-accoustic waves
are often turbulent [33]. For closure of set (4.6), we should use the equation of state for electrons.
In this derivation, we must assume the following ordering relation on the temperatures:
TFi< Ti < Te TFe
(4.7)
which leads us to the one-dimensional Fermi gas pressure in (3.36)
Pe =mev
2Fe
3n20
n3e. (4.8)
However, following the classical procedure, (4.8) is not an essential ingredient for the derivation,
since TF is much greater than the room temperatures only for highly dense systems, such as ordi-
nary metals and metalic clusters.
In this derivation, we follow the steps carried out in the case of classical ion-accoustic waves [9],
[8]. The small inertia forces the electronic fluid to attain equilibrium almost immediately, and so
∂ve
∂t ∂vi
∂t(4.9)
Hence, neglecting the left-hand side in the first equation of the set (4.6), we obtain an expression for
the self consistent electric field
E = −me
en0v2
Fe
(∂ne
∂x− ~2
4m2ev
2Fe
∂3ne
∂x3
). (4.10)
Comparing with the ionic fluid equation in (4.6), one should neglect the Bohm potential for ions in
respect to the second term of the right-hand side of (4.10), since me/mi ≈ 0, in order to keep this
derivation consistent. The approximated linearized equations for ions read
∂vi
∂t+ vi
∂vi
∂x=
e
miE, (4.11)
∂ni
∂t+ n0
∂vi
∂x= 0, (4.12)
∂E
∂x=
e
ε0(ni − ne). (4.13)
20
Putting equations (4.10), (4.11-4.13) and the continuity equation for electrons from the QHM, one
should derive the following equation
− iωni + ime
mi
k2v2Fe
ωmi
(1 +
~2
4m2ev
2Fe
)ω2
p
k2v2Fe
(1 + ~2
4m2ev2
F e
)+ ω2
p
ni = 0 (4.14)
By defining the sound speed velocity in an ultracold plasma
Cs =(
2kBT
mi
)1/2
=√me
mivFe, (4.15)
the ion plasma frequency
Ωp =(n0e
2
ε0mi
)1/2
(4.16)
and the adimensional parameter
H =(
~ωp
kBTFe
)1/2
, (4.17)
equation (4.14) leads finally to the ion-acoustic dispersion relation
ω2 = Ω2p
k2Ω2p
C2s
(1 + H2k2Ω2
p
C2s
)1 + k2Ω2
p
C2s
(1 + H2k2Ω2
p
C2s
) . (4.18)
Since H ∝ 1/~, the classical limit corresponds to H 1. In this case,
ω ≈ Ωp. (4.19)
Although this result can not be obtained directly from the relation dispersion (4.5), it can still be
achieved if one uses the Poisson equation instead of the Laplace equation, i.e., if one allows ni to
be different from ne. In this situation, the classical dispersion relation for the ion-acoustic waves
(IAW) is [9]
ωC =(kBTe
mi
11 + k2λ2
D
+ γikBTi
mi
)1/2
k. (4.20)
In the limit of short wavelenghts, λ2Dk
2 1 and the relation dispersion saturates
ωC ≈ Ωp, (4.21)
and one recovers the classical IAW directly from the classical limit of the quantum case. A remark
should be made, however, since the QHD model does not apply for very small wavelengths, and
for this reason, the physical meaning in these limits may not be completely true. This is why in
opposition to classical plasmas, in the quantum ones particles may not be considered pointlike.
For small wave numbers, equation (4.20) gives
ω ≈ Csk, (4.22)
which represents a wave propagating at the quantum velocity Cs, and for that reason this waves
should be called the quantum ion-acoustic mode. Just like in the classical case, this modes de-
scribes low frequency oscillations of both electrons and ions.
21
4.1.3 Quantum Alfvén waves
Let us consider a quantum plasma in the presence of an exterior magnetic field B. In the classi-
cal theory of plasmas, it is know the occurance of an ionic wave along the direction of the magnetic
field. Since Alfvén waves are low-frequency oscillations, we may assume that electrons are inertia-
less. Using the same approach of the last section, the linearization of the magnetohydrodynamical
equations (3.61) and (3.26) yields
∂vi
∂t=
e
mi
(E + vi ×B0
), (4.23)
0 = − e
me
(E + ve ×B0
)+
~2
4m2en0∇∇2ne, (4.24)
where B = B0 + B and vj × B ≈ 0. We neglected the quantum difraction effect for ions in (4.23). In
addition, let us consider the perturbed current
J = en0 (vi − ve) , (4.25)
and the Maxwell equation for the perturbed magnetic field
∇× B = µ0J +1c2∂E∂t. (4.26)
By setting (4.23-4.26) together, one obtains the following equation
∂vi
∂t=
e
mi
(1
en0µ0(∇× B)×B0 + 2vi ×B0
)+
~2
4memin0∇∇2ne. (4.27)
Here, we neglected the second term in (4.26), by using equation (4.24). By time derivation of the
continuity equation for ions
∂2ni
∂t2+ n0∇ ·
∂vi
∂t= 0, (4.28)
and by using the frozen-in field condition, i.e., that particles remain attatched to the magnetic field
lines [14]B
B0=ni
n0, (4.29)
we have, after computing the divergence of (4.26) (which justifies the use of the quasineutrality
condition), (∂2
∂t2+ v2
A∇2|| +
~2
4memin0∇4||
)ni = 0, (4.30)
where vA = B0/√min0µ0 represents the so-called Alfvén velocity along the magnetic field lines.
After Fourier transforming equation (4.30), one finally gets the relation dispersion for the Alfvén
waves
ω2 = v2Ak
2 +~2
4memin0k4. (4.31)
Neglecting the Bohm potential effect, ~→ 0, one recovers the classical relation dispersion for Alfvén
waves. The classical picture of (4.31) is the formation of a small ripple in the magnetic field lines,
produced by the oscilating term B. Electrons and ions experience mostly the effect of the drift
E × B0, and thus oscilate along the direction perpendicular to B0. The fluid and the field lines
oscillate together and these oscillations propagate in the same direction of B0.
22
4.2 Kinetic effects
As already referred to in the previous chapter, the hydrodinamical model does not describe all the
observed or predictable phenomena in plasmas. In this situation, the dynamical approach must be
coupled or even replaced by the kinetic one. In what concerns to quantum plasmas, it means that, in
some situation, the Schrödinger-Poisson description must be replaced by the Wigner-Poisson set of
equations. To do that, we must recall the distribution function f(x, v) introduced in the last chapter,
as well as its differential equations. In this section, we should bring up some of the most interesting
results obtained in the kinetic approach.
As an elementary illustration of the use of the quantum version of the Vlasov equation, we shall
derive the dispersion relation for the Langmuir waves. According to the previous discussions, the
quantum hydrodynamical model (QHM) is obtained by taking appropriate momenta from the Wigner
equation (3.51), which suggests that the results derived in the later section must be contained in the
set of the kinetic results. Let us assume a linear perturbation in both Wigner function and electric
field
f(x, v; t) ≈ f0(x, v; t) + f(x, v; t), φ = φ0 + φ (4.32)
Inserting in equation (3.46), and neglecting second-order terms in f , the following set of diferential
equations is obtained ∂f
∂t+ v · ∇f =
∫du K(u− v, r; t)f0(r,u; t)
∇2φ =e
ε0
(∫dv f − n0
),
(4.33)
where
K(u− v, r; t) = − qm
2iπ~2
∫ds exp (ims · (v − u)/~)
[φ(r + s/2)− φ(r− s/2)
](4.34)
By Fourier transforming the later equations, we may write the following dispersion relation
1 = −mω2
p
n0
∫f0(v + ~k/2m)− f0(v − ~k/2m)
~k2(ω − k · v)dv, (4.35)
For the sake of simplicity, let us use the one dimensional form of (4.35). With an appropriate change
of integration variable, one obtains
1 =ω2
p
n0
∫f0(v)
(ω − kv)2 − ~2k4/4m2dv. (4.36)
In the classical limit, the right-hand term in (4.35) approaches the derivative of f0(v) and one can
recover the Vlasov dispersion relation [9]
1 = −ω2
p
n0
∫1
ω − kv∂f0∂v
dv. (4.37)
The kinetic description is much more powerful because it allows the treatment of out-of-equilibrium
phenomena. Kinetic effects like Landau damping and instabilities strongly depend on the (quasi-)
equilibrium conditions and on the phase space region under study. Technically, it is related with the
integration over the complex plane.
In order to recover the dispersion relation of electron waves, we should treat the problem semi-
classically. First, we assume thermal equilibrium, but not necessarily the Fermi-Dirac one (in fact,
23
the last correspond to a full-degenerate quantum plasma). In this description, we may combine the
quasi-equilibrium function with the quantum mechanical corrections introduced by the Bohm poten-
tial, and justify this statement a posteriori. In this sense, let us assume the multistream distribution
function
f0(v) = n0
∑j
pjδ(v − 〈vj〉). (4.38)
In this situation, the complex dispersion relation (4.36) reads
1 = −ω2
p
n0
N∑j=1
pj
∫f0j(v)
(ω − kv)2 − ~2k4/4m2. (4.39)
Generally, the pole (ω−kv)2−~2k4/4m2 gives rise to both principal Chauchy value and an imaginary
residue. While the first is related to the propagation of waves, the later yields to collisionless Landau
damping and instabilities, depending on the sign of the residue. Accordingly, the frequency (or the
wave number k) can be split into its real and imaginary part
ω = ωr + iγω, (4.40)
where γω is the so-called growth rate. Since the perturbed part of a certain quantity, say, the density
is given by
n = Neikx−iωt, (4.41)
both damped and unstable solutions may occur, depending on the sign of γ. In fact, both damped
and unstable solutions occur, since complex solutions come in conjugate pairs. In the next few
sections, we present simple examples in order to concretize the quantum Vlasov dispersion relation
and thus show some results which are exclusive of a kinetic description.
4.2.1 Zero-temperature Fermi gas
Let f0(v) be the Fermi-Dirac distribution function. At T = 0, f0(v) reads
f0(v) =n0
2vFΘ(vF − v) , (4.42)
where Θ(x) denotes the usual Heaviside function. By inserting (4.42) in the relation dispersion in
(4.39), we have, by integrating from 0 to vF
ω2 = ω2p coth
(~k3vF
mω2p
)+ k2λ2
F +λ4
F k4
4ΓQ. (4.43)
Using the expansion x coth(x) ≈ 1 + x2/3− x4/45 + . . ., (4.43) yields
ω2 = ω2p + k2v2
F + ω2pΓQ
(k4λ4
F
4+λ6
F k6
3
)− k12λ12
F
45Γ2
Q + . . . . (4.44)
This corresponds to an expansion in power series of kλF and Γ2Q. As discussed previously, the
relation dispersion provided for the Wigner function is collisionless. Then, we would expect ΓQ to be
much less than one. But what happens in the case of metalic Fermi gases, where ΓQ ≈ 1? So, the
approximation should be done in powers of kλF , since the fluid model is characterized for reduced
wave lenghts (λF ≈ 0). Then, the relation dispersion in (4.44), in the order O(k6λ6F ) to the relation
24
dispersion found for the quantum Langmuir waves, by simply setting vth = vF . This result illustrates
that the computation of the dispersion from a kinetic via is much more rigorous.
4.2.2 One-stream instability
Let us assume the quasi-equilibrium distribution function of a single stream
f0(v) = n0δ(v − v0). (4.45)
It represents a simple but physically relevant out-of-equilibrium case of a mono-energetic electron
beam injected in the plasma (c.f. refs [11] and [12]). Replacing in equation (4.36), reads
1 =ω2
p
(ω − kv0)2 − ~2k4/4m2e
, (4.46)
which, in the static limit v0 = 0, recovers the dispersion relation for ultracold plasmas in (4.4). The
roots of (4.46) are
Ω = K(1±
√H2K2 + 1/4K2
), (4.47)
where Ω = ω/ωp, K = kv0/ωp and H = ΓQ = ~ωp/mev20 . The solutions of (4.47) have no imaginary
part. However, taking the stationary solutions Ω = 0, we have
K2 =2± 2
√1−H2
H2, (4.48)
which may lead to imaginary solutions: if H < 1, both solutions of k are real and the system can
sustain spatially oscillations; for H > 1, the solutions are unstable and grow exponentially, since
γk < 0. Thus, the separation line is given by K = 1/H.
In order to recover the dispersion due to thermal effects, we must replace the δ−correlation in
phase space by a more physical distribution function, with a nonzero bandwith σ. For example, let us
assume a certain incoherence in the electrom beam wave function, i.e., let us assume that electrons
are described by a certain wave function
ψ(x) = eiα(x)φ(x), (4.49)
where its phase α(x) varies randomly. This is an example of the random-phase-approximation
(RPA) theory for describing the dynamic electronic response of systems. Computing the correlation
function through a thermal average in the canonical ensemble, let us assume, for example
〈e−iα(x+y/2)eiα(x−y/2)〉 = e−vth|y|/me . (4.50)
By inspection, the later result is a first momentum of the distribution a lorentzian distribution function
f0(v) =n0
πme
vth
(v − vth)2 + v2th
. (4.51)
Inserting this function in equation (4.39), one obtains
1 = −ω2
p
π
vth
(ω − v0k)2 − ~2k4/4m4e
∫1
(v − vth)2 + v2th
. (4.52)
There are two poles in v = vth(1± i). Computing the Cauchy principal value of the integral, we have
25
(ωr − v0k)2 = ω2p +
~2k4
4m2e
(4.53)
and
γ = −vthk. (4.54)
In this case, the Landau damping is due to thermal effects, concerned with the stochastic nature
of the phase. The dispersion relation in (4.53) was already reported in previous works [12]. In
ref. [11], the thermal relation dispersion was obtained by spliting the velocity into its parallel and
perpendicular components and a marginal distribution function was obtained by integrating (4.35)
over the perpendicular one.
4.2.3 Two-stream instability
It is known from classical kinetic theory that an ion-electron plasma presents stream instability.
Considering that ions are motionless and assuming that electrons have uniform velocity v0, the
instability is characterized by the following growth rate
γ =(me
mi
)1/3
ωp. (4.55)
Let us consider a two-stream plasma composed by two species, 1 and 2. It may illustrate both
positron-electron and ion-electron plasmas. Without loss of generality, let us assume that they have
opposite velocities such that v01 = −v02 = v0/2 and that the quasi-neutrality condition is satisfied,
n1 = n2 = n0/2. In this situation, the distribution functions are
f01 =n0
2δ(v − v0/2) f02 =
n0
2δ(v + v0/2). (4.56)
Inserting these quasi-equilibrium distribution functions in equation (4.36), the dispersion relation
reads
ω2p = −
(1
(2ω − v0k)2 + ~2k4/m1+
1(2ω + v0k)2 + ~2k4/m2
)−1
. (4.57)
For the sake of simplicity, let us resume this study for the case of a two-streams electron plasma, by
setting m1 = m2 = me. Using the adimensional quantities Ω, K and H, one should rewrite the later
result and obtain
Ω4 = Ω2
(1 + 2K2 +
H2K4
2
)−K2
(1− H2K2
4
)(1−K2 +
H2K4
4
). (4.58)
The unstable and damped solutions occur simultaneously, since the solutions for Ω2 < 0 provides
complex conjugate pairs . After some algebra, we can find the two branches
Ω2± =
12
+K2 +H2K4
4± 1
2
√(1 + 8K2 + 4H2K6), (4.59)
where the unstable solutions Ω2 < 0 are provided for the following condition
(H2K2 − 4)(H2K4 − 4K2 + 4) < 0. (4.60)
26
Figure 4.1: Two stream-instability: (a): Plot of H = 1/K, H = 2/K and H2 = 2√K2 − 1/K2. The
thin line corresponds to the one-stream instability condition and the dashed area represents the
instability zone; (b): Square of the growth rate γ2. dashed line, H = 0; thin line, H = 0.5; bold line,
H = 1.
The classical condition for two-stream instability is obtained by setting H = 0, which leads to
K2 < 1, in agreement with the results presented in ref. [9]. In the quantum case, we have two
cases: for H > 1, the second factor is always positive and the plasma is stable for H < 2/K; for
H < 1, the instability condition reads for the following situations:
0 < H2K2 < 2− 2√
1−H2, 2 + 2√
1−H2 < H2K2 < 4. (4.61)
In figure (4.1) are ploted the stability conditions in (4.61) and the growing rate γ2, corresponding
to the negative solutions of (4.59). The instability zone is represented by the shadowed area. In the
right-handed plot, we observe that the growing rate is maximum for the classical case H = 0, and
that the full quantum case H = 1 presents two maxima.
27
Chapter 5
Nonlinear phenomena in quantumplasmas
In the last chapter, we reported some results of linear theory in quantum plasmas. Neverthe-
less, the full dynamics of a quantum plasma is described by nonlinear equations, in both fluid and
kinetic descriptions. In this chatper, we present some interesting nonlinear phenomena in quantum
plasmas.
5.1 Quantum Zakharov equations
In the context of the study of the collapse of Langmuir waves, an issue specially relevant in
plasma physics in the seventies, Zakharov showed the occurance of coupling between (classical)
Langmuir waves
ω2 = ω2p + 3v2
thk2 (5.1)
and the ion-accoustic waves
ω = vsk, (5.2)
presented in the last chapter. Zakharov showed that the slowly varying envelope (associated to the
high frequency part) of the electric field and the low-frequency ionic waves can be coupled through
a set of nonlinear equations [24] i∂E∂t
+∇2E = nE
∂2n
∂t2−∇2n = ∇2|E|
(5.3)
Here, the ionic density n is normalized by n0, time by ωp, the position by λD, and the perturbed
electric field is normalized by kBTe/λD. Solutions to the modified set (5.3), where the ponderomotive
force of an external laser field is included, have also been studied [25].
The quantum version of Zakharov set of equations should be derived by using the quantum
hydrodynamical model presented in chapter 2. Keeping the procedure used in the derivation of (5.1)
28
and (5.2), we assume ions to be pressureless and electrons to be isothermal, and thus, γ = 1. For
simplicity, we perform the derivation for the one-dimensional case which can be easily generalized
for higher dimensions. Since we are dealing with two time scale, we should follow the procedure
in [26] and separate the fast-varying parts (subscript h) from the low-varying (subscript `) for all the
quantities involved. Therefore, one should set
ne = n0 + n` + nh,
ni = n0 + nh,
ve = v` + vh,
vi = v`,
E = E` + Eh.
(5.4)
Because the inertia of electrons is much less than that of ions, the fast-varying parts hold only for
electrons. The quantum reduced fluid model derived in (3.26) and (3.27), reads:
∂ni
∂t+ n0
∂vi
∂x= 0,
∂ne
∂t+ n0
∂ve
∂x= 0,
∂vi
∂t+ v`
vi
∂x=
e
miE,
∂ve
∂t+ ve
ve
∂x= − e
meE + kBTene +
~2
4m4en0
∂3ne
∂x3.
(5.5)
In addition, one should include the Poisson equation
∂E
∂x=
e
ε0(ni − ne) (5.6)
By computing (5.5) and (5.6) for the fast-varying quantities, and writting the electric field in the form
E(x, t) = E(x, t)12(e−iωpt + eiωpt
), (5.7)
where E denotes the slowly-varying envelope of the fast-varying electric field, we must write
i∂E
∂t− 1ωp
∂2E
∂t2+v2
th
2ωp
∂2E
∂x2− ~2
8m2eωp
∂4E
∂x4=
ωp
2n0n`E. (5.8)
The second term of the later equation may be neglected, since the relation ∂2t E ωp∂tE holds.
The derivation of an equation for the low-frequency quantities is made by taking times averages
relatively to the high-frequency quantities. Therefore, if X`,h is one of the quantities in (5.4), the
following relation holds
〈Xh〉` ≡ Ωp
∫ 1/Ωp
0
Xh(t)dt ≈ 0 (5.9)
and the hydrodynamical equations for the ion-like quantities yields
29
∂n`
∂t+ n0
∂n`
∂x= 0,
∂v`
∂t+ v`
∂v`
∂x=
e
miE`,
∂vh
∂t+ vh
∂vh
∂x= − e
meE` +
~2
4m2en0
∂3n`
∂x3− e2
4m2eω
2p
∂E2
∂x.
(5.10)
Here, we neglected the convectice terms under the assumption that Langmuir waves are not turbu-
lent (in fact, as assumed in previous derivations, along chapters 2 and 3). Eliminating E` from the
later set of equation, using the approximation me mi, one obtains
∂2n`
∂t2− C2
s
∂2n`
∂x2+
~2
4mime
∂4n`
∂x4=
ε04mi
∂2E
∂x2, (5.11)
where Cs =√kBTe/mi is the ionic sound speed. By conveniently defining some dimensionless
variables
x∗ =√me
mi
x
λD, t∗ = 2
me
miωpt (5.12)
n∗` =14me
mi
n`
n0, E∗ =
14
√ε0mi
men0kBTeE, ΓQ =
~ωp
kBTe, (5.13)
we can rewrite equations (5.8) e (5.11) in the adimensional formi∂E∂t
+∇2E− Γ2Q∇4E = nE
∂2n
∂t2−∇2n+ Γ2
Q∇4n = ∇2|E|
, (5.14)
where we dropped all sub and superscripts in the notation. The later set of equations is the quantum
version of the Zakharov equations. One clearly recovers the classical result by taking the limit
ΓQ → 0. Solutions of one-dimensional version of Zakharov equations may be found in the Euler
coordinates ξ = x− v0t and choosing the ansatz
E = A(ξ)ei(kx−ωt), n = B(ξ)ei(kx−ωt) (5.15)
corresponding to a quasistationary envelope moving with constant speed v0, reducing (5.14) to a set
of ordinary differential equations, which yields, for the one-dimensional case−iv0
dA
dξ+d2A
dξ2− Γ2
Q
d4A
dξ4−BA = 0
−v20
d2B
dξ2+ Γ2
Q
d4B
dξ4− d2A
dξ2= 0
(5.16)
This approach is especially interesting when one seeks solitary waves. However, solutions of the
set of equations in (5.16) is still a hard job to perform here.
30
5.2 Numerical solutions to the nonlinear Schrödinger-Poisson
system
As discussed above and according to the equations derived in chapters 2 and 3, we can see
that the modulation of quantum plasmas involves nonlinear coupled equations, in both wave and
kinetic descriptions. For the sake of illustration, we present a preliminar and maybe not completely
correct simulation of the nonlinear Schrödinger-Poisson system for one and two-dimensional elec-
tron plasma. In the first case, we illustrate the temporal evolution of the Wigner function f(x, k, t);
secondly, we perform a simulation of the probability function |ψ|2. We remark that computational
solutions for the nonlinear Schrödinger equations (NLS) and the Wigner function have been studied
in very recent research works and are still under development.
5.2.1 Time dependent solution of the Wigner function
We start from the normalized one-dimensional version of (3.29) coupled to Poisson equation, for
a Fermi gas i∂Ψ∂t
+ΓQ
2∂2Ψ∂x2
+ ΦΨ− |Ψ|αΨ = 0
∂2Φ∂x2
= |Ψ|2 − 1
, (5.17)
where α = 4/d. The wave function Ψ is normalized by√n0, the self-consistent potential Φ by TF /e,
the time t by ~/TF , and the space x by λD. Shukla and Eliasson showed the propagation of solitary
waves in one-dimensional, by solving (5.17) numerically in the Eulerian frame [27]. Here, we solve
numerically, by using an implicit Crank-Nicholson scheme, the time dependent system above and
compute the discretized Wigner transform directly from its solutions. The details of the simulations
are presented in (appendix B). The first simulation was performed for a single particle with gaussian
profile at t = 0
Ψ(x, 0) = eik0e−(x−x0)2/σ2
, (5.18)
where we set k0 = 50, x0 = 1/2 and σ = L/10. The lenght of the simulation box was L = 1.
In figure (5.1) are presented some of the time steps of the simulation. It was performed a total
of 100 iteractions in steps of ∆t = 0.01, which corresponds to a total time of T = ~/TF . Also,
the coupling parameter used is ΓQ = 0.5. The Bohm potential is responsible for the diffusion in
the phase-space. We remark that the results presented in fig. (5.1) may be not completely correct
and admit the possibility of numerical incorrections, due to both error propagation and numerical
implementation itself.
5.3 Solitons in two-dimensional electron plasmas
We present next some numerical solutions of (5.17) for the two dimensional case, by fixing the
nonlinear exponent α = 2. Speciafically, we are seeking for solitary waves, which have already
been reported in numerical simulations of the nonlinear Schrödinger equation [28]. First, we should
remark that the extension of the wave funtion used in the previous simulation does not behave as a
31
Figure 5.1: Ploting the numerical Wigner function for t = 0, t = 0.2, t = 0.7, and t = 1 (from top to
bottom).
solitary wave. It is obvious in the one-dimensional case, where one clearly sees the diffusion of the
wave packet. The two-dimensional case of the test wave in (5.18) reads
Ψ(x, y, 0) = eik0xx+k0yy exp(− (x− x0)2
σ2x
− (y − y0)2
σ2y
). (5.19)
In the current simulation we set k0y = k0x = 50 and σx = σy = L/10, with L = 1. The calculation
occurs into 1000 steps with a time step of ∆t = 0.001, corresponding to a time interval of T =
0.2~/TF . In figure (5.2) we observe the dispersion of the wave packet, which clearly shows that the
trial function in (5.19) is not a soliton.
Our second trial is based on the trial function used by Shukla in [27], for the one-dimensional
solitary wave. In that letter, the authors recognize that the ansatz
Figure 5.2: Dispersion of the wave packet along the simulation box. Screen shots at t = 0, t = 0.1
and t = 0.2
32
Figure 5.3: Soliton in a two-dimensional plasma. ΓQ = 2. The time steps presented go from t = 0
until t = 0.9, in time steps of ∆t = 0.1.
Ψ(x, 0) = 0.18 + tanh[2L sin
( xL
)]eik0x, (5.20)
with the choice k0 = v0/ΓQ, suffers no dispersion during the simulation, which suggests it as
good candidate to a soliton wave in the plasma. However, also in ref. [28] it is reported that the
solitons in one-dimension are unstable and easily break into pairs of vortex-antivortex. According
to this, we should try an extension of the later trial function for the current case, by assuming the
separation of the wave function along x and y coordinates, i.e., by taking Ψ(x, y) = ψ(x)ψ(y), where
ψ(z) is assumed to take the form in (5.20). In figure (5.3) we can see that the complex structure given
by the ansatz above keeps its form. Again, the validity of the simulation must be questioned, since
solitons are usualy solitary waves, with a single maxima. In this case, we have several maxima,
which suggests that the simulation may not be correct or, at least, has no physical meaning.
33
Chapter 6
Complex plasmas
In the classical description, a laboratory plasma is a weakly damped system where a large num-
ber of particles interact through long-range forces. In a complex dusty plasma, dust particles interact
via electrostatic forces. A dusty plasma contains particles whose charge and mass can be much
greater than the electron ones. The mobility of the electrons is greater than that of ions, being thus
responsible for the dust particles acquiring negative rather than positive charge. In such systems,
gravity forces may not be neglected. Thus, after the discharge, when gravitational and electostatic
forces balance, dust particles congregate at the lower sheath edge. However, small particles may
fill the entire discharge volume, forming spherical agregates. These complex systems are often ob-
served in microgravity experiences [13]. In addition, as complex plasmas should be understood all
charged systems where collective effects take place, corresponding to a new and exciting branch
of plasma and condensed matter physics. In the last few years, some works have been dedicated
to the study of collective effects in dusty plasmas. Mendonça and Shukla predicted quantized os-
cillations in dusty spheres due to the Yukawa shielding [14], while Sheridan presented a continuum
model for the breathing oscillation of a spherical complex plasma [15]. Also, multiple plasmons and
anharmonic effects were reported by Gerchikov et al [16] and kinetic theory results by Fomichev [17]
in the study of metallic clusters.
In recent works, it has being given special relevance to the quasi-two-dimensional arrangements
of dusty particles. These layers often exhibit structural changes, depending on the experimental
conditions. Phase transitions in one-dimensional and two-dimensional systems, relevant to parti-
cle traps, were already theoretically studied and the formation of layers in Yukawa systems due to
one-dimensional force field confinement was investigated [19]. In this work, we should give special
attention to collective modes in two-dimensional Yukawa liquids, in which some results were already
given by Murillo and Gericke [20]. The ideal 2D systems exhibit two collective modes: the com-
pressional (longitudinal) mode and the shear (transverse) mode. The additional out-of-plane and
transverse collective mode, in strongly coupled liquid phase (ΓC 1), was reported by Qiao and
Hide [21] and was numerical simulated by Donkó et al [22]. In the later work, the authors showed
the evidence of a phase transition in the density function along the direction of the confinement force
field, as well as the occurance of optical-like dispersion relations. More recently, the same authors,
toghether with Goree and Kutasi, studied the shear viscosity in these quasi-two dimensional Yukawa
layers [23].
In the present work, we are specially interested in an analytical description of the collective
modes in complex plasmas. Our approach is based on the development of a plasma form factor
34
which should include corrections on the oscillations due to both geometry and nonuniform equi-
librium density profiles. In the following section, we will derive an expression that generalizes the
so-called Mie frequency, based on simple geometrical assumptions and linear theory. Secondly,
we will present the idea of how to insert this form factor in the description of collective modes in
two-dimensional Yukawa liquids, confined buy the one-dimensional parabolic potential.
6.1 Plasma form factor
In this section, we procede to a very first attempt in the derivation of a form factor which includes
the corrections to the usual plasma frequency ωp, obtained in the local description of a classical
plasma. The form factor f must be defined in such way that
ω = fωp. (6.1)
An application (still under study) of this theory is the studied of the coupling between this oscillations
and the electronic excitations in hidrogenoid species, where the corrected frequency ωp may be in
resonance with some electronic excitation, allowing the expectation value
〈i|H|j〉, (6.2)
where H = ~ωp(a†a), to be nonzero for some pair of quantum states i, j. Still in project is the study
of plasmonic resonances in complex molecules, such as the fluorene C60 and carbon nanotubes.
6.1.1 Integral formulation
Let us star from the zero-temperature colisionless plasma and let a certain carge q, in electro-
static equilibrium, to be perturbed. If r is the displacement of the charge q, it should obeys
∂2r∂t2
=q
mE, (6.3)
where E is the perturbation to the electric field. By integration of the Poisson equation, reads∫Sr
E.n dSr = − q
ε0
∫Vr
ndVr, (6.4)
where Sr is the surface defined by the displacement vector. Considering that the perturbation is
small enough for the surface Sr to be gaussean, one obtains, putting (6.3) and (6.4) toghether and
by Fourier transforming
ω2 = ω2p
1Suu
∫Vr
ndVr, (6.5)
where we define the normalizes density n = n/n0. However, the problem is not solved yet, since
the perturbation integral in (6.5) involves the displacement r itself. In addition, the later equation is
expected to provide an harmonic correction to the plasma frequency and thus should not depend
on the amplitude of the perturbation. The choice of r must be made taking in account firstly that it
must be a parameter of the plasma and secondly that it must be large enough to include all linear
perturbations, otherwise, this theory would not be self consistent. The idea is to fix r to be a certain
characteristic distance in a plasma. A first but incorrect attempt would be to fix r at the debye lenght
35
λD, but since thermal effects are not taken in account, this choice reveals to be incoherent. So, the
remaing and general enough quantity appears to be the Wigner-Seitz radius in the d− dimensional
lattice
rs =Γ(d/2)πd/2
nd0
, (6.6)
where Γ(x) is the usual gamma function. The choice of rs must be justified a posteriori, where some
simple and well-known results should be recovered. Putting (6.6) and (6.5) together, we obtain the
following form factor for a classical complex plasma
f =
(1
Srsrs
∫Vrs
n dVrs
)1/2
. (6.7)
A remark should be made at this point, since the later equation is only valid for small perturbations,
where the gauss symmetry is not broken. In a more general situation, the form factor f must be
derived from a differential formulation. In order to justify the validity of equation (6.7), let us compute
f for simple systems which satisfies to the conditions of the present derivation.
6.1.2 Mie resonance
Let us consider a charged sphere of radius R with bulk density n0. Accondingly, the normalized
perturbed density is n = 1 and the Wigner-Seitz radius is
rs =(
34πn0
)1/3
. (6.8)
By inserting the later in eq. (6.7), reads
f =1√3, (6.9)
which correspond to a corrected frequency ωp = ωM , where ωM = ωp/√
3 is the so-called Mie
frequency. This frequency correspond to the fundamental mode of surface plasmon in metallic
clusters.
6.1.3 Dust Balls
In microgravity experiments, dusty plasmas may occur in spherical arrangments [13]. In such
arrangements, dusty particles are confined by mean of the Yukawa potential. The Poisson’s equation
in such media reads
∇2φ = k2Dφ, (6.10)
where kD =√n0Q2/e0kBT represents the inverse of the (dust) Debye lenght and Q is the charge
of the dust particles. A solution of (6.10) provides
n(r) = e−kDr. (6.11)
The Wigner-Seitz radius depends only on geometry and on bulk density n0 and thus corresponds to
the same case of the Mie sphere. By computing the form factor, we shall have
36
f =(e−κ
κ3
(2e−κ − 2− κ(2 + κ)
))1/2
, (6.12)
where κ = kDrs denotes the screening parameter, already referred in previous works [22]. Notice
that f is always less than 1, which means that the effective charge that contributes for the oscillation
energy is less than an usual plasma. In other words, the plasmon energy is redshifted. Redshift on
the Mie resonance, due to spill-out corrections, was already reported in review papers of metallic
clusters [16].
6.1.4 Two-dimensional Yukawa liquids
The quasi-two-dimensional layers obtained in microgravity experiments are characterized for the
coupling parameter ΓC = Q2/(4πε0rskBT ). In the liquid phase, the (complex) plasma is highly
correlated, where the typical values for the classical coupling parameter are ΓC 100. According
to the discussion above, in the monolayer arrangements, dusty plasma may exhibit both in-plane
and out-of-plane modes. For symmetry arguments, and in the absence of thermal effects, so far
neglected, the correction to plasma frequency should be the same in both compressive (longitudinal,
L) and shear (transverse, T ) modes. Equation (6.7) then provides
fL,L =1κ
√e−κ (e−κ − 1− κ). (6.13)
Again, f obeys the inequality 0 < f < 1.
The situation for the out-of-plane mode is not straightforward and is currently in study. The
confinement force, along the perpendicular direction z, presented in ref. [22] derives from a parabolic
potential and obeys the following equation
Fz = −F0cz, (6.14)
where F0 appears as the third plasma parameter, toghether with κ and ΓC . According to ref. [22],
the constant c is ajusted in such way that at z = rs, the confinement force equals the Coulomb force.
By computing f by using (6.14), we can obtain a phenomenological form factor for the plasma
f =
√F0cε0Q2n0
. (6.15)
A complete description of these modes may involve a more complete knwoledge of the confinement
potential. This is still under construction and may involve the redefinition of the plasma factor, in
order to include more general geometrical distortions, rather than the gaussean one.
In resume, we are trying to find out a characteristic response function, initially in the frame of
the linear theory, which includes information about geometry and density profiles. Therefore, a first
step is to describe the eigen modes of each plasmon in the plasma: How does a plasmon reacts to
the collective field? Which are the eigen modes? Once understood this point, we should proceed
by trying to understand how those single plasmon modes are coupled or correlated and the how
those collective modes can be excited: are there multiple ressonances? How far is the system from
nonlinearity? All those questions are still under study and can not, unfortunately, be reported in this
thesis.
37
Chapter 7
Conclusion
In this thesis, I could review some of the most acttual features in quantum plasmas. In the first
chapter, I pointed out the relation between classical and quantum plasmas and how the regimes
can be distinguished, in thermal equilibria situations, by simply dimensional analysis. At the end of
chapter 1, I was able to separate both quantum and classical regimes into the respectives collisional
and collisionless subregimes by simple computation of the coupling parameter. As already referred,
this analysis is purely qualitative and does not holds for out-of-equilibrium scenaria.
The following step consisted in the derivation of the reduced fluid model, starting from the Hartree
approximation for the pure state in addition with the multi-stream approach. In fact, the reduced fluid
model is a very powerful and reliable model, for the majority of the applications. However, according
to previous discussions, the quantum hydrodynamical description of plasmas "hide", in some sense,
some effects. Nevertheless, this is not a surprising result, since it was a well-knwon fact from the
classical theory of plasmas. The kinetic effect of collisionless or Landau damping illustrate this fact
properly. So, it became necessary the description of quantum plasmas in the phase space. The ideia
of building a kinetic theory in quantum phase space is not exclusive from quantum plasmas. In fact,
the Wigner-Moyal formalism appears much before the generalized interest for quantum plasmas.
What is interesting, in some sense, is the equivalence between a fluid model, also formally equivalent
to a wave-like description, and a transport equation, in which the function is not even positive defined.
Beyond their importance and already proffed validity, the equivalence between three models is quite
exciting.
The linearization of both fluid and transport equations allows us to establish a quantitative parallel
between classical and quantum plasma physics. For a starter like me, it was a very interesting and
challenging job, trying to recover classical limits from simply (or not always that simple) setting the
quantum natural scale to zero, ~ → 0. It gives some confidence the fact I could derive some of the
results myself and compare with those which have been published. With special emphasis to the
results of kinetic theory: the linear response theory, already treated in some other courses, here
appears again. The question of few stream instabilities, barely treated in the classical theory of
plasmas, was revisited in the context of a quantum, and thus, more general theory.
In what concerns to complex plasmas, I am specially interested in applying some results derived
in chapter 6 to the study of collective modes in two-dimensional Yukawa liquids, which are formed
under microgravity conditions. In other words, when the electrostatic forces between the particles
and the gravitional forces balance, dust particles are arrenged in sort of thin films, which can be
assumed two-dimensional. The ideia, still under construction, is to understand how the single modes
38
of plasmon interact and how the collective resonaces depend on these single modes. This ideia must
be ready at the end of the summer, but the deliverance of this thesis does not allow to insert this job
here.
39
Appendix A
A.1 Fermi gas in d dimensions
Because fermions have spin 1/2, in each state of energy 0 < ε < EF , we can put, in 3 dimen-
sions, a maximun of 2 electrons. Let us decrease the dimension and set d = 2. In this case, only one
electron per state ε in allowed. Without demonstration, let us assume that generally, the following
relation is valid [30]:
N
d− 1= Vd
πd/2
Γ(
d2 + 1
)(2π)d
kd, (A.1)
where Vd = Ld is the volume in the direct space and Γ(x) is the usual gamma function. For free
electrons, the energy is ε = ~2k2/2m, and therefore, (A.1) reads
N
Vd≡ n =
(d− 1)(2π)d
πd/2
Γ(d/2 + 1)
(2m~2
d/2
εd/2
). (A.2)
The density of states is then given by
D(ε) =∂n
∂ε=d(d− 1)2(2π)d
πd/2
Γ(d/2 + 1)
(2m~2
)d/2
εd−22 . (A.3)
The Fermi energy EF should be evaluated by requiring integration of equation (A.3), in such way
that
n =∫ EF
0
D(ε) dε, (A.4)
which yields
EF =~2
2m4π(
Γ(d/2 + 1)d− 1
n
)2/d
. (A.5)
The average energy per volume unity is then
E0
Vd=
∫ EF
0
D(ε)ε dε
=d(d− 1)
(d+ 2)(2π)d
(2m~2
)2/dπd/2
Γ(d/2 + 1)E
d+2d
F (A.6)
which, using (A.2) yields
40
E0 =d
d+ 2NEF . (A.7)
Setting d = 3, we recover the well known result E0 = 3/5NEF , found in the literature.
A.2 Weyl transformation and the Liouville operator
A.2.1 The Wigner function and Weyl’s transformation
Following and resuming the steps of ref. [31], we start by defining the Weyl transformation of a
certain operator O. Here, we will be interested in the special case in which O is given in terms of a
commutation relation. Let |q〉 and |p〉 denote, respectively, the vector representation of the canonical
variables in the phase space for position and momentum, satisfying the unity partition relation∫|q〉〈q| dq =
∫|p〉〈p| dp = 1. (A.8)
The operator O can be decomposed in the basis of the phase space, by using (A.8)
O =∫dp′dp′′dq′dq′′|q′′〉〈q′′|p′′〉〈p′′|O|p′〉〈p′|q′〉〈q′|
=1
2π~
∫dp′dp′′dq′dq′′ exp
[i
~(p′′q′′ − p′q′)
]〈p′′|O|p′〉|q′′〉〈q′|, (A.9)
where we made use of the Fourier transformation between the canonical variables
|p〉 =1√2π~
∫dq exp
(i
~qp
)|q〉. (A.10)
By changing the variables p = p′+p′′
2
q = q′+q′′
2
η = p′′ − p′
ξ = q′′ − q′
(A.11)
the later equation reads
O =∫dp
dqfO(q, p)∆(q, p), , (A.12)
where were defined the operator
∆(q, p) =∫dξ exp
(i
~pξ
)|q + ξ/2〉〈q − ξ/2| (A.13)
and the so-called Weyl transformation of the operator O
fO(q, p) =∫dη exp
(i
~qη
)〈p+ η/2|O|p− η/2〉. (A.14)
It is easy to see, taking the definition of the trace, that the Weyl transformation of a certain operator
is given by
fO(q, p) = Tr(O∆(q, p)
). (A.15)
41
The Wigner function is defined as the Weyl transformation of the density operator ρ
W (q, p) =∫dη exp
(i
~qη
)〈p+ η/2|ρ|p− η/2〉. (A.16)
For a pure quantum state, ρ = |Ψ〉〈Ψ| and the Wigner function can be rewriten yielding
W (q, p) =∫dξ exp
(i
~qξ
)Ψ∗ (p+ ξ/2)Ψ (p− ξ/2) , (A.17)
where Ψ(x) = 〈x|Ψ〉 and Ψ∗ = 〈Ψ|x〉.
A.2.2 The Liouville operator
The density operator must satysfies the Liouville equation
i~∂ρ
∂t= −
[H, ρ
]. (A.18)
To derive the transport equation corresponding to (A.17), we should use (A.12) for the product of to
operators. Straightforward calculations involving the definitions above, yields
ρH =∫dp dq
2π~fH,ρ(q, p)∆(q, p) (A.19)
Iserting (A.15) in the later equation, we have
fρH =∫dq1 dq2 dp1 dp2
(2π~)2H(q2, p2)ρ(q1, p1) Tr
[∆(q1, p1)∆(q2, p2)∆(q, p)
]. (A.20)
The computation of the trace in the later equation is nothing but a tedious algebra job. Therefore,
we shall make use use of the following properties, directly derived from (A.13)
1. Tr[∆(q, p)
]= 1; (A.21)
2. Tr[∆(q1, p1)∆(q2, p2)
]=
∫dq′〈q′|∆(q1, p1)∆(q2, p2)|q′〉
=∫dq′dq′′〈q′|∆(q1, p1)|q′′〉〈q′′|∆(q2, p2)|q′〉
= dq′dp′′ exp[i
~(q′ − q′′)(p′ − p′′)
]δ
(q1 −
q′ + q′′
2
)δ
(q2 −
q′ + q′′
2
)= 2π~δ(p1 − p2)δ(q1 − q2), (A.22)
where were defined 2ξ = q′ + q′′ and η = q′ − q′′.
3. Tr[∆(q1, p1)∆(q2, p2)∆(q3, p3)
]=
∫dq′dq′′dq′′′〈q′|∆(q1, p1)|q′′〉〈q′′|∆(q2, p2)|q′′′〉〈q′′′|∆(q3, p3)|q′〉
=∫dq′dq′′dq′′′ exp
i
~[p1(q′ − q′′) + p2(q′′ − q′′′) + p3(q′′′ − q′)]
× δ
(q1 −
q′ + q′′
2
)δ
(q2 −
q′′ + q′′′
2
)δ
(q3 −
q′′′ + q′
2
)= 4 exp
2i~
[(q1 − p2)(p2 − p3)− (q2 − 13)(p1 − p3)]. (A.23)
42
Here, we have defined an extra auxiliar variable, namely 2ζ = q′′′ + q′, to computer the intermediate
integrals. Back to eq. (A.20), we are now able to write,
fρH = 4∫dp1 dq1 dp1 dp2
(2π~)2H(q1, p1)ρ(q2, p2) exp
2i~
[(q1 − q)(p2 − p)− (q2 − q)(p1 − p)]. (A.24)
Making the variable changing η = p2 − p e ξ = q2 − q, reads
fH,ρ = 4∫dp1 dq1 dξ dη
(2π~)2H(p1, q1)ρ(ξ + q, η + p) exp
2i~
[(η(q1 − p)− ξ(p1 − p))]. (A.25)
By tayloring the operator ρ(ξ + q, η + p), we get
ρ(ξ + q, η + p) =∞∑
k=0
1k!
(η∂
∂p+ ξ
∂
∂q
)k
ρ(q, p) = exp(η∂
∂p+ ξ
∂
∂q
)ρ(q, p). (A.26)
Whit the later result, and writing η and ξ in terms of the quantum differential operators
η = − ~2i
∂
∂ξand ξ =
~2i
∂
∂η, (A.27)
the Weyl transformation for the product of two operators follows, after some algebra,
fH,ρ = 4∫dq1 dq2 dp1 dp2
(2π~)2ρ(q1, p1) exp
2i~
[η(q1 − p)− ξ(p1 − p)]
× exp
[~2i
(←−∂
∂p
−→∂
∂q−←−∂
∂q
−→∂
∂p
)]H(q, p)
= · · ·
= ρ(q, p) exp
[~2i
(←−∂
∂p
−→∂
∂q−←−∂
∂q
−→∂
∂p
)]H(q, p). (A.28)
Finally, the Weyl transformation of the commutator in (A.18), reads
f[H,ρ] ≡ L(q, p) = 2i sin
[~2
(∂H∂p
∂ρ
∂q− ∂H
∂q
∂ρ
∂p
)]. (A.29)
The later expression defines the quantum version of the so-called Liouville operator, in such way
that Wigner function in (A.17) satysfies
∂W
∂t= −iL(p, q)W. (A.30)
43
Appendix B
B.1 Crank-Nicholson implicit scheme
The nonlinear Schrödinger equation may be writen in the form
i∂ψ
∂t= Hψ, (B.1)
which formal solution reads
ψ(t) = eiHt. (B.2)
For small time steps ∆t pequenos, we can write (B.2) in the form
eiH∆t ≈1 + i∆t
2 H
1− i∆t2 H
, (B.3)
known in the literature as the Cayley’s formula. Computing the wave function ψ(x, t) in the grid by
defining ψ(xj , tm) ≡ ψmj , equation (B.3) yields to the implicit Crank-Nicholson scheme(
1 +i∆t2H
)ψ(t+ ∆t) =
(1− i∆t
2H
)ψ(t), (B.4)
where the hamiltonian is given by
H = −A ∂2
∂x2− φ+ |ψ|4, (B.5)
with A = Γq/2. By using a scheme of finite centered differences for the spacial derivative, one
obtains the following system of equations
(1− i∆t
(A
∆x2− 1
2φm
j +12|ψm
j |4))
ψm+1j − iA∆t
2∆x2
(ψm+1
j+1 + ψm+1j−1
)=
=(
1 + i∆t(
A
∆x2+
12φm
j −12|ψm
j |4))
ψmj +
iA∆t2∆x2
(ψm+1
j+1 + ψm+1j−1
);
1∆x2
(φm+1
j+1 − 2φm+1j + φm+1
j−1
)= |ψm+1
j |2 − 1.
(B.6)
Once computed the wave function ψmj , it is possible to calculate the discrete Wigner function
Wmj,k
Wmj,k =
N∑`=1
ψm (j + `)ψ∗m (j − `) eik`∆k, (B.7)
44
where ∆k = ∆x = L/(N − 1) and the simulation box lenght is L = 1. In the current simulation, we
use peridodic boundary conditions.
B.2 Alternating Direction Implicit scheme (ADI)
For the two-dimensional case, equation (B.3) still holds, with a little change in the hamiltonian,
which now reads
H = −A(∂2
∂x2− ∂2
∂y2
)− φ− |ψ|2. (B.8)
The temporal evolution is done at two steps: first, we evolve the direction x between 0 and ∆t/2 and
then the y one is evoluted from ∆t/2 to ∆t. This is the so-called ADI (Alternating Direction Implicit)
method. In the current case, we have(1 + i∆t
4 Hx
)ψ(t+ ∆t/2) =
(1− i∆t
4 Hy
)ψ(t)
(1 + i∆t
4 Hy
)ψ(t+ ∆t) =
(1− i∆t
4 Hx
)ψ(t+ ∆t/2),
(B.9)
where Hα represents the part of the hamiltonean operator which acts on the variable α. For a simple
matter of choice, we first evolute the system in the x direction. The computational implementation
of the operators involved is done by adding a second auxiliar dimension to the arrays, resulting in
structures of 2N dimensions. Consequentely, the numerical operators result on 2N × 2N matrices.
45
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