1 Plasma Theory - Sektion Physik · 1 Plasma Theory Michael Bonitz Insitut für Theoretische Physik...

1

Plasma Theory

Michael Bonitz

Insitut für Theoretische Physik und AstrophysikKiel University

9. Juni 2010

preliminary lecture notes, not for distribution

Inhaltsverzeichnis

1 Plasma Dielectric Function 51.1 Vlasov Dielectric Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Dielectric function including collisions . . . . . . . . . . . . . . . . . . . . . 131.3 Vlasov dielectric function in a magnetic field . . . . . . . . . . . . . . . . . 17

1.3.1 Dielectric function for a two-dimensional magnetized plasma . . . . 201.3.2 Bernstein waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.4 Dielectric function of strongly correlated plasmas . . . . . . . . . . . . . . 221.4.1 Dielectric function and inverse dielectric function . . . . . . . . . . 221.4.2 Density response, RPA and local field corrections . . . . . . . . . . 231.4.3 Nonequilibrium Greens functions approach to a dielectric function

with correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.4.4 Quasilocalized charge approximation . . . . . . . . . . . . . . . . . 25

1.5 Dielectric function of a quantum plasma . . . . . . . . . . . . . . . . . . . 281.5.1 Classical limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.5.2 Quantum Dielectric function of fermions in thermodynamic equilibrium 34

1.6 Quantum dielectric function with collisions . . . . . . . . . . . . . . . . . . 35

Bibliography 41

3

4 INHALTSVERZEICHNIS

Kapitel 1

Plasma Dielectric Function1

As we have seen in Sec. ?? the effect of a medium on the electromagnetic field can becompletely described by the dielectric tensor εij (and the magnetic permeability). Theform of εij has to be determined separately – either experimentally, e.g. by scattering ofradiation off a medium, or theoretically – from kinetic theory or computer simulations.The approach via molecular dynamics simulations will be discussed in Sec. ??. Here weconsider the analytical theory of the dielectric function which is based on kinetic theoryand the linear response approximation. This means, we will analyze the medium responseto a weak external field.

The general form of a kinetic equation for the one-particle distribtution function ofspecies “α” is given by

∂fα(r,v, t)

∂t+ v · ∂fα(r,v, t)

∂r+

qαmα

(E +

v

c×B

)· ∂fα(r,v, t)

∂v= Iα(r,v, t), (1.1)

where the r.h.s. contains the collision integral which is determined by binary correlationsin the system, I = I[gαβ]. This equation is exact, provided the pair correlations are exactlyknown. In practice, one has to resort to approximations. The simplest approximation isgiven by the mean field ansatz – the Hartree or Vlasov approximation of many-body theory.As a next step we will include correlation effects by modeling collisions within the relaxationtime approximation. Finally, the results will be generalized to quantum plasmas.

1.1 Vlasov Dielectric Function

We begin the derivation of the dielectric function with the simplest case, the Vlasov diel-ectric function. The starting point is the Vlasov equation for the particle species α, cf.Sec.??,

∂fα∂t

+ v · ∂fα∂r

+qαmα

(E +

v

c×B

)· ∂fα∂v

= 0, (1.2)

1This chapter has been written together with Hanno Kählert.

5

6 KAPITEL 1. PLASMA DIELECTRIC FUNCTION

where, as usual, the distribution function is normalized to the density according to∫d3vfα(r,v, t) = nα(r, t).

To obtain ε(k, ω) we need to calculate the response of the distribution function (in parti-cular the density) to a small perturbation from an unperturbed state. In the following, wewill assume that, prior to the external excitation, the plasma was in local thermal equili-brium without any external field and, therefore, is spatially homogeneous, so the velocitydistribution is given by a Maxwellian2,

Φα(v) =

(mα

2πkBTα

)3/2

exp

(−mα|v − uα|2

2kBTα

), (1.3)

which is normalized to one,∫d3vΦα(v) = 1. For generality, we allow for the possibility of

a streaming plasma with a collective streaming velocity uα. In this case the particles areMaxwellian with the temperature Tα in a frame moving with uα.

The electromagnetic fields E and B must be determined self-consistently fromMaxwell’sequations with

ρ(r, t) =∑α

qαnα(r, t) = qα

∫d3vfα(r,v, t),

j(r, t) =∑α

jα(r, t) =∑α

qα

∫d3v vfα(r,v, t). (1.4)

This means that Eq. (1.2) is non-linear since the electric and magnetic fields are – via ρ andj – also functions of fα. In the local equilibrium state the plasma is electrically neutral, i.e.the total charge density and the total current of all plasma components, calculated withthe Maxwellians (1.3), cancel.

We now assume that a weak electric field is turned on and perturbs the local equilibriumstate. For the calculation of the response of fα to the external field we make the ansatz

E(r, t) = E(1)(r, t), limt→−∞

E(r, t) = 0,

B(r, t) ≡ 0,

fα(r,v, t) = f (0)α (v) + f (1)

α (r,v, t), (1.5)limt→−∞

fα(r,v, t) = f (0)α (v).

As discussed before, in the local equilibrium state the plasma is homogeneous and notexposed to external fields, and does not produce any internal fields, so we choose f 0

α =n0αΦα. Now the main approximation is that the perturbation of the distribution functioncaused by the field is weak, i.e. we require that, for any phase space point v, r and at anytime,

2In principle any function that satisfies the Vlasov equation could be used as equilibrium distribution.

1.1. VLASOV DIELECTRIC FUNCTION 7

1. |f (1)α | << f

(0)α , and

2. |E(1) · ∂f (1)α /∂v| << |E(1) · ∂f (0)

α /∂v|.

One has to be aware that no deep physics is behind these approximations and they certainlycannot always be satisfied3. The only motivation is the wish to linearize the Vlasov equationand apply the powerful mathematical tools of linear differential equations

Putting the ansatz into Eq. (1.2) and dropping terms of second order in the electricfield (E(1) · ∂f (1)

α /∂v) we get

∂f(1)α

∂t+ v · ∂f

(1)α

∂r+

qαmα

E(1) · ∂f(0)α

∂v= 0. (1.6)

The external perturbation can always be expanded in a sum of Fourier components, i.e.

E(1)(r, t) =

∫dω

2π

∫d3k

(2π)3E(1)(k, ω)e[i(k·r−(ω+iδ)t], (1.7)

and, therefore, the perturbed distribution function and all derived quantities will be of thesame structure. In Eq. (1.7) we introduced an infinitesimal small correction, δ > 0, whichassures the correct initial condition4 of vanishing field in the limit t → −∞. In the endwe will perform the limit δ → 0. In fact, the frequency integration is an inverse Laplacetransform and has to be performed in the complex frequency plane.

We now perform the space and time Fourier transform (1.7) of Eq. (1.6) and cancel thecommon exponential factors which yields an equation for a single Fourier component f (1)

α

[−i(ω + iδ) + ik · v]f (1)α +

qαmα

E(1) · ∂f(0)α

∂v= 0.

The perturbed particle density n(1)α (k, ω) can be obtained by integrating f (1)

α over velocityspace, see Eq. (1.4). Writing the electric field as E(1) = −ik φ we get the solution for thedistribution

f (1)α (k,v, ω) = − qα

mα

(k · ∂f(0)α

∂v) φ(k, ω)

(ω + iδ)− k · v(1.8)

from which the perturbed particle density follows as

n(1)α (k, ω) = − qα

mα

∫d3v

k · ∂f(0)α

∂v

(ω + iδ)− k · vφ(k, ω). (1.9)

3In particular the second condition is easily violated at the wings of the Maxwellian. Therefore, it isalways necessary to check the results against nonlinear approximations or numerical calculations.

4This assures causality of the results. In mathematical terms, the resulting response functions will be“retarded”.


Vlasov dielectric function

From the density response to the external field we can immediately compute all otherresponse functions. In particular, the dielectric tensor εij(k, ω) follows using the familiarrelation to the conductivity tensor σij(k, ω), cf. Sec. ??,

εij(k, ω) = δij +4πi

ωσij(k, ω). (1.10)

The conductivity tensor is easily eliminated by using the Fourier transform of the continuityequation and the linear response relation for the current density5,

ω ρ(1)(k, ω)− k · j(1)(k, ω) = 0,

j(1)i (k, ω) = σij(k, ω)E

(1)j (k, ω),

which allows us to transform

ω ρ(1)(k, ω) = kij(1)i = kiσij(k, ω)E

(1)j = −ikiσij(k, ω)kjφ. (1.11)

Inserting this result into Eq. (1.10) and introducing the longitudinal dielectric function weobtain

εl(k, ω) ≡ kikjk2

εij(k, ω) = 1− 4πρ(1)(k, ω)

k2φ(k, ω), (1.12)

which relates the dielectric function by the perturbation of the particle density. Finally,this result can be expressed by the unperturbed distribution function using Eq. (1.9),

εl,V(k, ω) = 1 +∑α

ω2pα

k2

∫d3v

k · ∂Φα∂v

ω + iδ − k · v(1.13)

where we have introduced the plasma frequency of species α

ωpα =

(4πq2

αn0α

mα

)1/2

(1.14)

and used f0α = n0αΦα.Result (1.13) is the famous (retarded) Vlasov dielectric function which describes the

density response of an unmagnetized plasma to a weak electric field and was first obtainedby Vlasov [Vla38]6. We use the superscript V to explicitly indicate the Vlasov approxima-tion.

5By explicitly using the continuity equation we assure particle number conservation in all subsequentresults.

6The correct causal solution was obtained by Landau [Lan46] who introduced the proper integration inthe complex frequency plane and obtained the collisionless (“Landau”) damping of the plasma oscillations.


Alternatively one can express the dielectric function in a general way7 by the longitu-dinal polarization function ΠV

α according to [Bon98]

εl,V(k, ω) = 1−∑α

Vαα(k)ΠVα (k, ω), (1.15)

where Vαβ(k) = 4πqαqβ/k2 is the Fourier transform of the Coulomb interaction between

particles α and β. This leads to the result for (retarded) Vlasov polarization function

ΠVα (k, ω) =

1

mα

∫d3v

k · ∂f(0)α

∂v

k · v − (ω + iδ)(1.16)

The physical meaning of the polarization function is the response of the density pertur-bation to an external potential energy U

(1)α = eαφ. The general definition of this linear

response function is via the following functional derivative [Bon98]

Πα(k, ω) =δn

(1)α (k, ω)

δU(1)α (k, ω)

∣∣U(1)→0

. (1.17)

One readily verifies that a functional derivation of the density, Eq. (1.9), yields in fact theVlasov result (1.16).

We will use the equation (1.15) frequently below as it allows to compare the differentmany-body approximations for the dielectric function and also makes the influence ofquantum effects transparant.

Cold particles

The integral in Eq. (1.13) can easily be evaluated in the limiting case of cold plasmaparticles, i.e. Φ(v) = δ(v − uα). Integrating by parts we obtain

∫d3v

k · ∂δ(v−uα)∂v

(ω + iδ)− k · v= −k2

∫d3v

δ(v − uα)

[(ω + iδ)− k · v]2= − k2

[(ω + iδ)− k · uα]2,

which leads to the following result for the dielectric function of cold particles

εl(k, ω) = 1−∑α

ω2pα

[(ω + iδ)− k · uα]2. (1.18)

7This is a general result of linear response theory and not limited to the Vlasov approximation. We willrecover it also in the case of a dielectric function with collisions and also for quantum plasmas.


Dielectric function of a Maxwellian plasma

Returning now to an equilibrium plasma at finite temperature for which the unperturbeddistribution function is given by a Maxwellian, we rewrite the dielectric function (1.13) ina different way by introducing the so-called plasma dispersion function

Z(ζ) = π−1/2

∫ ∞−∞

e−t2

t− ζdt. (1.19)

It fulfills the differential equation

Z ′(ζ) = −π−1/2

∫ ∞−∞

2 t

t− ζe−t

2

dt = −2 [1 + ζ Z(ζ)], (1.20)

which can easily be seen by taking the derivative of Eq. (1.19) and integrating by parts.Further properties of the dispersion function are listed separately in the section below.

We now rewrite the integral over the Maxwellian distribution function occuring in theexpression for ε(k, ω). We first introduce the thermal velocity vTα = (kBTα/mα)1/2 and thevariable ζα = (ω + iδ − k · uα)/(

√2kvTα), allowing us to rewrite the integral as∫

d3vk · ∂Φα

∂v

ω + iδ − k · v=

1

v2Tα

1√π

∫ ∞−∞

t

t− ζαe−t

2

dt =1

v2Tα

[1 + ζαZ(ζα)],

where we have used Eq. (1.20) and have chosen one of the axes parallel to k. This allowsus to rewrite Eq. (1.13) as

εl(k, ω) = 1 +∑α

1

k2λ2Dα

[1 + ζαZ(ζα)] (1.21)

Here λDα = vTα/ωpα = [kBTα/(4πn0αq2α)]

1/2 denotes the Debye screening length of speciesα.

Finally, we consider the static limit, ω → 0. If, further uα = 0, we note that the secondterm in the brackets vanishes [Z(0) is finite], and we recover the familiar static Debyedielectric function

εl,V(k, 0) = 1 +∑α

1

k2λ2Dα

.

For the sake of completeness we give an alternative result for the polarization (1.16) interms of the plasma dispersion function

ΠVα (k, ω) = − n0α

kBTα[1 + ζαZ(ζα)]. (1.22)


Properties of the plasma dispersion function

Here we summarize the main properties of the plasma dispersion function. The integrationin Z(ζ) must be performed such that the contour is below the pole. Using the Plemeljformula we can write the dispersion function as

Z(ζ) =1

π1/2

[P

∫ ∞−∞

e−t2

t− ζdt

]+ iπ1/2e−ζ

2

, (1.23)

where P denotes the principal value of the integral.There are two important limits for which we simply state the asymptotic limits of Z(ζ)

(see [Bel06]).

• |ζ| � 1:

Z(ζ) = −1

ζ

[1 +

1

2ζ2+

3

4ζ4+ . . .

]+ iπ1/2e−ζ

2

,

• |ζ| � 1:

Z(ζ) = −2ζ

[1− 2ζ2

3+ . . .

]+ iπ1/2e−ζ

2

.

Dispersion of plasma oscillations

The dispersion of longitudinal collective plasma modes follows from the dispersion relationεl(k, ω) = 0. Here we consider, as an example, a two-component electron-ion plasma, forwhich the dielectric function reads εl(k, ω) = 1 +χi +χe, where χe and χi are the electronand ion susceptibilities. Generalizations to multi-component systems are straightforward.

To obtain the dispersion of plasma oscillations we first recall that εl is a complexquantity, thus the dispersion relation in fact consists of two scalar equations for the realand imaginary part of εl: εr and εi, respectively. The solution of this system is in general acomplex function ω(k) consisting of a real and imaginary part as well, ω = ωr + iωi. In thefollowing we will assume that the imaginary part of the frequency is much smaller thanthe real part which is justified for weakly dissipative systems where the imaginary part ofεl is small. Then the dispersion relation can be expanded around ω ≈ ωr:

0 = εr(ω) + iεi(ω) ≈

≈ εr(ωr) + iωidεr(ω)

dω

∣∣∣∣ω=ωr

+ i

{εi(ωr) + iωi

dεi(ω)

dω

∣∣∣∣ω=ωr

}(1.24)

The real and imaginary parts of the complex frequency then follow from the first orderterm, i.e. εr(ωr), and the second order contributions – second and third terms, respectively,

εr(ωr) ' 0, ωi ' −εi(ωr)

dεr(ω)dω

∣∣∣ω=ωr

, (1.25)

whereas the last term is of higher order and is neglected.Let us now discuss the resulting dispersions ωr(k) and ωi(k) for three relevant cases.


1. ω/k � vT i, vTe.In the high-frequency limit electrons and ions are slow compared to the phase velocityof the wave. We use the asymptotics of the plasma dispersion function and obtain

χα = −ω2pα

ω2

(1 + 3

k2

ω2v2Tα + . . .

)+ i

ω

kvTα

1

k2λ2Dα

√π

2e−ω

2/(2k2v2Tα). (1.26)

Since ω2pe/ω

2pi = mi/me and, in general, vTe � vT i, the ion terms are small compared

to the electron terms and can be neglected8, i.e. ε ' 1 + χe. Then, from Eq. (1.25),we get the following dispersion for electron plasma waves,

ω2r(k) = ω2

pe

(1 + 3

k2

ω2r

v2Te

)≈ ω2

pe(1 + 3k2λ2De), (1.27)

and for the imaginary part,

ωi(k) = −√

8

π

ωpek3λ3

De

exp

(−1 + 3k2λ2

De

2k2λ2De

). (1.28)

The result is shown in Fig. 1.1. At zero wave number (spatially homogeneous excita-tion) the eigenfrequency of the plasma coincides with the electron plasma frequency.At finite k the frequency increases quadratically with k and proportional to the elec-tron temperature, but within the used expansion, this correction term has to be smallcompared to ωpe9.

In this frequency range the plasma can be regarded as a pure electron plasma wherethe ions act as a neutralizing background.

2. vT i � ω/k � vTe.In this range of intermediate frequencies, we have for the electron susceptibility |ζe| �1 in the plasma dispersion function and obtain

χα ≈1

k2λ2Dα

+ iω

kvTα

1

k2λ2Dα

√π

2e−ω

2/(2k2v2Tα). (1.29)

Using Eq. (1.26) for the ion susceptibility and (1.29) for the electrons, we obtain thedispersion relation

1 +1

k2λ2De

+ iω

kvTe

1

k2λ2De

√π

2e−ω

2/(2k2v2Te)

−ω2pi

ω2

(1 + 3

k2

ω2v2T i

)+ i

ω

kvT i

1

k2λ2Di

√π

2e−ω

2/(2k2v2Ti) = 0. (1.30)

8They only become important when Ti > Tem2i /m

2e.

9For larger wave numbers additional effects due to particle correlations and quantum effects – whichare neglected within the present classical meanfield approach – have to be taken into account leading to apossibly drastic change of the dispersion.

1.2. DIELECTRIC FUNCTION INCLUDING COLLISIONS 13

With the same technique as applied before we find the dispersion relation for ionacoustic waves as

ω2r(k) ≈ k2c2

s

1 + k2λ2De

+ 3k2v2T i, (1.31)

ωi(k) ≈ − ωr(1 + k2λ2

De)3/2

√8

π

[√me

mi

+

(TeTi

)3/2

exp

(− Te/2Ti

1 + k2λ2De

− 3

2

)],

where cs =√kBTe/mi. The dominant contribution to Landau damping is due to the

ions since the electron contribution is small,√me/mi � 1. If Te � Ti the ion term

becomes smaller as well, i.e. strong ion Landau damping occurs when Ti ≈ Te [Bel06].The complete dispersion relation is shown in Fig. 1.1.

3. ω/k � vT i, vTe.In this case no collective plasma oscillations are possible. Electrons and ions are ableto instaneously follow the external perturbation. Thus, in this low-frequency limitthe dielectric function can be approximated by the static limit

εl(0, k) = 1 + (k2λ2De)−1 + (k2λ2

Di)−1 = 1 +

κ2D

k2.

The longitudinal field is screened within a radius equal to the total Debye length ofthe plasma, λD = (λ2

Di + λ2De)

1/2 = κ−1D .

Before we proceed a few comments are in order. First, note the existence of a finitenegative value of ωi which clearly indicates damping of the plasma oscillations. This dam-ping is not due to collisions between plasmas particles which are entirely neglected in thepresent mean field approximation. This collisionless damping was discovered by Landau[Lan46] and is commonly called Landau damping. Second, we note that the behavior of atwo-component quantum plasma is very similar, see the right part of Fig. 1.1. Most import-antly, the zero wave number plasmon again starts at the plasma frequency which turns outto be a robust property. More details about plasmons in quantum plasmas will be discussedin Sec. 1.5. Finally we note that the plasmon spectra shown in Fig. 1.1 are valid only for athree-dimensional plasma. They change qualitatively in two and one-dimensional systems,see e.g. [Bon98] and references therein.

1.2 Dielectric function including collisions

The starting point for the derivation of the dielectric function with collisions is a kineticequation which goes beyond the Vlasov approximation and contains a collision integralI[f ],

∂fα∂t

+ v · ∂fα∂r

+qαmα

(E +

v

c×B

)· ∂fα∂v

= Iα[f ]. (1.32)


Abbildung 1.1: Wave number dispersion of longitudinal plasma oscillations in a two-component three-dimensional electron-ion plasma (arbitrary units). ωLe,i denote the elec-tron or ion plasma (Lengmuir) frequency, rDe,i the electron or ion Debye length. Uppermode is the electron plasmon (optical plasmon), lower mode the ion-acoustic oscillation.Left figure is for a classical plasma, right for a plasma with quantum degenerate elecctrons(vFe = pFe/me denotes the Fermi velocity). The figure also shows the imaginary part ofthe frequencies (damping), still to do....From ref. [ABR84].

In the following we employ the simple BGK model integral, e.g. [ABR84],

Iα[f ] = −ναn[fα(t)− nαΦα], (1.33)

with nα =∫d3vfα and a constant collision frequency ναn for collisions with neutral partic-

les10. This operator is linear in the distribution function and can be applied if the deviationfrom the equilibrium state is small. It is the simplest example of a so-called “relaxationtime approximation” (RTA), where the relaxation time is given by τ rel = ν−1

αn . The RTAis well suited for a qualitative analysis of collision effects in the final stage of relaxationtowards a stationary state11.

The time-dependence of the distribution function arising from this collision term iseasily seen for the case of a spatially homogeneous system without external field where∂fα∂t

= Iα[f ]. The solution of this equation together with the initial condition at fα(t =0) = fα0 is (the result is readily verified by inserting in the kinetic equation)

fα(t) = fα0e−ναnt + nαΦα[1− e−ναnt], (1.34)

showing the decay of the initial distribution and the approach to the equilibrium state.As in section 1.1 we now need to calculate the response of the distribution to a small

perturbation from equilibrium (which contains no external fields). In the present case as10This collision integral is appropriate for charge-neutral collisions.11For a discussion of the different time scales, see Ref. [Bon98].

1.2. DIELECTRIC FUNCTION INCLUDING COLLISIONS 15

well, the equilibrium distribution is given by f0α = n0αΦα since this is the only stationarysolution of Eq. (1.32). Making again the ansatz

E(r, t) = E(1)(r, t), limt→−∞

E(r, t) = 0, (1.35)

B(r, t) ≡ 0,

fα(r,v, t) = f (0)α (v) + f (1)

α (r,v, t), (1.36)limt→−∞

fα(r,v, t) = f (0)α (v),

and dropping terms of second order in the electric field in Eq. (1.32), we obtain

∂f(1)α

∂t+ v · ∂f

(1)α

∂r+

qαmα

E(1) · ∂f(0)α

∂v= −ναn(f (1)

α − n(1)α Φα). (1.37)

A space and time Fourier transform of Eq. (1.37), as in section 1.1, using the transform(1.7) for the field and the distribution function, yields

[−i(ω + iδ) + ik · v]f (1)α +

qαmα

E(1) · ∂f(0)α

∂v= −ναn(f (1)

α − n(1)α Φα).

To calculate the dielectric function we need the perturbed particle density n(1)α (k, ω). With

E(1) = −ik φ(k, ω) we first obtain

f (1)α (k,v, ω) = − qα

mα

(k · ∂f(0)α

∂v) φ(k, ω)

ωαn − k · v+

iναnΦα

ωαn − k · vn(1)α (k, ω), (1.38)

from which the perturbed particle density follows as

n(1)α (k, ω) = − qα

mα

∫d3v

k· ∂f(0)α∂v

ωαn−k·v

1− iναn∫d3v Φα

ωαn−k·vφ(k, ω). (1.39)

Here we have introduced the complex frequency ωαn = ω+ iδ+ iναn. In the case of a finitecollision frequency ναn we may safely omit the infinitesimal constant δ but have to restoreit if the collisionless limit is considered.

According to Eqs. (1.39, 1.12) the dielectric function is given by

εl(k, ω) = 1 +∑α

ω2pα

k2

∫d3v

k· ∂Φα∂v

ωαn−k·v


ωαn−k·v. (1.40)

This result has been obtained by Rostoker and Rosenbluth [RM60], see also Ref. [ABR84].In the limit ναn → +δ we recover the Vlasov dielectric function (1.13). Compared to (1.13)there are two changes: 1) the real frequency is replaced by a complex frequency ωαn which


includes a finite positive imaginary part describing a decay of the perturbation. 2) Thedielectric function contains an additional term in the denominator.

In terms of the polarization function we can again write, following (1.15),

εl(k, ω) = 1−∑α

Vαα(k)ΠBGKα (k, ω), (1.41)

where now a different expression of the polarization function can be identified, from (1.40)which we call ΠBGK

α and which is related to the collisionless (Vlasov) polarization accordingto,

ΠBGKα (k, ω) =

ΠVα (k, ω)


ωαn−k·v. (1.42)

This expression can be further simplified for a Maxwellian plasma, as we will show af-ter considering the expression of the dielectric function in terms of the plasma disperionfunction.

Cold particles

We again consider the limit of cold particles, i.e. Φ(v) = δ(v−uα). The integral involvingthe derivative of Φ(v) has been calculated in section 1.1. The evaluation of the integral inthe denominator is even simpler. Here we obtain for the dielectric function

εl(k, ω) = 1−∑α

ω2pα

(ω − k · uα)(ωαn − k · uα). (1.43)

Note that the complex frequency only appears in one of the terms in the denominator.This shows again that the simple replacement ω ↔ ω+ iναn to account for collisions is notcorrect.

Dielectric function with collisions in terms of the plasma dispersion function

Once again the dielectric function (1.40) can be expressed in terms of the plasma dispersionfunction (1.19). For the integral in the denominator of Eq. (1.40) we obtain

∫d3v

Φα

ωαn − k · v=(2πv2

Tα

)−3/2∫d3v

e− |v−uα|2

2v2Tα

ωαn − k · v

=1

π3/2√

2vTα

∫d3t

e−|t|2

kζα − k · t= − 1√

2vTαk

1√π

∫ ∞−∞

e−t2

t− ζαdt. (1.44)

1.3. VLASOV DIELECTRIC FUNCTION IN A MAGNETIC FIELD 17

Here we have modified the variable ζα = (ωαn− k ·uα)/(√

2kvTα), which now includes thecomplex frequency ωαn. Collecting terms, we can write for the dielectric function12

εl(k, ω) = 1 +∑α

1

k2λ2Dα

1 + ζαZ(ζα)

1 +(

iναn√2kvTα

)Z(ζα)

(1.45)

Taking the static limit, ω → 0, and also uα = 0 we get

εlo(k, 0) = 1 +∑α

1

k2λ2Dα

,

regardless of the value of the damping ναn. This shows that collisions do not alter the staticscreening in a plasma.

Finally, going back to arbitrary frequencies, we rewrite the polarization function of acollisional Maxwellian plasma, Eq. (1.42), in terms of the plasma dispersion function

ΠBGKα (k, ωαn) =

ΠVα (k, ωαn)

1− iναnωαn−k·uα

[1 + ΠVα (k,ωαn)

n0α/kBTα

] , (1.46)

which is easily achieved by comparing Eq. (1.45) with (1.21) and expressing the plasmadispersion function in the denominator in terms of ΠV

α , see Eq. (1.22). Thus we have obtai-ned a closed expression of the correlated polarization function in terms of the collisionlessrestult. For future reference we note that for uα = 0 this result can be rewritten in theform

ΠBGKα (k, ωαn) =

ΠVα (k, ωαn)

1− iναnωαn

[1− ΠVα (k,ωαn)

ΠVα (k,0)

] . (1.47)

1.3 Vlasov dielectric function of a plasma in an externalmagnetic field

We now turn to the dielectric properties of a plasma in a magnetic field13. While the fieldis assumed to be stationary and spatially homogeneous, it is allowed to have any strength.We again start with the collisionless case. In the presence of a constant external magneticfield B0 the Vlasov equation reads

∂fα∂t

+ v · ∂fα∂r

+qαmα

(E +

v

c×B0

)· ∂fα∂v

= 0. (1.48)

In the initial state, t → −∞, we assume that there is no electric field and the plasma isin equilibrium, described by a Maxwellian f (0)

α . Turning on a weak electric field leads to a12We mention that this result has also given by Jenko et al. [JJT05]. They, however, define the thermal

velocity with an additional factor√

2.13We follow, in part, the derivation in the book of Bellan [Bel06].


small perturbation f (1)α of the distribution and we can linearize the Vlasov equation around

f(0)α

∂f(1)α

∂t+ v · ∂f

(1)α

∂r+

qαmα

(v

c×B0

)· ∂f

(1)α

∂v=

qαmα

∇φ(1) · ∂f(0)α

∂v. (1.49)

In order to solve Eq. (1.49) we employ the method of characteristics. First we rewriteEq. (1.49) as

d

dtf (1)α (r(t),v(t), t) =

qαmα

∇φ(1) · ∂f(0)α

∂v, (1.50)

where r(t′) and v(t′) are the characteristics of Eq. (1.49) which are nothing but Newton’sequations

dr(t′)

dt′= v(t′),

dv(t′)

dt′=

qαmαc

v(t′)×B0, (1.51)

with the boundary condition

r(t′ = t) = r, v(t′ = t) = v. (1.52)

They can be regarded as the unperturbed particle orbits in phase space. The first ordercorrection to the distribution function is then given by

f (1)α (r,v, t) =

qαmα

∫ t

−∞dt′

[∇φ(1) · ∂f

(0)α

∂v

]r=r(t′),v=v(t′)

. (1.53)

Due to the linearity of Eq. (1.49) it can be solved via Fourier transform, i.e. it is sufficientto solve it for a single Fourier component of the potential, φ(1)(r, t) = φ(1)eik·r−iωt, for whichEq. (1.53) yields 14

f (1)α (r,v, t) =

qαφ(1)

mα

f (0)α

∫ t

−∞dt′[−ik · v(t′)

v2Tα

exp (ik · r(t′)− iωt′)]. (1.54)

Usingd

dt′exp[ik · r(t′)] = ik · v(t′) exp[ik · r(t′)],

we find, by partial integration and, using v2Tαmα = kBTα,

f (1)α (r,v, t) = − qα

kBTαφ(1)f (0)

α

{[exp(ik · r(t′)− iωt′)]t−∞ + iωIphase(r, t)

}, (1.55)

where the contribution from −∞ vanishes, and we introduced the short notation for theintegral

Iphase(r, t) =

∫ t

−∞dt′ exp[ik · r(t′)− iωt′]. (1.56)

14we again introduce a small positive imaginary part δ of the frequency, i.e. ω = ω + iδ, to ensurecausality, i.e. vanishing of φ and f (1)

α for t→ −∞.


At this point we require the explicit form of the unperturbed orbits. One can readilycheck that the solution of the equations of motion in the magnetic field B0, Eq. (1.51) is

v(t′) = vz ez + v⊥ cos[ωcα(t′ − t)]− ez × v⊥ sin[ωcα(t′ − t)], (1.57)

r(t′) = r + vz(t′ − t)ez +

1

ωcα{v⊥ sin[ωcα(t′ − t)]+

ez × v⊥ (cos[ωcα(t′ − t)]− 1)} , (1.58)

where ωcα = qαB0/(mαc) is the cyclotron frequency and the magnetic field was chosen asB0 = B0ez. In Eq. (1.57) the velocity components parallel and perpendicular to the ma-gnetic field are time-independent. At t′ = t they conincide with the respective componentsof the initial velocity, whereas for t′ 6= t the perpendicular component which consists oftwo mutually orthogonal contributions performs a rotation around the magnetic field. Nowdefine the angle ϕ such that k⊥ · v⊥ = k⊥v⊥ cosϕ and k⊥ · (ez × v⊥) = k⊥v⊥ sinϕ. Thephase in the exponent of Eq. (1.55) can then be written as

k · r(t′) = k · r + k‖v‖(t′ − t) + sα {sin[ωcα(t′ − t) + ϕ]− sinϕ} ,

where we defined sα = k⊥v⊥/ωcα, and the phase integral turns into (we introduce τ = t′−t)

Iphase(r, t) = eik·r−iωt∫ 0

−∞dτ exp

[i(k‖v‖ − ω)τ + isα{sin[ωcατ + ϕ]− sinϕ}

]. (1.59)

Using the Bessel function identity

eiz sin θ =∞∑

n=−∞

Jn(z)einθ,

we obtain

Iphase(r, t) = eik·r−iωt∞∑

n=−∞

Jn (sα) e−isα sinϕ

∫ 0

−∞dτ ei(k‖v‖−ω)τ+in(ωcατ+ϕ).

The integral is easily calculated (the t = −∞ contribution vanishes due to the imaginarypart of the frequency), and we obtain for the distribution function (1.55)

f (1)α (r,v, t) = − qα

kBTαφ(1)f (0)

α eik·r−iωt

[1− e−isα sinϕ

∞∑n=−∞

ωJn (sα) einϕ

ω − k‖v‖ − nωcα

]. (1.60)

The next step is to compute the first order density which follows from integration ofthe distribution function over v which is conveniently performed in cylindrical coordinates

n(1)α (r, t) =

∞∫−∞

dv‖

∫ 2π

0

dφ

∫ ∞0

dv⊥v⊥f(1)α (r,v, t). (1.61)


Using the Bessel identity∫ ∞0

zJ2n(βz)e−α

2z2

dz =1

2α2e−β

2/2α2

In

(β2

2α2

), (1.62)

for the v⊥ integration, and the relation

Jn(z) =1

2π

∫ 2π

0

eiz sin θ−inθdθ, (1.63)

for the φ integration, we are left with

n(1)α = −qαφ

(1)nα0

kBTα

[1 + γ0αe

−zα∑n

In(zα)

π1/2

∫ ∞−∞

dte−t

2

t− γnα

]

= −qαφ(1)nα0

kBTα

[1 + γ0αe

−k2⊥r

2Lα

∑n

In(k2⊥r

2Lα)Z(γnα)

], (1.64)

where we definedγnα =

ω − nωcα√2k‖vTα

, zα = k2⊥r

2Lα, (1.65)

and we introduced the Larmor radius

rLα =vTαωcα

. (1.66)

Z(γ) denotes the plasma dispersion function of a Maxwellian plasma, see Eq. (1.19). Thedielectric function is then obtained from Eqs. (1.15, 1.17) as

εl,V (k, ω) = 1 +∑α

1

k2λ2Dα

[1 + γ0αe

−zα∞∑

n=−∞

In(zα)Z(γnα)

]. (1.67)

This is the retarded longitudianal dielectric function of a Maxwellian plasma in an externalmagnetic field at finite temperature. It is the mean field result which neglects collisions.The result (1.67) allows to compute the plasmon spectrum of a magnetized plasma whichis rather complex and presented in detail in many text books, see e.g. [ABR84]. Below wewill analyze several limiting cases and, in particular, the Bernsetin modes.

1.3.1 Dielectric function for a two-dimensional magnetized plasma

If the plasma motion is confined to a plain, the dielectric function follows straightforwardlyfrom the above 3D results, see e.g. [Tot75]. We simply use

f (0)α =

n(0)2D

2πv2Tα

e−v2⊥/2v

2Tαδ(vz)δ(z), (1.68)

where the unperturbed 2D density is n(0)2D = 1/(πa2). The derivation of the dielectric

function is even simpler in this case, and the result is

εl,V,2D(k⊥, ω) = 1−∑α

Vαα(k⊥)2n

(0)2D

kBTαe−zα

∞∑n=1

n2ω2pα

ω2 − n2ω2cα

In(zα). (1.69)


Abbildung 1.2: Bernstein modes in a three-dimensional plasma. Left: classical (Maxwell)plasma. Right: Fermi plasma. From ref. [ABR84].

1.3.2 Bernstein waves

It was shown by Bernstein [Ber58] that the plasma in a magnetic field supports longi-tudinal waves propagating strictly perpendicular to the magnetic field, i.e. kz → 0. Thenthe dispersion relation simplifies and we have

εl,V (k⊥, ω) = 1−∑α

e−zα

zα

∞∑n=1

2n2ω2pα

ω2 − n2ω2cα

In(zα) = 0. (1.70)

This equation contains, in general, an infinite number of coupled modes, in particularmodes with frequencies close to multiples of the cyclotron frequency (Bernstein waves).This dispersion relation becomes particularly simple to solve in the limits of small andlarge wave numbers, respectively.

i.) Small wave numbers

ii.) Large wave numbers


1.4 Dielectric properties of strongly correlated plasmas.The quasilocalized charge approximation

We now turn to a more systematic analysis of correlation effects on the dielectric properties.

1.4.1 Dielectric function and inverse dielectric function

Before proceeding further we discuss the relation between the dielectric function ε(k, ω)and its inverse, ε−1(k, ω). While this may seem a triviality, it is not if approximations arebeing developed. Then it is not always trivial to translate approximations for the formerinto the corresponding approximations for the latter. Traditionally, in plasma physics thedielectric function is used whereas in condensed matter and in quantum systems, whereone starts from more general linear response theory, the inverse dielectric function is thestarting point15.

Assume that an approximation for the dielectric function in terms of the polarizationfunction is known, for example the mean field result with the non-correlated polarizationΠ0 (all functions are retarded),

ε(k, ω) = 1− V (k)Π0(k, ω). (1.71)

On the other hand, the inverse function is frequently written in the following form

ε−1(k, ω) = 1 + V (k)χ0(k, ω), (1.72)

and it remains to establish the relation between χ0 and Π0. To this end we rewrite expression(1.71)

ε−1(k, ω) =1

1− V (k)Π0(k, ω)=

1− V (k)Π0(k, ω) + V (k)Π0(k, ω)

1− V (k)Π0(k, ω)

= 1 + V (k)Π0(k, ω)

1− V (k)Π0(k, ω),

from which we immediately read off the result for χ in the expression (1.72)

χ0(k, ω) =Π0(k, ω)

1− V (k)Π0(k, ω). (1.73)

If Π0 is the simplest approximation for ε – the Vlasov polarization [or Lindhard polarizationin the quantum case], then χ0 is the simplest approximation for ε−1 – the random phaseapproximation. Mathematically, χ0 depends on Π0 in a nonlinear way. If VΠ0 < 1 we canexpand the denominator in (1.73) and obtain the sum of ring or “bubble” diagrams

χ0(k, ω) = Π0(k, ω)

{1 + V (k)Π0(k, ω) +

1

2![V (k)Π0(k, ω)]2 + . . .

}. (1.74)

15In addition to these two different approaches, there is also some confusion in the literature regardingthe terminology, for example regarding the termin “random phase approximation” (RPA), which we clarifybelow.

1.4. DIELECTRIC FUNCTION OF STRONGLY CORRELATED PLASMAS 23

In other words, expression (1.73) is the sum of an infinite series (geometric progression) ofthe form (1.74). Alternatively, the solution (1.73) can be transformed into an equation forχ0, the Bethe-Salpeter equation (or one of its versions),

χ0(k, ω) = Π0(k, ω) + V (k)Π0(k, ω)χ0(k, ω). (1.75)

The particular use of these expressions arises from their generality: we may straightforward-ly extend this concept to the (inverse) dielectric function with correlations. To this end wefirst need to find an improved expression for the polarization which generalizes Π0 −→ Πand, via Eq. (1.71), the dielectric function. Secondly, the corresponding approximation forthe inverse dielectric function is then obtained from equation (1.72) with χ0 −→ χ, whereχ is determined by Π via the same relation (1.73),

χ(k, ω) =Π(k, ω)

1− V (k)Π(k, ω). (1.76)

Thus, the problem of computing a correlated inverse dielectric function is reduced to findingsuitable approximations for the retarded polarization Π. For this, many methods have beendeveloped.

1.4.2 Density response, RPA and local field corrections

In order to better understand some of the condensed matter theory approaches to thecorrelated dielectric function let us recall the definition of the longitudinal polarization asa density response function, which we already encountered, cf. Eq. (1.17),

χ(k, ω) =δn(k, ω)

δU(k, ω), (1.77)

where δn is the density response of the system to the external potential perturbation δφwhere the latter is associated with the potential energy perturbation δU = qδφ, and thederivative in Eq. (1.77) is to be evaluated at U = 0. Together with a density modulation,the external potential gives rise to an induced potential and potential energy, so the totalpotential energy in the system becomes U tot = U +U ind. The tree main approximations tothe density response are then readily classified:

1. Ideal system: the induced potential is neglected, and χ ≡ χ0.

2. Mean field approximation: the induced potential is approximated by a mean field(solution of Poisson’s equation),

δU tot(k, ω) = δU(k, ω) + V (k)δn(k, ω). (1.78)

According to Eq. (1.77), the density response is now given by

δn(k, ω) = χ(k, ω)δU(k, ω) = χ0(k, ω)δU tot(k, ω)

= χ0(k, ω){δU(k, ω) + V (k)δn(k, ω)

}=

χ0(k, ω)

1− V (k)χ0(k, ω)δU(k, ω). (1.79)


Comparison with the first line yields the RPA polarization

χRPA(k, ω) =χ0(k, ω)

1− V (k)χ0(k, ω)(1.80)

in terms of the ideal (Lindhard) polarization.

3. Interaction effects beyond mean field. Correlations and exchange: The true potentialinduced by the external field is sometimes called “polarization potential” which wasintroduced by Pines [Pin66]

δUpol = V (k)[1−G(k, ω)]δn(k, ω) (1.81)

and deviates from the mean field approximation by some functionG called “dynamicallocal field correction”. It accounts for correlations and exchange effects missing in themean field approximation. The appearance of G is the same as that of the paircorrelation function in classical density functional theory, cf. Sec.... The derivation ofthe polarization is completely analogous to the previous case and yields

χLFC(k, ω) =χ0(k, ω)

1− V (k)[1−G(k, ω)]χ0(k, ω), (1.82)

expressing χ in terms of the ideal polarization χ0 and an additional function G. Itnow remains to find the corresponding expression for the dielectric function on thelevel of the local field correction. To this end we use the relation (1.76) between χ andΠ and solve it for Π, using expression (1.82) for χ. A straightforward caluculationyields

Π(k, ω) =χ0(k, ω)

1 + V (k)χ0(k, ω)G(k, ω). (1.83)

and the dielectric function

ε(k, ω) = 1− V (k)Π(k, ω) = 1− V (k)χ0(k, ω)

1 + V (k)χ0(k, ω)G(k, ω). (1.84)

Expressions (1.83, 1.84) are formally exact would the function G be known. Thus thesolution of the response problem of the correlated many-body system has been shifted tothe determination of the dynamical local field correction. There have been many successfulattempts to derive expressions for G, most notably the early results by Singwi, Tosi, Land,Sjolander and others [STLS68]. The first apprximation used by them is to replace thedynamics (frequency-dependent) function by its static limit, G(k, ω) → G(k, ω = 0) =G(k). Then all exchange and correlation effects should be contained in the pair distributionfunction or the static structure factor S(k). Indeed, Singwi et al. derived the followingapproximation

G(k) = − 1

n

∫dq

(2π)3

k · qq2

[S(|k− q|)− 1] . (1.85)


This approximation turned out to be very successful and substantially improves the RPAresults. In particular, it accurately reproduces the quantum Monte Carlo data for theground state energy of strongly coupled electrons at metallic densities of 2 ≤ rs ≤ 6, e.g.[Ich94]. Among its problems is that it fails to reproduce the results of strongly coupledclassical systems and violates the compressibility sum rule. Further discussions and someexamples for improved local field correction can be found in the text book of Ichimaru[Ich94].

1.4.3 Nonequilibrium Greens functions approach to a dielectricfunction with correlations

A powerful approach to incorporate correlation effects in the polarization is provided bynonequilibrium Greens functions theory (NEGF) where Π is expressed via a particle-holeT-matrix....diagrams from PRL here

The T-matrix obeys a Lippmann-Schwinger equation involving a general two-particlekernel K for which approximations can be derived in a systematic way. Special care has tobe taken for obeying consistency requirements such as sum rules for the dielectric function,many approximation schemes do not fulfill this requirement. Within NEGF this problemcan be solved via solution of the Keldysh-Kadanoff-Baym equations, for details see ref.[KB00].

1.4.4 Quasilocalized charge approximation

For strongly correlated plasmas and liquids a powerful method is the quasi-localized chargeapproximation (QLCA) derived by Kalman and Golden in 1990 [KG90], for a recent over-view, see [KGDH05]. The idea is to use results known from solid state theory. The groundstate of a classical system (T = 0) is a lattice configuration with the equilibrium particlecoordinates R0 = (r10, . . . rN0). These coordinates correspond to a minimum of the totalpotential energy

U =N∑i=1

Φ(ri) +∑i 6=j

V (ri − rj), (1.86)

which could be the ground state (absolute minimum) or a metastable configuration (localminimum). In standard lattice theory one considers small excitations ξi of the particlesaround their minimum positions. Expanding the ξi into a complete set of eigenfunctionsξk with eigenfrequencies ωk allows to diagonalize the hamiltonian.

In a strongly correlated liquid the situation is different. There are no fixed lattice sitesri0 of the particles, no strict periodicity and no stable phonon spectrum. However, closeto the freezing point the deviations from crystal behavior should be small. This meanseach particle experiences a local crystal-like environment for some finite “caging” time τcage.After this time the surrounding particles have left their positions and the particle can moveuntil it is trapped in a new “cage” at a new (quasi-)stationary position r′i0. This is a verycomplicated process and an analytical treatment seems impossible. The key idea of QLCA


is that, in the liquid phase, close to the freezing point, one should expect a time scaleseparation: a particle in its cage has eneough time to perform many phonon oscillationcycles before it leaves its cage, i.e. τcage � 2π/minωk. The second assumption is that,while the particle motion in its cage is random, over a sufficiently long time the particlemotion will exhibit some regularities in a statistical sense and each particle will exhibit –on average – the same behavior (ergodicity). Thus we may expect that the key properties ofthis behavior are captured by suitable probabilistic quantities of the N -particle system suchas the pair distribution function or the static structure factor. In fact, these assumptionsare the basis for the QLCA

Let us now proceed with the mathematical formalism of the QLCA. We consider smallexcitations around the stationary state R0, i.e. particles have the coordinates ri(t) =ri0 + ξi(t). As discussed above, the stationary coordinates may change slowly in time. Thiswill be accounted for by considering them as random functions the statistical propertiesof which are determined by the thermodynamic state of the system. Let us expand thepotential energy to second order

U(R)− U(R0) =1

2!

∑i,j

∑αβ

∂2U(R0)

∂xiα∂xjβξiαξjβ +O(ξ3)

≈ 1

2!

∑i,j

∑αβ

Kij,αβ(R0)ξiαξjβ, (1.87)

where α and β denote the cartesian coordinates and the total hamiltonian is given by

H =N∑i=1

p2i

2m+ U. (1.88)

We now introduce a complete set of normal modes {ξk} and expand the coordinatesand momenta pi = mξi,

ξiα =1

(Nm)1/2

∑k

ξkα eikri0 (1.89)

piα =(mN

)1/2∑k

πkα eikri0 (1.90)

with the completeness and orthogonality propertiesWe now show that under certain conditions, ξkα and πkα, are canonically conjugate

variables and can be used to diagonalize the hamilton function. To this end we have tocompute the Poisson bracket and to show that and when

[ξpµ, π−qν ] = δµνδp−q. (1.91)

Here, the Poisson bracket is defined, as usually, by

[ξpµ, π−qν ] =∑i

∂ξpµ∂ξiα

∂π−qν

∂πiα. (1.92)


To compute the partial derivatives, we need the invers of the transforms (1.89) and (1.90):

ξkα =1

(Nm)1/2

∑i

Θki,αβ ξiβ , (1.93)

πkα =(mN

)1/2∑i

Θki,αβ πiβ , (1.94)

where we definedΘki,αβ = δαβ

{e−ikri0 + ∆ki

}, (1.95)

which is a random function due to the last term. Would the stationary positions be fixed∆ would be absent and we would recover just the exponential, i.e. the regular latticecontribution. The exponential is defined as the ensemble average, e−ikri0 = 〈e−ikri0〉, and∆ki describes the deviations from this exponential due to the randomness of the stationarypositions in the fluid phase. With these definitions and assumptions it is straightforwardto compute the ensemble average of the Poisson brackets:⟨

[ξpµ, π−qν ]⟩

=1

N

∑i

⟨Θpi,µαΘ−qi,να

⟩= δµν

∑i

{⟨e−i(p−q)ri0

⟩+⟨e−ipri0∆−qi

⟩=

⟨eiqri0∆pi

⟩+⟨

∆−qi∆pi

⟩}≈ δµνδp−q. (1.96)

In the last line we made use of the key assumption of QLCA that not only the expectati-on value of ∆ki vanishes (second and third terms) but also the correlation function of thedeltas (last term) may be neglected. This means it is assumed that the ∆ki for different wa-venumbers are uncorrelated. For the first (non-random) term the ensemble average yieldsjust the exponential and, due to the completeness of the basis we obtain the Kroneckerdelta. Thus, we may conclude that under these assumption on the randomness of the fluc-tuations in the fluid phase the normal coordinates ξpµ and π−qν are canonically conjugatein the sense of the ensemble average.

Consequently, they can be used to diagonalize the hamilton function which, nevert-heless, will retain a trace of the randomness of the stationary positions. To this end wecompute the ensemble average of kinetic and potential energy in the normal mode repre-sentation: ⟨∑

i

p2i

2m

⟩(1.97)


1.5 Dielectric function of a quantum plasma

We now extend the analysis of dielectric properties of a plasma to quantum systems. Thiswill be of relevance for plasmas at low temperature or/and high density. We will proceedin close analogy to the classical derivation of the previous sections. However, instead of theVlasov equation we will need to use, as a starting point, a quantum kinetic equation16. Suchequations were derived and discussed in Sec. ??. For the analysis of dielectric propertiesit will be convenient to use the momentum representation where the kinetic equation hasthe form (??).

As in the classical case we will start with the collisionless case und generalize the Vlasovresult to quantum systems. Let us rewrite Eq. (??) neglecting the collision integral andgeneralize it to the multi-component case

i~∂fk,k′,α∂t

− (εk,α − εk′,α)fk,k′,α −VNa

〈k|[U eff1,α, F1,α]|k′〉 = 0, (1.98)

with the operator of the effective potential and the mean field potential (??)

U eff1,α = U1,α +W1,α,

W1,α =∑β

nβTr2V1α,2βF2β,

and the normalization conditions for the density operator and the distribution function

V〈k|F1,α|k′〉 = Nαfk,k′,α,

Nα =Nα

VTr1F1,α(t) = 2

∑k

fkk,α(t),

where the kinetic energy of particle species “α” in momentum state |k〉 is given by εk,α =~2

2mαk2. Eq. (1.98) is the nonlinear quantum Vlasov equation or the time-dependent Hartree

equation. Here we will concentrate on an unmagnetized plasma, so the external potential Uis due to an electrostatic potential and U eff

1,α = eα(φext + φind), where the induced potentialis the solution of Poisson’s equation, ∆φind = −4πρind with the plasma particles acting asthe source. As in the classical case φind functionally depends on the particle density andthus, on the distribution function, giving rise to a nonlinear kinetic equation.

We again start with a plasma state without external fields, U ≡ 0. Thus the plasmawill be homogeneous and all contributions to the induced potential cancel, due to chargeneutrality, φind ≡ 0. Then, we expect that the plasma will be in a (local) thermodynamic

16Details of the derivation are given in [LBKD10]

1.5. DIELECTRIC FUNCTION OF A QUANTUM PLASMA 29

equilibrium state which, for fermions, e.g. electrons, is given by the Fermi function17

fk,k′,α = f(0)k,k′,α = f

(0)k,αδk,k′ = fEQk,α δk,k′

fEQk,α =[eβ(εkα−µα) + 1

]−1, (1.99)

where β = 1/kBT and µα(nα, T ) is the chemical potential, and the momentum deltafunction arises due to spatial homogeneity.

We now assume that a weak electrostatic potential φext is turned on perturbing theequilibrium state. Weakness of the potential allows us to linearize the kinetic equation andsolve it with the linear response ansatz

U effα (r, t) = U eff(1)

α (r, t), limt→−∞

U effα (r, t) = 0,

fk,k′,α(t) = f(0)k,k′,α + f

(1)k,k′,α(t), (1.100)

limt→−∞

fk,k′,α(t) = f(0)k,k′,α.

Using this ansatz we can now evaluate the matrix elements of the commutator in Eq. (1.98).Since the potential is of first order, for the density operator the zeroth order gives thedominant contribution which is diagonal,

〈k|[U eff1,α, F1,α]|k′〉 ≈ 〈k|[U eff(1)

1,α , F(0)1,α]|k′〉

= Ueff(1)k,k′,α

Na

V

(f

(0)k′,α − f

(0)k,α.)

Inserting this result into the kinetic equation (1.98) and taking into account that f (0)k,k′α is

time-independent and diagonal, we obtain a closed equation for f (1)k,k′,α(t)

i~∂f

(1)k,k′,α

∂t− (εk,α − εk′,α)f

(1)k,k′,α − U

eff(1)k,k′,α

(f

(0)k′,α − f

(0)k,α

)= 0, (1.101)

which is already a linear equation. Expanding the field into a Fourier integral of mono-chromatic oscillations18

Uk,k′,α(t) =

∫dt Uk,k′,α(ω) e−i(ω+iδ)t, (1.102)

the same expansion will also apply to f (1)k,k′,α and U eff(1)

k,k′,α.

17The only remaining contribution in Eq. (1.98) is the time derivative, i.e. ∂fk,k′,α∂t = 0. Therefore,

as in the classical case, any stationary distribution function can be used. The choice of the equilibriumdistribution is based on experience. Only in the case of collisions, relaxation processes (scattering) willlead to a unique selfconsistent stationary solution.

18For the discussion of the constant δ, see Sec. 1.1.


Fourier transforming Eq. (1.101) and cancelling the common exponent yields the resultfor the perturbation of the distribution,

f(1)k,k′,α(ω) =

f(0)k′,α − f

(0)k,α

~(ω + iδ)− (εk,α − εk′,α)U

eff(1)k,k′,α(ω). (1.103)

This is the general result for the linear perturbation to the distribtution function, and thetime-dependent expression follows by a back transform. In deriving (1.103) we have madeno assumptions on the space dependence of the external perturbation U .

In the following we consider an exciting field which is purely periodic in space withoutany macroscopic spatial modulation. Then the field can be expanded in a Fourier seriesUα(r) =

∑q Uqαe

iqr. We now have to establish the relation of the Fourier components Ukαto the momentum matrix elements Uk,k′,α. To this end we introduce center of mass andrelative momenta by (all quantities are vectors)

Q =k + k′

2, q = k − k′, or, vice versa, (1.104)

k = Q+q

2, k′ = Q− q

2.

While the center of mass variable Q is related to spatial inhomegeneities, the relativemomentum q is directly related to small scale spatially periodic modulations. Thus, in thepresent case, there will be no dependence on Q, i.e.

Uk,k′,α ≡ UQ+ q2,Q− q

2,α −→ Uq

and similarly for U eff(1)k,k′,α(ω). In contrast, in the unperturbed quantities εk and f

(0)k,α the

momentum arguments k and k′ remain. Therefore, to shorten the notation, in the followingwe will use the notation for the momentum arguments Q± q/2→ ±. Via εk and f (0)

k,α, thedependence on Q remains also in the perturbed distribution function. Thus, we can rewritethe result (1.103) for a single Fourier component q of a periodic monochromatic excitation

f(1)+,−,α(ω) =

f(0)−,α − f

(0)+,α

~(ω + iδ)− (ε+,α − ε−,α)U eff(1)q,α (ω) (1.105)

This is still not an explicit result for f (1) because the function also appears in the effec-tive potential. To make further progress we consider the density disturbance and computeits Fourier components

n(1)q,α(ω) = 2

∫d3Q

(2π)3f

(1)

Q+ q2,Q− q

2,α

(ω), (1.106)

and the prefactor 2 accounts for the spin summation. Below, we will also need the currentdensity which is calculated in similar way

j(1)q,α(ω) = 2

∫d3Q

(2π)3

~Qmα

f(1)

Q+ q2,Q− q

2,α

(ω). (1.107)


Particle and current density are connected via the continuity equation, ∂n(r,t)∂t

+div j(r, t) =0. Since the unperturbed expressions are time and space independent we can replace thetotal density and current density by the perturbed functions. The Fourier transform of thisexpression then reads [cf. Eq. (1.11)]

ω n(1)q,α(ω) = q · j(1)

q,α(ω). (1.108)

We can now explicitly compute the density perturbation by inserting the distributionfunction (1.105) into Eq. (1.106) and obtain

n(1)q,α(ω) = ΠMF

0α (q, ω) U eff(1)q,α (ω), (1.109)

ΠMFnα (q, ω) = 2

∫d3Q

(2π)3

(~Qmα

)n f(0)−,α − f

(0)+,α

~ω − [ε+,α − ε−,α], (1.110)

where we defined ω = ω + iδ.The simplest way to obtain the longitudinal quantum dielectric function is to use the

continuity equation and relation (1.12) which is valid for classical and quantum plasmas.Computing the total electric charge density perturbation

ρ(1)q (ω) =

∑α

qαn(1)q,α(ω), (1.111)

and using the relation of the effective potential to the total electrostatic potential

U eff(1)q,α (ω) = qαφq(ω), (1.112)

we only need to insert (1.111) with the result (1.109) into (1.12) and obtain

εl,MF(k, ω) = 1−∑α

Vαα(k)ΠMF0α (k, ω). (1.113)

This is the longitudinal dielectric function for a collisionless quantum plasma. In usingthe superscript “MF” for the dielectric function and the polarization (1.110) we indicatedthat this is the result in mean field approximation which is also frequently called Lindhardpolarization [Lin54]. Note, however, that this result was obtained by many authors inde-pendently. The first, apparantly were Klimontovich and Silin [KS52a, KS52b]. Using thisresult for χ0 in Eq. (1.80) yields the RPA approximation for ε−1 19.

Finally, we separate the real and imaginary parts of the mean field dielectric function(1.113). Employing the Plemlj formula (Dirac identity) for δ → +0 (P denotes the principalvalue) we find

Re εl,MF(q, ω) = 1− 2∑α

Vαα(q) P∫

d3Q

(2π)3

f(0)−,α − f

(0)+,α

~ω − [ε+,α − ε−,α], (1.114)

Im εl,MF(q, ω) = 2π∑α

Vαα(q)

∫d3Q

(2π)3δ[~ω − (ε+,α − ε−,α)]

[f

(0)−,α − f

(0)+,α

]. (1.115)

19The term random phase approximation was introduced by Bohm and Pines [BP53], for an overviewsee [EC59, Bon98]


The result for the imaginary part is particularly transparent: Only for those combinationsof ω and q energy exchange between field and plasma is possible for which the argumentof the delta function can be equal to zero. This condition can be understood as an inelasticscattering process where a particle loses (gains) energy ~ω and momentum q to (from) acollective excitation, i.e. a longitudianal plasmon:

~ω = ε+,α − ε−,αq = Q +

q

2−(Q− q

2

), (1.116)

where the momentum conservation is fulfilled by construction (we have assumed a spatiallyhomogeneous system, so the center of mass momentum drops out). The sign of Im εl,RPA isthen simply determined by the difference of the distribution functions, i.e. depends on therelative occupation of the states |Q− q

2〉 and |Q + q

2〉. If the occupation of low momentum

states (“-”) is higher than that of high momentum state then the imaginary part of thedielectric function is positive which corresponds to dissipation, i.e. damping of plasmons.This is in particular the case in thermodynamic equilibrium where f (0) is a Maxwell, Boseor Fermi function which are all monotonically decaying with momentum.

The opposite case that f (0)−,α < f

(0)+,α in the vicinity of some momentum Q is possible

only in nonequilibrium and may lead to negative “damping”, i.e. to a plasma instability– amplification of plasma oscillations, for discussion of plasma instabilities in quantumplasmas see e.g. [BBK93, Bon98]. Note that this is only a necessary condition for aninstability but not a sufficient one. For example, in isotropic 3D plasmas even with thiscondition instabilities are impossible [Bon94], and only undamping of plasmons is possible[SBBK94], whereas in 1D or 2D plasmas plasma instabilities do exist [Bon98]. For classicalplasmas the sufficient condition for an instability have been worked out by Penrose andothers [Pen60]. For low-dimensional quantum plasmas some results have been obtaine inRef. [BBS+94, BBS+93].

1.5.1 Classical limit

It is natural to verify that the quantum result for the dielectric function has the cor-rect classical limit. For this it is sufficient to consider the classical limit of the Lindhardpolarization (1.110). We first compute the energy difference

ε+,α − ε−,α =~2

2mα

[(Q +

q

2

)2

−(Q− q

2

)2]

=~2Q · qmα

= vα · ~q,


and, further, the classical limit of the difference of the unperturbed distribution functions.This limit is performed by assuming, in all momentum arguments, q << Q,

f(0)−,α − f

(0)+,α = f

(0)Q,α −

1

2

∂f(0)Q,α

∂Q· q−

(f

(0)Q,α +

1

2

∂f(0)Q,α

∂Q· q

)+O

(q2

Q2

)

= −∂f

(0)Q,α

∂Q· q +O

(q2

Q2

),

where f (0)Q,α denotes that the limit q → 0 has been taken. Inserting these results in the mean

field polarization and performing the classical limit, we obtain

limqQ→0

2

∫d3Q

(2π)3

f(0)−,α − f

(0)+,α

~ω − [ε+,α − ε−,α]

= 2

∫d3Q

(2π)3

∂f(0)Q,α

∂Q· q

vα · ~q− ~ω=

2

mα

∫d3Q

(2π)3

q · ∂f(0)Q,α

∂v

q · vα − ω= ΠV

α (q, ω),

i.e. we recover the Vlasov polarization, taking into account the different normalizationswhich have been used in the classical (c) and quantum (q) cases

Nα = 2

∫d3Q

(2π)3f

(0),qQ,α =

∫d3vf (0),c

α (v).

In the classical limit we immediately recover the known expression for the imaginarypart of the Vlasov dielectric function. In particular it is clear that its sign is now determinedby the sign of the derivative of f (0). A negative sign (monotonically decaying distribution)corresponds to damping and vice versa.

Functional derivatives. Before proceeding we mention another approach to the diel-ectric function which uses the linear response definition via partial derivatives [Bon98].Here one computes the Fourier component of the effective potential for the case of Cou-lomb interacting particles, with the result (for details see the problem 1)

U eff(1)q,α (ω) = Uq,α(ω) +

∑β

Vαβ(q) n(1)q,β(ω), (1.117)

where Vαβ(k) = 4πqαqβ/k2 is the Fourier transform of the Coulomb potential. Eliminating

the density, using the result (1.109), we can express the external potential via the effectivepotential,

Uq,α(ω) =∑β

Ueff(1)q,β (ω)

{δα,β − Vαβ(q) Π0β(q, ω)

},


and, in the same way, the external electrostatic potential φextq,α = Uq,α/qα via the totalpotential,

φextq (ω) =∑β

qβqαφq(ω)

{δα,β − Vαβ(q) Π0β(q, ω)

},

= φq(ω)

{1−

∑β

Vββ(q) Π0β(q, ω)

}. (1.118)

Using the linear response definition of the dielectric function

ε(q, ω) =δφextq (ω)

δφq(ω), (1.119)

we immediately recover the mean field result (1.113).Lindhard dielectric function for an isotropic distribtution function. If the

distribution function depends only on the modulus of the momentum p, the angle integralsin the real and imaginary parts of the dielectric function can be performed and we obtain,for the real part (see problem xxx),

Re εl,MF(q, ω) = 1− 4π∑α

Vαα(q) P∫ ∞

0

dQ

(2π)3Q2f (0)

α (Q)

×∫ 1

−1

dz

{1

~ω − (εQ+q − εQ)− 1

~ω − (εQ − εQ−q)

}(1.120)

= 1 +4πmα

~2q

∑α

Vαα(q)

∫ ∞0

dQ

(2π)3Qf (0)

α (Q) ln|κ− q

2−Q||κ+ q

2+Q|

|κ− q2

+Q||κ+ q2−Q|

,

where we defined the wavenumber κ = ωmα/~q. For the imaginary part we find analogously,taking into account that the cosine of the angle between q and Q is restricted to [−1, 1](see problem xxxx)

Im εl,MF(q, ω) =1

~2q

∑α

mαVαα(q)

∞∫|κ+q/2|

dQ

2πQf (0)

α (Q)−∞∫

|κ−q/2|

dQ

2πQf (0)

α (Q)

(1.121)

1.5.2 Quantum Dielectric function of fermions in thermodynamicequilibrium

The expressions (1.120) and (1.121) are readily evaluated for a Fermi function f(0)α =

[eβ(p2/2mα−µα) + 1]−1. For the imaginary part the momentum integration can be carried outwith the result [Glu71]

Im εl,MF(q, ω) =kBT

2π~q∑α

mαVαα(q) ln1 + eβ(µα−E−α )

1 + eβ(µα−E+α ), (1.122)

1.6. QUANTUM DIELECTRIC FUNCTION WITH COLLISIONS 35

Abbildung 1.3: Wave number – frequency plane for an electron plasma at T = 0. Regionsof Imε = 0 correpond to damping of plasmon due to excitation of particle hole pairs (paircontinuum). From ref. [KSK05].

with the defintion E±α = ~2

2mα(± q

2− κ).

Ground state. A particularly simple result follow for zero temperature. Using againthe general formulas (1.120) and (1.121), we obtain...

The region of non-zero imaginary part of the dielectric function is shown by the shadearea in Fig. 1.5.2.

1.6 Quantum dielectric function including collisions

After having obtained the simplest quantum result for the dielectric function – the randomphase approximation – we now proceed to improve this result by including correlationeffects. This requires to go beyond the mean field kinetic equation and to include a collisionintegral. As we did in the classical case, we will use a relaxation time approximation, cf.Sec. 1.2. The quantum kinetic equation in momentum representation, with collision integralreads

i~∂fk,k′,α∂t

− (εk,α − εk′,α)fk,k′,α −VNα

〈k|[U eff1,α, F1,α]|k′〉 = Ik,k′,α, (1.123)

Prior to external perturbation, we again assume a homogeneous state with the distribu-tion given by a diagonal matrix fk,k′,α = f

(0)k,k′,α = f

(0)k,αδk,k′ , and the same applies to the

collision integral, Ik,k′,α = I(0)k,k′,α = I

(0)k,αδk,k′ . Then the kinetic equation (1.123) simplifies to


a diagonal equation

i~∂f

(0)k,α

∂t= I

(0)k,α ≡ Ik,α[f

(0)k,α], (1.124)

where in the collision integral the unperturbed distribution function has to be used. Therelaxation time approximation for the collision integral is constructed by the ansatz (wedrop all arguments)

IRTA[f ] ≡ −1

τ

(f − fEQ

), f(0) = f0. (1.125)

The solution of this equation together with the initial condition at t = 0 is f(t) = f0e−t/τ +

fEQ[1− e−t/τ ], showing the decay of the initial state and the approach to the asymptoticstate. Here, τ is the relaxation time which has to be computed from a separate kinetic theoryor taken from experiment. We use a simple static approximation where τ is frequency-independent. Despite its simplicity, this approximation allows one to achieve a selfconsistentrelaxation of the distribution function to the equilibrium state (in contrast to the meanfield approximation). While we expect that fEQ will be a Fermi function, we will not needthe explicit form of the equilibrium distribution below.

We now again consider a weak perturbation by a longitudinal field Ua = qaφext,

U effα (r, t) = U eff(1)

α (r, t), limt→−∞

U effα (r, t) = 0,

fk,k′,α(t) = f(0)k,k′,α + f

(1)k,k′,α(t), (1.126)

limt→−∞

fk,k′,α(t) = f(0)k,k′,α.

The equation for the perturbation of the distribution reads, in first order,

i~∂f

(1)k,k′,α

∂t− (εk,α − εk′,α)f

(1)k,k′,α − U

eff(1)k,k′,α ·

(f

(0)k′,α − f

(0)k,α

)= I

(1)k,k′,α, (1.127)

limt→−∞

fk,k′,α(t) = f(0)k,αδk,k′ .

Here, U eff(1) is again obtained by replacing fk,k′ by f(1)k,k′ , whereas I

(1) is obtained by keepingin all appearances of the electron distribution functions only terms of first order in f (1)

[KB00]. In case of the relaxation time apprximation the collision integral is linear in f , andwe just have to use f (1)

k,k′ . In order to find the explicit expression for I(1) we consider thelong time limit of the system. The complete asysmptotic solution of the original kineticequation will be the sum of the zero and first order terms, i.e. fEQ and the asymptoticsolution for f (1)

k,k′ which we will denote f (1)∞k,k′ . To find this solution assume a purely periodic

space dependence of U . Using again center of mass and relative momenta [cf. Sec. 1.5],Eq. (1.127) becomes

i~∂f

(1)∞k,k′,α

∂t− (ε+,α − ε−,α)f

(1)∞k,k′,α − U

eff(1)∞q,α ·

(f

(0)−,α − f

(0)+,α

)= I

(1)∞k,k′,α,


Since we are looking for a stationary solution, the time-dependence should vanish. Also,we expect that collisions have relaxed to zero, i.e. I(1)∞

k,k′,α = I(1)k,k′,α[f (1)∞] = 0. As a result we

obtain

limt→∞

f(1)k,k′,α = f

(1)∞k,k′,α =

f(0)+,α − f

(0)−,α

ε+,α − ε−,αU eff(1)∞q,α . (1.128)

Note that also the effective potential carries the superscrtipt “∞” since it also depends onthis solution.

Below we will again work in Fourier space and perform a Fourier transform with respectto time according to Eq. (1.102)

f(1)∞k,k′,α =

f(0)+,α − f

(0)−,α

ε+,α − ε−,αU eff(1)∞q,α , ω = 0, (1.129)

where, as a result of the long-time limit, this form is restricted to zero frequency.Now we can construct the first order correction to the collision term in Eq. (1.127).

Since f (0) relaxes towards fEQ, in Eq. (1.125) only the difference of f (1) and f (1)∞ remains.Multiplying by i~ we obtain

I(1)k,k′,α(ω) = −i~

τ

{f

(1)k,k′,α(ω)−

f(0)+,α − f

(0)−,α

ε+,α − ε−,αU eff(1)∞q,α

}, (1.130)

which can be inserted in Eq. (1.127) after Fourier transform to frequency space20 yieldingthe solution

f(1)k,k′,α(ω) =

{U eff(1)q,α − i~ν U

eff(1)∞q,α

ε+,α − ε−,α

}f

(0)−,α − f

(0)+,α

~ω − [ε+,α − ε−,α](1.131)

where we defined ω = ω + iν and used δ → 0 due to the existence of a finite collisionaldamping ν = τ−1. This result is a straightforward extension of the collisionless randomphase approximation, cf. Eq. (1.105). Scattering effects (terms proportional to ν) are con-tained in two places: first, the frequency in the denominator is replaced by a complexfrequency and, second, there appears an additional contribution proportional to U eff(1),∞

in the numerator which renormalizes the Fourier component of the effective potential.Eq. (1.131) is not an explicit solution for f (1) since this function also appears in the

effective potential. To solve this problem and compute the dielectric function, we proceed asin the collisionless case, in Sec. 1.5. We first compute the Fourier component of the densitydisturbance according to Eq. (1.106) and also of the current density, using Eq. (1.107),

nq,α(ω) = ΠRPA0α (q, ω)U eff(1)

q,α (ω)− i~ν U eff(1)∞q,α Πν0α(q, ω), (1.132)

jq,α(ω) = ΠRPA1α (q, ω)U eff(1)

q,α (ω)− i~ν U eff(1)∞q,α Πν1α(q, ω), (1.133)

20We again assume a spatially periodic excitation.


where we used the definition (1.113) of the RPA polarization and introduced a modifiedpolarization function which arises from the collisions

Πνnα(q, ω) = 2

∫d3Q

(2π)3

(~Qmα

)n1

ε+,α − ε−,α×

f(0)−,α − f

(0)+,α

~ω − [ε+,α − ε−,α], (1.134)

Eq. (1.132) indicates that collisions change the particle density (second term) comparedto the RPA result. As a consequence of the local density conservation law also the currentdensity has to change, cf. Eq. (1.133).

Eqs. (1.132) and (1.133) contain the still unknown function U eff(1)∞ which we determineusing the continuity equation (1.108). We now transform ~q times the integral Π1, byadding and subtracting under the integral ~ω. Taking into account that ε+,α − ε−,α =~2Q · q/mα, we obtain the identity

~q · ΠRPA1α (q, ω) = 2

∫d3Q

(2π)3

(f

(0)−,α − f

(0)+,α

)+ (~ω + i~ν)ΠRPA

0α (q, ω). (1.135)

Assuming that the field-free distribution depends only on the modulus of the momentum,i.e. f (0)

−k,α = f(0)k,α the integrals over f− and f+ cancel. The same transformation is possible

for the integral Πν1α with the result

~q · Πν1α(q, ω) = ΠRPA0α (q, 0) + (~ω + i~ν)Πν0α(q, ω). (1.136)

Collecting the results (1.135) and (1.136) together we may rewrite the expression for thecurrent density, Eq. (1.133),

~q · jq,α(ω) = ~(ω + iν)ΠRPA0α (q, ω)U eff(1)

q,α

−i~ν[ΠRPA

0α (q, 0) + (~ω + i~ν)Πν0α(q, ω)]U eff(1)∞q,α =

= ~ω · nq,α(ω) +

+ i~ν{

ΠRPA0α (q, ω)U eff(1)

q,α − U eff(1)∞q,α

[ΠRPA

0α (q, 0) + i~νΠν0α(q, ω)]}.

Evidently, the continuity equation (1.108) is fulfilled if the terms on the last line (in thecurley brackets) vanish which yields the required condition on U eff(1)∞

U eff(1)∞q,α (ω) =

ΠRPA0α (q, ω)U

eff(1)q,α (ω)

ΠRPA0α (q, 0) + i~νΠν0α(q, ω)

=nq,α(ω)

ΠRPA0α (q, 0)

. (1.137)

With this result we can now eliminate U eff(1)∞ from all expressions. Inserting it in-to (1.132), we can solve for the density perturbation which – as in the collisionless case,cf. Eq. (1.109) – is proportional to the effective potential, however, with a modified coeffi-cient

nq,α(ω) = ΠMα (q, ω)U eff(1)

q,α (ω), (1.138)


Using the result (1.138) we immediately obtain the dielectric function. Computing the totalcharge density to ρq(ω) = Φq(ω)

∑α q

2αΠM

α (q, ω) and inserting the result into Eq. (1.12) weobtain

εl,M(k, ω) = 1−∑α

Vαα(k)ΠMα (k, ω) (1.139)

This is the longitudinal quantum dielectric function including collisions in relaxation timeapproximation. This result was first obtained by Mermin [Mer70], therefore we use thesuperscript “M ”. The formal structure of this expression is the same as in the collisionlessand classical cases, it is a general property of linear response theory. The different physicalapproximations are solely contained in the longitudinal polarization function, the Merminpolarization (see problem 3),

ΠMα (q, ω) ≡ ΠRPA

0α (q, ω)

1 + i~νΠν0α(q, ω)(1.140)

where we introduced the definition

Πν0α(q, ω) ≡ Πν0α(q, ω)

ΠRPA0α (q, 0)

=1

~ω

[ΠRPA

0α (q, ω)

ΠRPA0α (q, 0)

− 1

]. (1.141)

In the last equality we have eliminated the function Πν0α and expressed it in terms of theRPA polarization where use has been made of the identity ~ωΠν0α(q, ω) = ΠRPA

0α (q, ω) −ΠRPA

0α (q, 0), see problem 2. Obviously, the collisionless limit of the Mermin dielectric func-tion, i.e. ν → 0 and ω → ω + iδ, is given by the random phase approximation.

Let us summarize our results on the dielectric function for quantum plasmas.

1. The present results have been obtained in linear response, with the general key rela-tions coupling the density response, on one hand, and total potential, on the other,with the external potential,

nq,α(ω) = Πα(q, ω)U eff(1)q,α (ω),

Φq(ω) =[εl(q, ω)

]−1Φextq (ω),

where U effa = qaφ and Ua = qaφ

ext. This defines Π and ε via functional derivatives.

2. In all approximations, the longitudinal dielectric function is coupled with the longi-tudinal polarization and the total charge density via two fundamental relations

εl(k, ω) = 1−∑α

Vαα(k)Πα(k, ω).

= 1− 4π

k2

ρ(k, ω)

Φ(k, ω).


3. Inclusion of scattering effects into the dielectric function cannot be done by thesimple replacement ω → ω + iν since this violates particle number conservation(the continuity equation), cf. the second term in Eq. (1.132). Local particle numberconservation is directly related to the preservation of the f-sum (frequency sum) ruleby the dielectric function.

4. The Mermin dielectric function (polarization) is the direct quantum generalization ofthe classical BGK result, cf. Sec. 1.2. The classical limit of the Mermin polarizationis exactly the BGK polarization, Eq. (1.42), see problem 4.

Further improved dielectric functions

Further improvements of the Mermin result have been considered by various groups. Röpkeet al. have derived a Mermin-type expression which, besides particle conservation containsenergy conservation [SRW02]. However, they found that the effect was small. Anothermodification by this group was to include a frequency dependent collision frequency intothe relaxation time collision integral [MSW+03]. Finally, we mention that a selfconsistentnonequilibrium calculation within Nonequilibrium Green’s functions, e.g. [Bon98, Bon09]which fully included sum rule preservation has been recently performed [KB00].

Problems to Chapter 1

Problem 1: prove Eq. (1.117)Problem 2: prove Eq. (1.141).Problem 3: Derive the Mermin polarization, Eq. (1.140).Problem 4: Show that the classical limit of the Mermin polarization, Eq. (1.140), is theBGK polarization (1.42).

Solutions to Problemss

Solution of problem 1. To prove Eq. (1.117) we start from the operator definition (??),

U eff1,α − U1,α =

∑β

nβTr2V1α,2βF2β,

Solution of problem 4. To prove the statement, use the form (1.47) of the BGK polari-zation. It has exactly the functional form of the Mermin polarization, Eq. (1.140), with thereplacement of the Vlasov polarization by the RPA result. Since the former is the classicallimit of the latter, the statement is verified.

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