3.3.4 Force-force correlation function and static (dc) conductivity

As an example for the generalized linear response theory, we calculate the conductivity of a plasma of charged particles (electrons and ions) that is exposed to a static homogeneous electric field in x-direction: ω ¼ 0, E ¼ Eex,

$$\mathbf{H}\_{\rm F} = -e\mathbf{E}\mathbf{X}, \qquad \mathbf{X} = \sum\_{i}^{N\_{\epsilon}} \mathbf{x}\_{i}. \tag{170}$$

Instead of hj, we have only one constant external field E. For the treatment of arbitrary ω to obtain the dynamical (optical) conductivity see [11, 13, 16, 17]. The conjugated variable A from Eq. (149) that couples the system to the external field is <sup>A</sup> <sup>¼</sup> <sup>e</sup>X. The time derivative follows as <sup>A</sup>\_ <sup>¼</sup> ð Þ <sup>e</sup>=<sup>m</sup> P, with P <sup>¼</sup> <sup>∑</sup>Ne <sup>i</sup> px,i denoting the total momentum in x direction.

For simplicity, the ions are considered here as fixed in space because of the large mass ratio (adiabatic approximation). Then, the transport of charge is owing to the

motion of the electrons. In general, the ions can also be treated as moving charged particles that contribute to the current.

A stationary state will be established in the plasma where the electrons are accelerated by the external field, but loose energy (and momentum) due to collisions with the ions. This nonequilibrium state is characterized by an electrical current that is absent in thermal equilibrium. We can take the electric current density je<sup>l</sup> <sup>¼</sup> ð Þ <sup>e</sup>=m<sup>Ω</sup> <sup>P</sup> <sup>¼</sup> ð Þ <sup>e</sup>=<sup>Ω</sup> X as a relevant observable that characterizes the \_ nonequilibrium state. Instead, we take the total momentum B <sup>¼</sup> <sup>P</sup> <sup>¼</sup> <sup>m</sup>X. The \_ generalized linear response equations (165) and (167) read

$$E\left[\left<\dot{\mathbf{P}}|\mathbf{P}\right> + \left<\dot{\mathbf{P}};\dot{\mathbf{P}}\right>\_{i\epsilon}\right] = \frac{e}{m}E\left\{\left<\mathbf{P}|\mathbf{P}\right> + \left<\mathbf{P};\dot{\mathbf{P}}\right>\_{i\epsilon}\right\},\tag{171}$$

P ¼ ∑ p

DOI: http://dx.doi.org/10.5772/intechopen.84707

i<sup>ϵ</sup> ¼

ð 0

�∞

ð 0 dte<sup>ϵ</sup><sup>t</sup> ð1

δ Ep � E<sup>p</sup>þ<sup>q</sup>

integrand are not depending on the direction in space.

0 dλe i

� �<sup>f</sup> <sup>p</sup> <sup>1</sup> � <sup>f</sup> <sup>p</sup>

�∞

<sup>y</sup> <sup>þ</sup> <sup>q</sup><sup>2</sup> z

<sup>d</sup>Ep ¼ f <sup>p</sup> 1 � f <sup>p</sup>

so that the Ziman-Faber formula is obtained,

<sup>R</sup> <sup>¼</sup> <sup>m</sup><sup>2</sup>Ω<sup>3</sup> 12π<sup>3</sup>ℏ<sup>3</sup>

Debye potential (146). The evaluation yields

<sup>σ</sup>dc <sup>¼</sup> <sup>3</sup> 4 ffiffiffiffiffi <sup>2</sup><sup>π</sup> <sup>p</sup>

e<sup>2</sup>N<sup>2</sup>

∞ð

0

P\_; P\_ � �

Nonequilibrium Statistical Operator

<sup>i</sup><sup>ϵ</sup> ¼ � ∑ <sup>p</sup>, <sup>p</sup><sup>0</sup> , <sup>q</sup>, <sup>q</sup><sup>0</sup>

> ¼ ∑ <sup>p</sup>, <sup>q</sup> Vq � � � �2

<sup>x</sup> <sup>¼</sup> <sup>q</sup><sup>2</sup>

β <sup>d</sup>f Eð Þ<sup>p</sup>

<sup>x</sup> <sup>þ</sup> <sup>q</sup><sup>2</sup>

P\_; P\_ � �

replace q<sup>2</sup>

P; P\_ � � iϵ

interaction.

relations � <sup>1</sup>

with ℏ<sup>2</sup>

39

p2

<sup>ℏ</sup>px <sup>a</sup>†

dte<sup>ϵ</sup><sup>t</sup> ð 1

0 dλ i

<sup>p</sup>ap, ½ �¼� HS; P ∑

<sup>ℏ</sup> ½ � HS; <sup>P</sup>ð Þ <sup>t</sup> � <sup>i</sup>λβ<sup>ℏ</sup> <sup>i</sup>

in Born approximation with respect to Vq. In lowest order, the force-force correlation function is of second order so that in the time evolution exp i ½ � ð Þ =ℏ HSð Þ t � iλβℏ the contribution Hint of interaction to HS, Eq. (176), can be dropped as well as in the statistical operator. The averages are performed with the noninteracting ρ0. The product of the two commutators is evaluated using Wick's theorem. One obtains

<sup>ℏ</sup>ð Þ <sup>E</sup>p�Epþ<sup>q</sup> ð Þ <sup>t</sup>�<sup>i</sup> <sup>ℏ</sup>βλ VqVq0qxq<sup>0</sup>

x:

� �πℏq<sup>2</sup>

Because the x direction can be arbitrarily chosen in an isotropic system, we

Evaluating Eq. (175) in Born approximation, the correlation function

strength. For the resistance, this term contributes only to higher orders of the

The force-force correlation function (179) is further evaluated using the

integration has to be performed in the limits 0 ≤q≤2p. Finally the resistance can be calculated by inserting the previous expressions Eqs. (172) and (179) into Eq. (175)

<sup>d</sup>E pðÞ � <sup>d</sup>f Eð Þ

The expression for the resistance depends on the special form of the potential Vq. For a pure Coulomb potential <sup>e</sup><sup>2</sup><sup>=</sup> <sup>Ω</sup>ϵ0q<sup>2</sup> ð Þ, the integral diverges logarithmically as typical for Coulomb integrals. The divergency at very small values of q is removed if screening due to the plasma is taken into account. Within a manyparticle approach, in static approximation the Coulomb potential is replaced by the

> ð Þ 4πϵ<sup>0</sup> 2

m<sup>1</sup>=<sup>2</sup>e<sup>2</sup>

where the Coulomb logarithm is approximated by the value of the average p,

therm=2m ¼ 3kBT=2. In the low-density limit, the asymptotic behavior of

ð Þ k<sup>B</sup> 3=2

dE � � ð

� � and <sup>δ</sup> Ep � <sup>E</sup>pþ<sup>q</sup>

� �=<sup>3</sup> <sup>¼</sup> <sup>q</sup><sup>2</sup>=3 if the remaining contributions to the

ð Þ β=mN can be neglected in relation to 1 because it contains the interaction

� � <sup>¼</sup> <sup>m</sup>

ℏ2qp

2p

0

1

<sup>Λ</sup> <sup>p</sup>therm � � (181)

<sup>δ</sup> cos <sup>θ</sup> � <sup>q</sup>

dq q<sup>3</sup> Vq � � � � 2

2p � �. The <sup>q</sup>

: (180)

� �

We calculate the force-force correlation function (only x component)

<sup>p</sup>, <sup>q</sup>

Vq ℏqx a †

<sup>ℏ</sup> ½ � HS; <sup>P</sup>

<sup>x</sup> a † <sup>p</sup>þ<sup>q</sup>apa † p0 þq0a<sup>p</sup><sup>0</sup> D E

eq

<sup>p</sup>þ<sup>q</sup>ap: (177)

eq

(178)

(179)

The term <sup>P</sup>\_j<sup>P</sup> � � <sup>¼</sup> h i ½ � <sup>P</sup>; <sup>P</sup> <sup>e</sup><sup>q</sup> vanishes as can be shown with Kubo's identity, see Eq. (158). With the Kubo identity, we also evaluate the Kubo scalar product

$$\mathbf{P}(\mathbf{P}|\mathbf{P}) = m \int\_0^1 \mathrm{d}\lambda \left< \dot{\mathbf{X}} (-\mathbf{i}\,\hbar \boldsymbol{\beta}\lambda) \mathbf{P} \right>\_{\mathrm{eq}} = -\frac{\mathrm{i}m}{\hbar \beta} \mathrm{Tr} \left\{ \rho\_{\mathrm{eq}}[\mathbf{X}, \mathbf{P}] \right\} = \frac{m\mathcal{N}}{\beta}.\tag{172}$$

The solution for response parameter F is

$$F = \frac{e}{m} E \frac{m \mathbf{N} / \beta + \langle \mathbf{P}; \dot{\mathbf{P}} \rangle\_{ie}}{\langle \dot{\mathbf{P}}; \dot{\mathbf{P}} \rangle\_{ie}}.\tag{173}$$

With Eq. (169) we have

$$
\langle \mathbf{j}\_{\rm el} \rangle = \frac{e}{m\Omega} \langle \mathbf{P} \rangle\_{\rm rel} = \frac{e\beta}{m\Omega} F(\mathbf{P}|\mathbf{P}) = \sigma\_{\rm dc} E. \tag{174}
$$

The resistance R in the static limit follows as

$$R = \frac{\mathbf{1}}{\sigma\_{\rm dc}} = \frac{\Omega \beta}{e^2 N^2} \frac{\langle \dot{\mathbf{P}}; \dot{\mathbf{P}} \rangle\_{ie}}{\mathbf{1} + \langle \mathbf{P}; \dot{\mathbf{P}} \rangle\_{ie} \beta / mN}. \tag{175}$$

### 3.3.5 Ziman formula for the Lorentz plasma

To evaluate the resistance R, we have to calculate the correlation functions P\_; P\_ � � <sup>i</sup><sup>ϵ</sup> and P; <sup>P</sup>\_ � � iϵ . For this, we have to specify the system Hamiltonian HS, which reads for the Lorentz plasma model (136)

$$\mathbf{H}\_{\rm S} = \mathbf{H}\_0 + \mathbf{H}\_{\rm int} = \sum\_{\mathbf{p}} E\_{\mathbf{p}} \mathbf{a}\_{\mathbf{p}}^{\dagger} \mathbf{a}\_{\mathbf{p}} + \sum\_{\mathbf{p}, \mathbf{q}} V\_{\mathbf{q}} \mathbf{a}\_{\mathbf{p} + \mathbf{q}}^{\dagger} \mathbf{a}\_{\mathbf{p}}, \qquad E\_{\mathbf{p}} = \frac{\hbar^2 p^2}{2m}. \tag{176}$$

We consider the ions at fixed positions R<sup>i</sup> so that Vð Þ¼ r ∑<sup>i</sup> Veið Þ r � R<sup>i</sup> . The Fourier transform Vq depends for isotropic systems only on the modulus q ¼ ∣q∣ and will be specified below. A realistic plasma Hamiltonian should consider also moving ions and the electron-electron interaction so that we have a two-component plasma Hamiltonian with pure Coulomb interaction between all constituents. This has been worked out [14], but is not the subject of our present work so that we restrict ourselves mainly to the simple Lorentz model.

The force P on the electrons follows from the \_ x component of the total momentum (p is the wave-number vector)

Nonequilibrium Statistical Operator DOI: http://dx.doi.org/10.5772/intechopen.84707

motion of the electrons. In general, the ions can also be treated as moving charged

A stationary state will be established in the plasma where the electrons are accelerated by the external field, but loose energy (and momentum) due to collisions with the ions. This nonequilibrium state is characterized by an electrical current that is absent in thermal equilibrium. We can take the electric current density je<sup>l</sup> <sup>¼</sup> ð Þ <sup>e</sup>=m<sup>Ω</sup> <sup>P</sup> <sup>¼</sup> ð Þ <sup>e</sup>=<sup>Ω</sup> X as a relevant observable that characterizes the \_ nonequilibrium state. Instead, we take the total momentum B <sup>¼</sup> <sup>P</sup> <sup>¼</sup> <sup>m</sup>X. The \_

m

The term <sup>P</sup>\_j<sup>P</sup> � � <sup>¼</sup> h i ½ � <sup>P</sup>; <sup>P</sup> <sup>e</sup><sup>q</sup> vanishes as can be shown with Kubo's identity, see

eq ¼ � <sup>i</sup><sup>m</sup> ℏβ

mN=<sup>β</sup> <sup>þ</sup> <sup>P</sup>; <sup>P</sup>\_ � �

P\_; P\_ � � iϵ

mΩ

P\_; P\_ � � iϵ

iϵ

. For this, we have to specify the system Hamiltonian HS, which

<sup>p</sup>þ<sup>q</sup>ap, E<sup>p</sup> <sup>¼</sup> <sup>ℏ</sup><sup>2</sup>

<sup>1</sup> <sup>þ</sup> <sup>P</sup>; <sup>P</sup>\_ � �

To evaluate the resistance R, we have to calculate the correlation functions

Fourier transform Vq depends for isotropic systems only on the modulus q ¼ ∣q∣ and will be specified below. A realistic plasma Hamiltonian should consider also moving ions and the electron-electron interaction so that we have a two-component plasma Hamiltonian with pure Coulomb interaction between all constituents. This has been worked out [14], but is not the subject of our present work so that we

\_ x component of the total

Eq. (158). With the Kubo identity, we also evaluate the Kubo scalar product

<sup>E</sup> ð Þþ <sup>P</sup>j<sup>P</sup> <sup>P</sup>; <sup>P</sup>\_ � �

Tr ρeq½ � X; P n o

iϵ

n o

iϵ

<sup>¼</sup> mN

Fð Þ¼ PjP σdcE: (174)

<sup>β</sup>=mN : (175)

p2

<sup>2</sup><sup>m</sup> : (176)

Veið Þ r � R<sup>i</sup> . The

: (173)

, (171)

<sup>β</sup> : (172)

particles that contribute to the current.

Non-Equilibrium Particle Dynamics

ð Þ¼ PjP m

With Eq. (169) we have

P\_; P\_ � �

38

<sup>i</sup><sup>ϵ</sup> and P; <sup>P</sup>\_ � �

ð 1

0

The solution for response parameter F is

j el � � <sup>¼</sup> <sup>e</sup>

3.3.5 Ziman formula for the Lorentz plasma

iϵ

HS ¼ H0 þ Hint ¼ ∑

reads for the Lorentz plasma model (136)

The resistance R in the static limit follows as

<sup>R</sup> <sup>¼</sup> <sup>1</sup> σdc

> p Epa†

restrict ourselves mainly to the simple Lorentz model. The force P on the electrons follows from the

momentum (p is the wave-number vector)

generalized linear response equations (165) and (167) read

<sup>F</sup> <sup>P</sup>\_j<sup>P</sup> � � <sup>þ</sup> <sup>P</sup>\_; <sup>P</sup>\_ � �i<sup>ϵ</sup> � � <sup>¼</sup> <sup>e</sup>

<sup>d</sup><sup>λ</sup> <sup>X</sup>\_ð Þ �iℏβλ <sup>P</sup> � �

<sup>F</sup> <sup>¼</sup> <sup>e</sup> m E

<sup>m</sup><sup>Ω</sup> h i <sup>P</sup> rel <sup>¼</sup> <sup>e</sup><sup>β</sup>

<sup>¼</sup> <sup>Ω</sup><sup>β</sup> e<sup>2</sup>N<sup>2</sup>

<sup>p</sup>a<sup>p</sup> þ ∑ <sup>p</sup>, <sup>q</sup> Vqa †

We consider the ions at fixed positions R<sup>i</sup> so that Vð Þ¼ r ∑<sup>i</sup>

$$\mathbf{P} = \sum\_{\mathbf{p}} \hbar p\_x \, \mathbf{a}\_{\mathbf{p}}^{\dagger} \mathbf{a}\_{\mathbf{p}}, \qquad [\mathbf{H}\_{\rm S}, \mathbf{P}] = -\sum\_{\mathbf{p}, \mathbf{q}} V\_{\mathbf{q}} \hbar q\_x \, \mathbf{a}\_{\mathbf{p} + \mathbf{q}}^{\dagger} \mathbf{a}\_{\mathbf{p}}.\tag{177}$$

We calculate the force-force correlation function (only x component)

$$\left<\dot{\mathbf{P}};\dot{\mathbf{P}}\right>\_{ic} = \int\_{-\infty}^{0} \mathbf{d}t \,\mathbf{e}^{ct} \int\_{0}^{1} \mathbf{d}\lambda \left<\frac{\mathbf{i}}{\hbar}[\mathbf{H}\_{\rm S},\mathbf{P}(t-\mathbf{i}\lambda\theta\hbar)] \frac{\mathbf{i}}{\hbar}[\mathbf{H}\_{\rm S},\mathbf{P}]\right>\_{\rm eq} \tag{178}$$

in Born approximation with respect to Vq. In lowest order, the force-force correlation function is of second order so that in the time evolution exp i ½ � ð Þ =ℏ HSð Þ t � iλβℏ the contribution Hint of interaction to HS, Eq. (176), can be dropped as well as in the statistical operator. The averages are performed with the noninteracting ρ0. The product of the two commutators is evaluated using Wick's theorem. One obtains

$$\begin{split} \left< \dot{\mathbf{P}}; \dot{\mathbf{P}} \right>\_{ic} &= -\sum\_{\mathbf{p}, \mathbf{p}', \mathbf{q}, \mathbf{q}'} \int\_{-\infty}^{0} \mathrm{d}\mathbf{t} \, \mathbf{e}^{\mathbf{t}} \Big|\_{-\infty}^{1} \mathrm{d}\lambda \, \mathbf{e}^{\mathbf{t}} (\mathrm{E}\_{p} - \mathrm{E}\_{p+q}) (\mathrm{t} - \mathrm{i}\, \hbar \ell \lambda) \, V\_{q} \, V\_{q'} q\_{\mathbf{x}} q\_{\mathbf{x}}^{\prime} \Big\langle \mathbf{a}\_{p+q}^{\dagger} \mathbf{a}\_{p} \mathbf{a}\_{p'+q'}^{\dagger} \mathbf{a}\_{p'} \Big\rangle\_{\mathrm{eq}} \\ &= \sum\_{\mathbf{p}, \mathbf{q}} \left| V\_{q} \right|^{2} \delta(\mathbf{E}\_{p} - \mathrm{E}\_{\mathbf{p}+\mathbf{q}}) f\_{p} \left( \mathbf{1} - f\_{p} \right) \pi \hbar q\_{\mathbf{x}}^{2} . \end{split} \tag{179}$$

Because the x direction can be arbitrarily chosen in an isotropic system, we replace q<sup>2</sup> <sup>x</sup> <sup>¼</sup> <sup>q</sup><sup>2</sup> <sup>x</sup> <sup>þ</sup> <sup>q</sup><sup>2</sup> <sup>y</sup> <sup>þ</sup> <sup>q</sup><sup>2</sup> z � �=<sup>3</sup> <sup>¼</sup> <sup>q</sup><sup>2</sup>=3 if the remaining contributions to the integrand are not depending on the direction in space.

Evaluating Eq. (175) in Born approximation, the correlation function P; P\_ � � iϵ ð Þ β=mN can be neglected in relation to 1 because it contains the interaction strength. For the resistance, this term contributes only to higher orders of the interaction.

The force-force correlation function (179) is further evaluated using the relations � <sup>1</sup> β <sup>d</sup>f Eð Þ<sup>p</sup> <sup>d</sup>Ep ¼ f <sup>p</sup> 1 � f <sup>p</sup> � � and <sup>δ</sup> Ep � <sup>E</sup>pþ<sup>q</sup> � � <sup>¼</sup> <sup>m</sup> ℏ2qp <sup>δ</sup> cos <sup>θ</sup> � <sup>q</sup> 2p � �. The <sup>q</sup> integration has to be performed in the limits 0 ≤q≤2p. Finally the resistance can be calculated by inserting the previous expressions Eqs. (172) and (179) into Eq. (175) so that the Ziman-Faber formula is obtained,

$$R = \frac{m^2 \Omega^3}{12\pi^3 \hbar^3 e^2 N^2} \int\_0^\infty \mathrm{d}E(p) \left( -\frac{\mathrm{d}f(E)}{\mathrm{d}E} \right) \int\_0^{2p} \mathrm{d}q \, q^3 \left| V\_q \right|^2. \tag{180}$$

The expression for the resistance depends on the special form of the potential Vq. For a pure Coulomb potential <sup>e</sup><sup>2</sup><sup>=</sup> <sup>Ω</sup>ϵ0q<sup>2</sup> ð Þ, the integral diverges logarithmically as typical for Coulomb integrals. The divergency at very small values of q is removed if screening due to the plasma is taken into account. Within a manyparticle approach, in static approximation the Coulomb potential is replaced by the Debye potential (146). The evaluation yields

$$
\sigma\_{\rm dc} = \frac{3}{4\sqrt{2\pi}} \frac{(k\_{\rm B})^{3/2} (4\pi c\_0)^2}{m^{1/2} c^2} \frac{1}{\Lambda \left( p\_{\rm thermal} \right)} \tag{181}
$$

where the Coulomb logarithm is approximated by the value of the average p, with ℏ<sup>2</sup> p2 therm=2m ¼ 3kBT=2. In the low-density limit, the asymptotic behavior of the Coulomb logarithm Λ is given by �ð Þ 1=2 ln n. However, this result for σd<sup>c</sup> is not correct and can only be considered as an approximation, as discussed below considering the virial expansion of the resistivity.
