3.2.9 Example: conductivity of the Lorentz plasma

In the Lorentz plasma model, the electron-electron collisions are neglected, and only electron-ion collisions are considered, interaction potential Veið Þr . In the adiabatic approximation where the ions are regarded as fixed at positions R<sup>i</sup> (elastic collisions), the interaction part of the Hamiltonian reads

$$\mathbf{H}' = \sum\_{i} V\_{ei}(\mathbf{r} - \mathbf{R}\_i). \tag{136}$$

In Born approximation (or time-dependent perturbation theory), the transition rate is given by Fermi's Golden rule:

$$\boldsymbol{w}\_{\mathbf{p'p}} = \frac{2\pi}{\hbar} \left| \mathbf{H'\_{p'p}} \right|^2 \delta(E\_p - E\_{p'}) = \boldsymbol{w}\_{\mathbf{p}\mathbf{p'}}; \qquad E\_p = p^2/2m. \tag{137}$$

<sup>σ</sup> <sup>¼</sup> <sup>2</sup>e<sup>2</sup> m<sup>2</sup>kBTΩ

eið Þ¼ <sup>r</sup> <sup>e</sup><sup>2</sup>

Λð Þ¼ p

V<sup>D</sup>

ð d3 pp<sup>2</sup> zf 0 <sup>1</sup> Ep � � <sup>1</sup> � <sup>f</sup>

Nonequilibrium Statistical Operator

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

ð 2p=ℏ

0

3.2.10 Conclusions

<sup>σ</sup> <sup>¼</sup> <sup>25</sup><sup>=</sup><sup>2</sup>

evaluation can be performed. With

1 <sup>q</sup><sup>2</sup> <sup>þ</sup> <sup>κ</sup><sup>2</sup> ð Þ<sup>2</sup> <sup>q</sup><sup>3</sup>

mation, hydrodynamic equations, etc.

collision term have been investigated.

3.3 Linear response theory

3.3.1 Response to an external field

33

acting on the particles, see [4, 11–16],

we finally obtain for the conductivity [5].

ð Þ <sup>k</sup>B<sup>T</sup> <sup>3</sup>=<sup>2</sup>

0 <sup>1</sup> Ep � � � � <sup>2</sup><sup>π</sup>

ℏ ð d3 p<sup>0</sup> Hp<sup>0</sup> p � � � � 2

<sup>1</sup> <sup>þ</sup> <sup>b</sup> <sup>p</sup> � <sup>1</sup>

<sup>4</sup>πϵ0∣r<sup>∣</sup> <sup>e</sup>�κ∣r<sup>∣</sup> with the Debye screening parameter <sup>κ</sup><sup>2</sup> <sup>¼</sup> <sup>e</sup>2N=ð Þ <sup>ϵ</sup>0kBT<sup>Ω</sup> , the

2

The method of the nonequilibrium statistical operator gives not only the derivation of the Boltzmann equation (quantal and classical), but indicates also possible improvements such as conservation of total energy, inclusion of bound state for-

The solution of the general Boltzmann equation is not simple, in addition to numerical simulations different approximations have been worked out. For the linearized Boltzmann equation, the relaxation time approximation can be used for elastic scattering, but for the general case (inclusion of electron-electron collisions in a plasma), the Kohler variational principle [11] can be applied. Landau-Vlasov equations for mean-field effects as well as Fokker-Planck equations for the

The basic assumption to derive the Boltzmann equation is the selection of the single-particle distribution as relevant observable. Correlations are neglected and have to be built up in higher orders of approximation or extending the set of relevant observables. The most appropriate systems for kinetic theory are dilute gases where the collision time is short compared with the time of free flight. Irreversibility is owing to the Stoßzahlansatz for the intrinsic interaction.

A third example, which allows the explicit elimination of the Lagrange multipliers to fulfill the self-consistency conditions, is a system near to thermodynamic equilibrium which is under the influence of mechanical (external forces) or thermodynamic (gradients of temperature, pressure, chemical potentials, etc.) perturbations. As response, currents appear in the system. Assuming linearity for small perturbations, transport coefficients are defined. Fluctuations in equilibrium are considered as a nonequilibrium state which relaxes to equilibrium, see Eq. (7).

We consider a system under the influence of external (time dependent) fields

<sup>H</sup><sup>t</sup> <sup>¼</sup> HS <sup>þ</sup> <sup>H</sup><sup>t</sup>

b

Considering the screened interaction potential (Debye potential)

<sup>d</sup><sup>q</sup> <sup>¼</sup> ln ffiffiffiffiffiffiffiffiffiffiffi

ð Þ 4πϵ<sup>0</sup> 2 <sup>π</sup><sup>3</sup>=<sup>2</sup>m<sup>1</sup>=<sup>2</sup>e<sup>2</sup><sup>Λ</sup> ; <sup>Λ</sup>≈<sup>Λ</sup> <sup>p</sup><sup>2</sup> δ Ep � Ep<sup>0</sup>

� ��<sup>1</sup>

� �ð Þ <sup>1</sup> � cos <sup>ϑ</sup>

<sup>1</sup> <sup>þ</sup> <sup>b</sup> , b <sup>¼</sup> <sup>4</sup>p2kBTΩϵ<sup>0</sup>

e2ℏ<sup>2</sup>

<sup>=</sup>2<sup>m</sup> <sup>¼</sup> <sup>3</sup>kBT=<sup>2</sup> � �: (147)

<sup>F</sup>, (148)

: (145)

<sup>N</sup> , (146)

To solve the Boltzmann equation Eq. (135), we make use of the ansatz

$$f\_1(\mathbf{p}) = f\_1^0(E\_p) + \Phi(\mathbf{p}) \frac{\mathrm{d}f\_1^0(E\_p)}{\mathrm{d}E\_p} k\_\mathrm{B} T = f\_1^0(E\_p) \left\{ 1 + \Phi(\mathbf{p}) \left( 1 - f\_1^0(E\_p) \right) \right\}. \tag{138}$$

For equilibrium distributions, we have the detailed balance condition

$$w\_{\mathbf{p}\mathbf{p}'}f\_1^0(E\_{\mathbf{p}'})\left(\mathbf{1} - f\_1^0(E\_{\mathbf{p}})\right) = w\_{\mathbf{p}'\mathbf{p}}f\_1^0(E\_{\mathbf{p}})\left(\mathbf{1} - f\_1^0(E\_{\mathbf{p}'})\right). \tag{139}$$

Insertion of Eq. (138) into the Boltzmann equation Eq. (135) yields with Eq. (139)

$$\frac{e}{2m\mathbb{k}\_B T} \mathbf{E} \cdot \mathbf{p} f\_1^0(\mathbf{E}\_p) \left[ 1 - f\_1^0(\mathbf{E}\_p) \right] = \int \frac{\mathbf{d}^3 p' \Omega}{(2\pi\hbar)^3} w\_{\mathbf{pp}} f\_1^0(\mathbf{E}\_{p'}) \left[ 1 - f\_1^0(\mathbf{E}\_p) \right] [\Phi(\mathbf{p'}) - \Phi(\mathbf{p})],\tag{140}$$

where we have neglect terms with higher order of E and have used the fact that Φð Þ p ∝E. With the definition of the relaxation time tensor ^τð Þ p , according to Φð Þ¼ p e=ð Þ mkBT E � ^τð Þ� p p, the equation reads

$$\mathbf{e}\_{\rm E} \cdot \mathbf{p} = \int \frac{\mathrm{d}^3 \mathbf{p}' \Omega}{\left(2\pi\hbar\right)^3} w\_{\mathbf{p}\mathbf{p}'} \frac{f\_1^0(E\_{\mathbf{p}'})}{f\_1^0(E\_{\mathbf{p}})} \mathbf{e}\_E \cdot (\hat{\boldsymbol{\tau}}(\mathbf{p}') \cdot \mathbf{p}' - \hat{\boldsymbol{\tau}}(\mathbf{p}) \cdot \mathbf{p}),\tag{141}$$

e<sup>E</sup> ¼ E=E. The electric current density Eq. (133) depends only on the deviation of the distribution function since f 0 <sup>1</sup> is an even function in p (isotropy). We obtain by insertion of Eq. (138) into Eq. (133)

$$j\_{\rm el} = \frac{e}{\Omega} 2 \int \frac{\mathrm{d}^3 \mathbf{p} \Omega}{(2\pi\hbar)^3} \frac{\mathbf{p}}{m} \Phi(\mathbf{p}) f\_1^0(E\_p) \left[1 - f\_1^0(E\_p)\right]. \tag{142}$$

The conductivity σ is the proportionality factor between the current density and the effective field E:

$$\sigma = \frac{e^2}{m^2 k\_B T} 2 \int \frac{d^3 \mathbf{p}}{(2\pi\hbar)^3} p\_x(\hat{\mathbf{r}}(\mathbf{p}) \cdot \mathbf{p})\_x f\_1^0(E\_p) \left[1 - f\_1^0(E\_p)\right]. \tag{143}$$

We have derived an analytical expression for the conductivity of a Lorentz plasma in terms of the relaxation time tensor ^τð Þ p . For isotropic systems, ^τij ¼ τδij, the well-known Ziman formula στ <sup>¼</sup> <sup>τ</sup>ne<sup>2</sup>=<sup>m</sup> for the conductivity results.

The solution of Eq. (141) for a momentum-dependent relaxation time is

$$\pi(E\_p) = \left\{ \int \frac{\mathrm{d}^3 \mathbf{p'} \Omega}{\left(2\pi\hbar\right)^3} w\_{pp'} (\mathbf{1} - \cos\theta) \right\}^{-1} \tag{144}$$

as can be verified by insertion. Now, the conductivity reads with Eq. (137)

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

$$\sigma = \frac{2\varrho^2}{m^2 k\_B T \Omega} \left\{ \mathrm{d}^3 p \, p\_\sigma^2 f\_1^0(E\_p) \left[ 1 - f\_1^0(E\_p) \right] \left\{ \frac{2\pi}{\hbar} \left[ \mathrm{d}^3 p' \middle| \mathrm{H}\_{\mathbf{P}'\mathbf{P}} \right]^2 \delta(E\_p - E\_{\mathbf{P}'}) (1 - \cos \theta) \right\}^{-1} . \tag{145}$$

Considering the screened interaction potential (Debye potential) V<sup>D</sup> eið Þ¼ <sup>r</sup> <sup>e</sup><sup>2</sup> <sup>4</sup>πϵ0∣r<sup>∣</sup> <sup>e</sup>�κ∣r<sup>∣</sup> with the Debye screening parameter <sup>κ</sup><sup>2</sup> <sup>¼</sup> <sup>e</sup>2N=ð Þ <sup>ϵ</sup>0kBT<sup>Ω</sup> , the evaluation can be performed. With

$$\Lambda(p) = \int\_0^{2p/h} \frac{1}{\left(q^2 + \kappa^2\right)^2} q^3 \mathrm{d}q = \ln\sqrt{1+b} - \frac{1}{2} \frac{b}{1+b}, \qquad b = \frac{4p^2 k\_B T \Omega \varepsilon\_0}{e^2 \hbar^2 N}, \tag{146}$$

we finally obtain for the conductivity [5].

$$\sigma = \frac{2^{5/2} (k\_{\rm B} T)^{3/2} (4 \pi \epsilon\_0)^2}{\pi^{3/2} m^{1/2} e^2 \Lambda}; \qquad \Lambda \approx \Lambda \left( p^2 / 2m = 3k\_{\rm B} T / 2 \right). \tag{147}$$
