3.3.7 Kubo formula

The most simple choice of relevant observables is the empty set. There are no response parameters to be eliminated. According Eq. (182), the Kubo formula

$$
\sigma\_{\rm dc}^{\rm Kubo} = \frac{\sigma^2 \beta}{m^2 \Omega} \langle \mathbf{P}; \mathbf{P} \rangle\_{ie}^{\rm irred} \tag{183}
$$

perturbation theory can be applied, and in Born approximation a standard result of transport theory is obtained, the Ziman formula (180). We conclude that the use of relevant observables gives a better starting point for perturbation theory. In contrast to the Kubo formula that starts from thermal equilibrium as initial state, the correct current is already reproduced in the initial state and must not be created by

However, despite the excellent results using the Ziman formula in solid and liquid metals where the electrons are strongly degenerated, we cannot conclude that the result (181) for the conductivity is already correct for low-density plasmas (nondegenerate limit if T remains constant) in the lowest order of perturbation theory considered here.

contributions arise. These divergences can be avoided performing a partial summation, that will also change the coefficients in Eq. (181) which are obtained in the lowest order of the perturbation expansion. The divergent contributions can also be avoided

Besides the electrical current, also other deviations from thermal equilibrium can occur in the stationary nonequilibrium state, such as a thermal current. In general, for homogeneous systems, we can consider a finite set of moments of the

> ℏpx βEp � �<sup>n</sup>=<sup>2</sup>

as set of relevant observables Bf g<sup>n</sup> . It can be shown that with increasing number

ð ÞÞ 4πϵ<sup>0</sup> 2

is improved, as can be shown with the Kohler variational principle, see [11, 15].

Kinetic equations are obtained if the occupation numbers n<sup>ν</sup> of single-(quasi-) particle states ∣νi is taken as the set of relevant observables Bf g<sup>n</sup> . The single-particle state ν is described by a complete set of quantum numbers, e.g., the momentum, the spin and the species in the case of a homogeneous multi-component plasma. In thermal equilibrium, the averaged occupation numbers of the quasiparticle states

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

3.3.10 Single-particle distribution function and the general form of the linearized

These equilibrium occupation numbers are changed under the influence of the

They describe the fluctuations of the occupation numbers. The response equations, which eliminate the corresponding response parameters Fν, have the

are given by the Fermi or Bose distribution function, nh i<sup>ν</sup> eq ¼ f

external field. We consider the deviation Δn<sup>ν</sup> ¼ n<sup>ν</sup> � f

a†

1

=π<sup>3</sup><sup>=</sup>2. For details see [5, 15, 16], where also other thermo-

0

<sup>2</sup><sup>π</sup> � � <sup>p</sup> obtained from the single moment approach is increasing to

<sup>p</sup>a<sup>p</sup> (185)

<sup>Λ</sup> <sup>p</sup>therm � � (186)

0

<sup>ν</sup> ¼ Tr ρeqn<sup>ν</sup> n o.

<sup>ν</sup> as relevant observables.

extending the set of relevant observables Bf g<sup>n</sup> , see below.

3.3.9 Higher moments of the single-particle distribution function

P<sup>n</sup> ¼ ∑ p

> ð Þ k<sup>B</sup> 3=2

σdc ¼ s

<sup>2</sup><sup>π</sup> � � <sup>p</sup> is wrong. If we go to the next order of interaction, divergent

the dynamical evolution.

Nonequilibrium Statistical Operator

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

The prefactor 3= 4 ffiffiffiffiffi

of moments the result

The value <sup>s</sup> <sup>¼</sup> <sup>3</sup><sup>=</sup> <sup>4</sup> ffiffiffiffiffi

41

the limiting value <sup>s</sup> <sup>¼</sup> 25<sup>=</sup><sup>2</sup>

Boltzmann equation

electric effects in plasmas are considered.

single-particle distribution function

follows [18, 19]. The index "irred" denotes the irreducible part of the correlation function, because the conductivity is not describing the relation between the current and the external field, but the internal field. We will not discuss this in the present work. A similar expression can also be given for the dynamical, wave-number vectordependent conductivity σ q;ω which is related to other quantities such as the response function, the dielectric function, or the polarization function, see [5, 11, 16, 17]. Equation (183) is a fluctuation-dissipation theorem; equilibrium fluctuations of the current density are related to a dissipative property, the electrical conductivity.

The idea to relate the conductivity with the current-current auto-correlation function in thermal equilibrium looks very appealing because the statistical operator is known. The numerical evaluation by simulations can be performed for any densities and degeneracy. However, the Kubo formula (183) is not appropriate for perturbation theory. In the lowest order of interaction, we have the result σKubo,<sup>0</sup> dc <sup>¼</sup> ne<sup>2</sup>=m<sup>ϵ</sup> (conservation of total momentum) which diverges in the limit ϵ ! 0.

### 3.3.8 Force-force correlation function

The electrical current can be considered as a relevant variable to characterize the nonequilibrium state, when a charged particle system is affected by an electrical field. We can select the total momentum as the relevant observable, B<sup>n</sup> ! P. Now, the character of Eq. (182) is changed. According the response equation (162), we have

$$-\left<\mathbf{P};\dot{\mathbf{P}}\right>\_{ie}\mathbf{F}+\left<\mathbf{P};\mathbf{P}\right>\_{ie}\frac{e}{m}E=\mathbf{0}\tag{184}$$

so that these contributions compensate each other. As a relevant variable, the averaged current density is determined by the response parameter F which follows from the solution of the response equation (184). We obtain the inverse conductivity, the resistance, as a force-force auto-correlation, see Eq. (175). Now,
