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

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 con-

After fully linearizing the statistical operator (161) with (155), we have for the

 <sup>i</sup><sup>ϵ</sup> Fn <sup>þ</sup> h i <sup>P</sup>; <sup>P</sup> <sup>i</sup><sup>ϵ</sup>

After deriving the Ziman formula from the force-force correlation function in the previous section, we investigate the question to select an appropriate set of

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

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 pertur-

bation theory. In the lowest order of interaction, we have the result σKubo,<sup>0</sup>

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

F þ h i P; P <sup>i</sup><sup>ϵ</sup>

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 conductiv-

e m

(conservation of total momentum) which diverges in the limit ϵ ! 0.

� <sup>P</sup>; <sup>P</sup>\_ iϵ

ity, the resistance, as a force-force auto-correlation, see Eq. (175). Now,

3.3.8 Force-force correlation function

40

<sup>m</sup><sup>2</sup><sup>Ω</sup> h i <sup>P</sup>; <sup>P</sup> irred

e m E

<sup>i</sup><sup>ϵ</sup> (183)

E ¼ 0 (184)

¼ σdcE: (182)

dc <sup>¼</sup> ne<sup>2</sup>=m<sup>ϵ</sup>

ð Þ� <sup>P</sup>jB<sup>n</sup> <sup>P</sup>; <sup>B</sup>\_ <sup>n</sup>

σKubo dc <sup>¼</sup> <sup>e</sup><sup>2</sup><sup>β</sup>

sidering the virial expansion of the resistivity.

<sup>m</sup><sup>Ω</sup> <sup>∑</sup> n

3.3.6 Different sets of relevant observables

electrical current density

relevant observables Bf g<sup>n</sup> .

3.3.7 Kubo formula

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

Non-Equilibrium Particle Dynamics

j el <sup>¼</sup> <sup>e</sup> 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 the dynamical evolution.

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. The prefactor 3= 4 ffiffiffiffiffi <sup>2</sup><sup>π</sup> � � <sup>p</sup> is wrong. If we go to the next order of interaction, divergent 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 extending the set of relevant observables Bf g<sup>n</sup> , see below.
