3.3.9 Higher moments of the single-particle distribution function

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 single-particle distribution function

$$\mathbf{P}\_n = \sum\_{\mathbf{p}} \hbar p\_x \left(\beta E\_p\right)^{n/2} \mathbf{a}\_\mathbf{p}^\dagger \mathbf{a}\_\mathbf{p} \tag{185}$$

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

$$\sigma\_{\rm dc} = s \frac{(k\_{\rm B})^{3/2} (4 \pi \epsilon\_0))^2}{m^{1/2} \epsilon^2} \frac{1}{\Lambda \left( p\_{\rm thermal} \right)} \tag{186}$$

is improved, as can be shown with the Kohler variational principle, see [11, 15]. The value <sup>s</sup> <sup>¼</sup> <sup>3</sup><sup>=</sup> <sup>4</sup> ffiffiffiffiffi <sup>2</sup><sup>π</sup> � � <sup>p</sup> obtained from the single moment approach is increasing to the limiting value <sup>s</sup> <sup>¼</sup> 25<sup>=</sup><sup>2</sup> =π<sup>3</sup><sup>=</sup>2. For details see [5, 15, 16], where also other thermoelectric effects in plasmas are considered.
