4.2.3 Free Term and Initial Distribution

Assuming an impulse like excitation of the electric field, $\vec{E}_{1}(t)=\delta(t)\vec{E}_\mathrm{im}$, gives:

$$f_{1}(\vec{K}(t),t)= \int_{0}^{t}\,dt^{'}\int \,d\vec{k}^{'}f_{1}(\vec{k}^{'},t^{'})\w... ...))\cdot \exp\biggl(-\int_{t^{'}}^{t}\widetilde{\lambda}[\vec{K}(y)]\,dy\biggr)+$$

$$+G(\vec{K}(0))\exp\biggl(-\int_{0}^{t}\widetilde{\lambda}[\vec{K}(y)]\,dy\biggr),$$ (4.23)

where

$$G(\vec{k})=-\frac{q}{\hbar}\vec{E}_\mathrm{im}\cdot\nabla f_{s}(\vec{k}).$$ (4.24)

The essential difference of this integral representation from the one of the non-degenerate approach consists in the appearance of the new differential scattering rate $\widetilde{S}(\vec{k}^{'},\vec{k})$ and of the total scattering rate $\widetilde{\lambda}(\vec{k})$. The Boltzmann-like equation (4.11) differs from the Boltzmann equation (4.4) by the additional free term on the right hand side which in general cannot be treated as an initial distribution because it also takes negative values.

S. Smirnov: