4.4.1.1 Initial Distributions of the Two Ensembles

Using the same method as suggested in [74] the free term in (4.24) is split into two positive functions $G^{+}$ and $G^{-}$ which are related to $G$ through the relation: $G = G^{+}-G^{-}$. These two positive functions are considered as initial distributions of two carrier ensembles which contain the same numbers of particles. This follows from the following equality:

$$\int G(\vec{k})\,d\vec{k}=0.$$ (4.54)

To find the initial distributions for the case of a longitudinal perturbation the zeroth order equation (4.10) is used. This equation gives together with (4.13):

$$G(\vec{k})=\frac{E_\mathrm{im}}{E_{s}}\cdot\biggl(\lambda(\vec{k}... ...int f_{s}(\vec{k}^{'})\widetilde{S}(\vec{k}^{'},\vec{k})\, d\vec{k}^{'}\biggr),$$ (4.55)

where $\lambda(\vec{k})=\int S(\vec{k},\vec{k}^{'})\,d\vec{k}^{'}$. The last expression provides a splitting of $G$ into two positive functions. From the balance condition stated by the zeroth order equation (4.10) it follows $\langle\widetilde{\lambda}\rangle_{s}=\langle\lambda\rangle_{s}$, where the stationary statistical average is defined as $\langle\cdots\rangle_{s} = \int f_{s}(\vec{k})\cdots\,d\vec{k}$. Then the initial distributions can be written as:

$$G^{+}=\frac{E_\mathrm{im}}{E_{s}}\langle\widetilde{\lambda}\rangl... ...iggl\{\frac{\lambda(\vec{k})f_{s}(\vec{k})}{\langle\lambda\rangle_{s}}\biggr\},$$

$$G^{-}=\frac{E_\mathrm{im}}{E_{s}} \langle\widetilde{\lambda}\rang... ...tilde{S}(\vec{k},\vec{k}^{'})}{\widetilde{\lambda}(\vec{k})}\biggr\}\,d\vec{k}.$$ (4.56)

In (4.57), $G^{+}$ represents the normalized before-scattering distribution function for a particle trajectory whose free-flight times are determined by the conventional scattering rate $\lambda(\vec{k})$, while $G^{-}$ gives the normalized after-scattering distribution function for a particle trajectory constructed using $\widetilde{S}(\vec{k},\vec{k}^{'})$ and $\widetilde{\lambda}(\vec{k})$, respectively. S. Smirnov: