4.2.1 New Differential and Total Scattering Rates

For this purpose a new differential scattering rate and new total scattering rate are introduced, as defined by the following expressions:

$$\widetilde{S}(\vec{k}^{'},\vec{k})=[1-f_{s}(\vec{k})]S(\vec{k}^{'},\vec{k})+f_{s}(\vec{k})S(\vec{k},\vec{k}^{'}),$$ (4.12)

$$\widetilde{\lambda}(\vec{k})=\int ([1-f_{s}(\vec{k})]S(\vec{k},\v... ...\vec{k}))\,d\vec{k}^{'}= \int \widetilde{S}(\vec{k},\vec{k}^{'})\,d\vec{k}^{'}.$$ (4.13)

The differential scattering rate and total scattering rate are now functionals of the stationary distribution function which is the solution of the equation of zero order (4.10).

With these definitions the scattering operator of the first order $Q^{(1)}[f](\vec{k},t)$ takes the form:

$$Q^{(1)}[f](\vec{k},t)=\int f_{1}(\vec{k}^{'},t)\widetilde{S}(\vec{k}^{'},\vec{k})\,d\vec{k}^{'}-f_{1}(\vec{k},t)\widetilde{\lambda}(\vec{k}),$$ (4.14)

and the Boltzmann-like equation can be rewritten as follows:

$$\frac{\partial f_{1}(\vec{k},t)}{\partial t}+ \frac{q}{\hbar}\vec{E}_{s}\cdot\nabla f_{1}(\vec{k},t)= \int f_{1}(\vec{k}^{'},t)\widetilde{S}(\vec{k}^{'},\vec{k})\,d\vec{k}^{'}-$$ (4.15)

$$-f_{1}(\vec{k},t)\widetilde{\lambda}(\vec{k})-\frac{q}{\hbar}\vec{E}_{1}(t)\cdot\nabla f_{s}(\vec{k}).$$

S. Smirnov: