4.1.2 The First Order Equation

Collecting terms of the first order gives the following equation:

$$\frac{\partial f_{1}(\vec{k},t)}{\partial t}+\frac{q}{\hbar}\vec{... ...-\frac{q}{\hbar}\vec{E}_{1}(t)\cdot\nabla f_{s}(\vec{k})+Q^{(1)}[f](\vec{k},t),$$ (4.10)

where the notation $Q^{(1)}[f](\vec{k},t)$ has been introduced for the first order scattering operator which has the form

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

$$-f_{1}(\vec{k},t)\int [1-f_{s}(\vec{k}^{'})]S(\vec{k},\vec{k}^{'}... ...}- f_{1}(\vec{k},t)\int f_{s}(\vec{k}^{'})S(\vec{k}^{'},\vec{k})\,d\vec{k}^{'}+$$ (4.11)

$$+f_{s}(\vec{k})\int f_{1}(\vec{k}^{'},t)S(\vec{k},\vec{k}^{'})\,d\vec{k}^{'}.$$

Equation (4.11) is linear with respect to $f_{1}(\vec{k},t)$, but it is a kinetic equation different from the usual form of the Boltzmann equation. The first difference is the additional term on the right hand side being the term proportional to $\vec{E}_{1}$ which additionally depends on the stationary distribution $f_{s}(\vec{k})$ determined by (4.10). The second difference is the expression for the scattering operator which now has a more complex form and also depends on the stationary distribution $f_{s}(\vec{k})$.

S. Smirnov: