4.4.2 Solution of the First Order Equation

(4.11) contains terms which depend on the stationary distribution function $f_{s}(\vec{k})$. These are the free term and the scattering term. The stationary distribution function is the solution of (4.10). This fact prevents an analytical solution for $\widetilde{\lambda}$, and a numerical integration is necessary. However, in this work a rejection technique is applied to solve (4.11). In Section 4.2.1 a new differential scattering rate $\widetilde{S}$ has been introduced (see (4.13)). Here another differential scattering rate is defined according to the following expression:

$$\widetilde{S}_{0}(\vec{k}^{'},\vec{k})=S(\vec{k}^{'},\vec{k})+S(\vec{k},\vec{k}^{'}).$$ (4.71)

The corresponding total scattering rate is

$$\widetilde{\lambda}_{0}(\vec{k})=\lambda(\vec{k})+\lambda^{*}(\vec{k}),$$ (4.72)

where $\lambda^{*}$ stands for the total backward-scattering rate

$$S^{*}(\vec{k},\vec{k}^{'})=S(\vec{k}^{'},\vec{k}),$$

$$\lambda^{*}(\vec{k})=\int S^{*}(\vec{k},\vec{k}^{'})\,d\vec{k}^{'}.$$ (4.73)

From (4.13) and (4.72) it follows that

$$\widetilde{S}_{0}(\vec{k}^{'},\vec{k})\geq\widetilde{S}(\vec{k}^{'},\vec{k}).$$ (4.74)

To solve (4.11) a wave vector $\vec{k}$ is generated using the differential scattering rate $S_{0}(\vec{k}^{'},\vec{k})$. The condition of acceptance takes the following form

$$r\widetilde{S}_{0}(\vec{k}^{'},\vec{k})<\widetilde{S}(\vec{k}^{'},\vec{k}),$$ (4.75)

where $r$ is a random number evenly distributed between 0 and $1$. The last inequality may be rewritten as follows:

$$r[S(\vec{k}^{'},\vec{k})+S(\vec{k},\vec{k}^{'})]<(1-f_{s}(\vec{k}))S(\vec{k}^{'},\vec{k})+f_{s}(\vec{k})S(\vec{k},\vec{k}^{'}).$$ (4.76)

When the scattering process can be split into the sum of the emission and absorption of some quasi-particles such as phonons and plasmons, this condition can be rewritten. Considering a forward transition from $\vec{k}^{'}$ to $\vec{k}$ it can be easily shown that one of the following rejection conditions has to be checked depending on whether an absorption or emission process has occurred. For absorption processes it takes the form:

$$r[1+\frac{N_\mathrm{eq}}{N_\mathrm{eq}+1}]<[1-f_{s}(\vec{k})]\frac{N_\mathrm{eq}}{N_\mathrm{eq}+1}+f_{s}(\vec{k}),$$ (4.77)

whereas for emission processes the following condition is checked

$$r[1+\frac{N_\mathrm{eq}}{N_\mathrm{eq}+1}]<1-f_{s}(\vec{k})+f_{s}(\vec{k})\frac{N_\mathrm{eq}}{N_\mathrm{eq}+1},$$ (4.78)

where $N_\mathrm{eq}$ denotes the equilibrium number of quasi-particles. For example, when $N_\mathrm{eq}/(N_\mathrm{eq}+1)\ll1$ it follows from (4.78) and (4.79) that for the non-degenerate case, $f_{s}\ll1$, emission processes will be dominantly accepted while absorption processes will be mostly rejected. This means that the kinetic behavior is determined by emission processes. On the other side for the degenerate case, when $f_{s}\sim 1$, it follows from the same relations that emission processes will be mostly rejected while the probability of the acceptance of absorption processes increases. Finally, it should be noted that for elastic processes, $S(\vec{k},\vec{k}^{'})=S(\vec{k}^{'},\vec{k})$, the rejection condition (4.77) takes the following form:

$$r<\frac{1}{2}$$ (4.79)

This means that one half of the elastic scattering events will not be accepted in the rejection scheme given above. S. Smirnov: