4.3.4 Expression for the Initial Distribution

The additional free term in (4.24) cannot be considered an initial distribution because the function $G(\vec{k})$ may take on negative values. However, in the case of zero electric field the stationary distribution is known analytically and $G(\vec{k})$ can be evaluated explicitly:

$$G(\vec{k})=\frac{q}{k_{B}T_{0}}\vec{E}_\mathrm{im}\cdot\vec{v}\fr... ...igr)} {\biggl(\exp\bigl(-\frac{E_{f}-\epsilon}{k_{B}T_{0}}\bigr)+1\biggr)^{2}},$$ (4.50)

where $\vec{v}$ denotes the group velocity. This expression can be rewritten in the following manner:

$$G(\vec{k})=\frac{q\vec{E}_\mathrm{im}\langle\widetilde{\lambda}\r... ...k})f_\mathrm{FD}(\epsilon(\vec{k}))}{\langle\widetilde{\lambda}\rangle}\biggr\}$$ (4.51)

where the term in curly brackets represents the normalized distribution function of the before-scattering states.

The Monte Carlo algorithm contains the same steps as that in [73] except that the whole kinetics is now determined by $\widetilde{S}(\vec{k},\vec{k}^{'})$ and $\widetilde{\lambda}(\vec{k})$ instead of $S(\vec{k},\vec{k}^{'})$ and $\lambda(\vec{k})$.

Another difference from the non-degenerate zero field algorithm is that the weight coefficient $\vec{v}(\vec{k})/\widetilde{\lambda}(\vec{k})$ must be multiplied by the factor $[1-f_\mathrm{FD}(\epsilon(\vec{k}))]$.

With the modifications described above the zero-field algorithm for the time discrete impulse response of a physical quantity $A(\vec{k})$ of interest becomes:

1. Follow a main trajectory for one free flight and store the before-scattering state $\vec{k}$.
2. Compute the weight $w=[1-f_{FD}(\epsilon(\vec{k}))]\vec{v}(\vec{k})/\widetilde{\lambda}(\vec{k})$.
3. Start a trajectory $\vec{K}(t)$ from $\vec{k}$ and follow it for time $T$. At equidistant times $t_{i}$ add $wA(\vec{K}(t_{i}))$ to a histogram $\nu_{i}$.
4. Continue with the first step until $N$ $\vec{k}$-points have been generated.
5. Calculate the time discrete impulse response as $\langle A\rangle_\mathrm{im}(t_{i})=q\vec{E}_\mathrm{im}\langle\widetilde{\lambda}\rangle\nu_{i}/k_{B}T_{0}N$.

This algorithm is also depicted in Fig. 4.5.

Figure: Schematic representation of the zero field Monte Carlo algorithm. Here $\vec{k}_{b}$ and $\vec {k}_{a}$ denote before- and after-scattering states, respectively.

S. Smirnov: