2.1.3 Dimensional Reduction

To derive a numerically tractable model from the full (3+3)-dimensional Boltzmann equation we assume that the distribution $f({\mathbf{x}},{\mathbf{k}})$ is one-dimensional in space ${\mathbf{x}}$ and has cylinder symmetry in ${\mathbf{k}}$. That is

$$f({\mathbf{x}},{\mathbf{k}}) = f(x_3,k_3,k_r), \quad k_r^2 = k_1^2 + k_2^2 .$$ (2.11)

This is known as the ``1+2''-Boltzmann equation. Note that also the input data has to fulfill this symmetry. Consequently the cylinder radial components of the electrical field $E$ vanish.

With this assumption we have

$$\frac{\partial f}{\partial x_1} = \frac{\partial f}{\partial x_2} = 0$$ (2.12)

and

$$E_{x_1} = E_{x_2} = 0 .$$ (2.13)

Hence the BTE reduces for parabolic bands to

$$\frac{\partial f}{\partial t} + v_3 \cdot \nabla_{x_3} f + \frac{q}{m^{*}} E_{x_3} \cdot \nabla_{v_3} f = Q(f) .$$ (2.14)