The spherical harmonics expansion (SHE) model can be derived from the Boltzmann equation by the three-step procedure introduced in Sect. 2.4. In contrast to the previous chapters, we do not integrate the Boltzmann equation over the whole wave vector space but only over the isoenergetic wave-vector space. As a result, the variable is still a distribution function, but depending on the position-energy space (x,ε) (and time) only and not on the position-wave-vector space (x,k) (and time). Thus, we are able to reduce the seven-dimensional Boltzmann equation to a five-dimensional problem.
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Jüngel, A. (2009). Spherical Harmonics Expansion Equations. In: Transport Equations for Semiconductors. Lecture Notes in Physics, vol 773. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89526-8_7
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