(3.11) |

and assume that the coefficients are small, that is, we stay in the diffusion limit. Then we linearize the second factor and get

(3.12) |

which is of the linear-isotropic type. Given the even moments, the parameters and can be determined. However, not for every combination of and from a distribution function we can find such parameters. In particular, it can be shown, that

which limits the range of , where is defined as

In Equation 3.13 the value comes from a Gaussian distribution. The value of 1 is reached from a distribution function of the form in the limit .

Likewise one can show that

must hold. Our implementation of the diffusion closure uses one lookup table for and one for

(3.16) |

both parameterized by the single parameter . From the study of Monte Carlo data we know that the range of parameters and in the results largely exceeds the range of and given by Equations 3.13 and 3.15.

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R. Kosik: Numerical Challenges on the Road to NanoTCAD