For probability distributions () the kurtosis is defined as
Other invariants exist and can be used to define closure relations. One such family of invariants was used in [GKGS01] to express the sixth moment as a function of the lower moments.
Similar to the kurtosis is the dimensionless quantity
Then we approximate by ([Gri02], p.24)
For this type of closure the sixth moment is a rational function in the even moments. When this is combined with the use of mobilities to describe the scattering integral this method can be easily implemented by extending an existing drift-diffusion code.
A suitable value of is found by comparison with Monte Carlo data. This approach was developed in [GKGS01], [Gri02], where was assumed integer.
For we get from a Maxwellian with temperature by the relation:
For a dimensionless parameter
For we get another Gaussian invariant and introduce a parameter as
Finally, for we get
It is only for the value of that the sixth moment goes with the first power of .
In general the given moments , and cannot be represented as moments of a Gaussian distribution. Yet all closures of the family impose some type of gaussianity condition on the moments of the distribution function.
As the generalized invariant approach does not make an ansatz for the distribution function it has the advantage that the domain of skewness and kurtosis which can be represented is in principle not restricted, which improves the fourth order closure 3.18.
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