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# 3.4 Gaussian Invariant Closure

If the first three moments of a one-dimensional distribution function are known, a unique normal distribution with parameters and is defined. Similarly, by assuming that the diffusion approximation holds (that is, the odd moments are assumed to be small) and fix a distribution function in three dimensions, which gives the relation (3.18)

Hence, naturally, the assumption of gaussianity gives a closure for the hierarchy of moment equations at order 4.

For probability distributions ( ) the kurtosis is defined as (3.19)

Note that here denote the central moments of a one-dimensional probability distribution. The kurtosis is one invariant of the class of Maxwellian distributions. Deviations from this constant value are measures of nongaussianity.

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  (3.20)

Here the subscript denotes the Maxwellian distribution with local temperature . In terms of moments is given by Equation 3.14.

Then we approximate by ([Gri02], p.24) (3.21)

In terms of moments this closure is written as (3.22)

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: (3.23)

For a dimensionless parameter (3.24)

can be introduced. The closure becomes (3.25)

This closure was studied in [SYT+96].

For we get another Gaussian invariant and introduce a parameter as (3.26)

which is (up to a constant factor) the quotient between and . The definition of is a generalization of the kurtosis. This gives the closure (3.27)

Note that only for this value of the closure does not depend on .

Finally, for we get (3.28)

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.