In the parabolic case the energy is

(2.20) |

In this special case we can choose the observables to be polynomials in .

We set

(2.21) |

and we define the set of velocity observables:

The moments derived from these observables are denoted by . This set of moments naturally corresponds to the use of a ``shifted'' distribution function (diffusion approximation) as defined in Equation 2.17.

For the non-parabolic case the following set of observables
is more appropriate and is used in [GJK^{+}04]:

(2.23) |

We define the ``energy'' moments as

and the fluxes

In the parabolic band case the energy and the velocity set of moments are equivalent descriptions. In this case the quotient between the velocity moments and the energy moments depends on . We have

where .

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