2.2.1.1 Sets of Moments

The description of the distribution function $f$ by a set of moments eliminates the $k$-coordinates at the expense of information loss concerning the details of the distribution function, as the common simplification is to investigate only a few moments of the distribution function.

In the parabolic case the energy $\varepsilon$ is

$$\varepsilon = \frac{m^*}{2} v^2 .$$ (2.20)

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

We set

$$u = v^2 = v_3^2 + v_r^2.$$ (2.21)

and we define the set of velocity observables:

$$O = \{O_0, O_1, O_2, O_3, O_4, O_5, O_6\} = \{1,v_3,u,v_3 u,u^2,v_3u^2,u^3\} .$$ (2.22)

The moments derived from these observables are denoted by $M_i$. 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]:

$$P = \{P_0, P_1, P_2, P_3, P_4, P_5, P_6\} = \{1, v_3, \varepsilon, v_3 \varepsilon, \varepsilon^2, v_3 \varepsilon^2, \varepsilon^3\}$$ (2.23)

We define the ``energy'' moments as

$$V_{2i} = \langle \varepsilon^i \rangle$$ (2.24)

and the fluxes

$$V_{2i+1} = \langle v_3 \varepsilon^i \rangle.$$ (2.25)

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 $i=0,1,2$. We have

$$V_{2i+k} = \bigl( \frac{m^*}{2}\bigr)^i M_{2i+k} , \qquad M_{2i+k}= V_{2i+k} \bigl( \frac{2}{m^*} \bigr)^i ,$$ (2.26)

where $k=0,1$.