2.3 Closure Relations for the Scattering Operator

In general the terms involving the scattering operator cannot be expressed as a simple function of, for instance, the parabolic moments $M_i$. Here we use a macroscopic relaxation time approximation for the closure which gives the system of moment equations a structure resembling the drift diffusion system.

In a deliberate way the scattering integral stemming from an observable $O$ is modeled as

$$\int Q(f) O d(v_3,v_r) = - \frac{\langle O \rangle - \langle O \rangle_{\mathrm{eq}}}{\tau}$$ (2.52)

Here $\langle \rangle_{\mathrm{eq}}$ denotes the moment from the equilibrium solution. A discussion on the validity of this approximation is given in [Lun00]. Note that the odd equilibrium moments vanish and hence the expression simplifies in this case.