3.2.3 Diffusion Closure

The diffusion closure is a linearized version of the maximum entropy closure. We rewrite a distribution function

$$\begin{gather*}\begin{split}\exp(a_0 + a_1 k_x + a_2 r^2 + a_3 k_x r^2 + a_4 r^4... ...a_2 r^2 + a_4 r^4) \exp(k_x(a_1 + a_3 r^2 + a_5 r^5)) & \end{split}\end{gather*}$$ (3.11)

and assume that the coefficients $a_{2i+1}$ are small, that is, we stay in the diffusion limit. Then we linearize the second factor and get

$$\exp(a_0 + a_2 r^2 + a_4 r^4) (1 + k_x(a_1 + a_3 r^2 + a_5 r^5))$$ (3.12)

which is of the linear-isotropic type. Given the even moments, the parameters $a_0, a_2,$ and $a_4$ can be determined. However, not for every combination of $M_0, M_2,$ and $M_4$ from a distribution function $f$ we can find such parameters. In particular, it can be shown, that

$$1 < \frac{M_0 M_4}{M_2^2} < \frac{5}{3}$$ (3.13)

which limits the range of $\beta$, where $\beta$ is defined as

$$\beta = \frac{3}{5}\frac{M_0 M_4}{M_2^2} .$$ (3.14)

In Equation 3.13 the value $5/3$ comes from a Gaussian distribution. The value of 1 is reached from a distribution function of the form $\exp(-a x^4 + x^2)$ in the limit $a \rightarrow 0$.

Likewise one can show that

$$1 < \frac{M_0 M_6}{M_2 M_4} < \frac{7}{3}$$ (3.15)

must hold. Our implementation of the diffusion closure uses one lookup table for $\beta$ and one for

$$\gamma = \frac{M_0 M_6}{M_2 M_4}$$ (3.16)

both parameterized by the single parameter $\mathrm{signum}(a_2)a_2^2/a_4$ . From the study of Monte Carlo data we know that the range of parameters $\beta$ and $\gamma$ in the results largely exceeds the range of $\beta$ and $\gamma$ given by Equations 3.13 and 3.15.