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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

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

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

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

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

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

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

$\displaystyle \gamma = \frac{M_0 M_6}{M_2 M_4}$ (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.

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