A gallery of maximum-entropy distributions: 14 and 21 moments

Abstract

This work explores the different shapes that can be realized by the one-particle velocity distribution functions (VDFs) associated with the fourth-order maximum-entropy moment method. These distributions take the form of an exponential of a polynomial of the particle velocity, with terms up to the fourth-order. The 14- and 21-moment approximations are investigated. Various non-equilibrium gas states are probed throughout moment space. The resulting maximum-entropy distributions deviate strongly from the equilibrium VDF, and show a number of lobes and branches. The Maxwellian and the anisotropic Gaussian distributions are recovered as special cases. The eigenvalues associated with the maximum-entropy system of transport equations are also illustrated for some selected gas states. Anisotropic and/or asymmetric non-equilibrium states are seen to be associated with a non-uniform spacial propagation of perturbations.

Figures (19)

• Figure 1: Physical realizability boundary (paraboloid, Eq. \ref{['eq:realizability-14mom-Riijj-dimensionless']}) and Junk subspace (red dashed line, Eq. \ref{['eq:Junk-subspace-14mom']}) for the case of an isotropic gas, with $P_{ij} = P\delta_{ij}$, and for $Q_{zii}=0$. The minimum of the paraboloid is at location $Q^\star_{xii}=Q^\star_{yii} = 0$ and $R^\star_{iijj}=9$. The realizability paraboloid has been cropped to permit the visualization of the equilibrium point (red cross), situated at $Q^\star_{xii}=Q^\star_{yii} = 0$ and $R^\star_{iijj}=15$.
• Figure 2: Physical realizability boundary from Eq. \ref{['eq:realizability-14mom-Riijj-dimensionless']} and Junk subspace (red dashed line and red transparent surface, from Eq. \ref{['eq:Junk-subspace-14mom']}) in the presence of pressure anisotropy, with $P_{xx}=P_{zz} \neq P_{yy}$. Here, only the $Q_{yii}$ component of the heat flux non-zero. The Maxwellian distribution, $\mathcal{M}$, is located at the lowest point of the Junk subspace (red symbol), for $P_{yy}^\star=1$ (isotropic state). For anisotropic states, the lower boundary of the Junk subspace corresponds to a Gaussian distribution, $\mathcal{G}$.
• Figure 3: Top-Left: dimensionless moment space; the black parabola represents the physical realizability boundary, and the vertical dashed red line is the Junk subspace. Other panels: case (14a) $R_{iijj}^\star = 10$; (14b) $R_{iijj}^\star = 9.2$; (14c) $Q_{xii}^\star = 1$; (14d) $Q_{xii}^\star = 2$; (14e) $Q_{xii}^\star = 2.4$. As the physical realizability boundary is approached, the VDF collapses on a sphere (white dashed line in cases (14b) and (14e)), with radius from Eq. \ref{['eq:radius-sphere-velspace-14mom']}. The coloring of each distribution is rescaled for a better representation.
• Figure 4: 14-moment distribution functions obtained for different values of $R_{iijj}^\star$, with $Q_{ijj}^\star = 0$ and $P_{ij}^\star = \delta_{ij}$. Left: slices of the three-dimensional VDFs, extracted along the line $v_y^\star = v_z^\star = 0$. Right: EDFs, obtained by numerical evaluation of Eq. \ref{['eq:integral-fE']}. The cases $R_{iijj}^\star=10$ and $R_{iijj}^\star=9.2$ correspond to the VDFs (14a) and (14b) of Fig. \ref{['fig:14mom-R-Q-VDFs']}. The two vertical dashed lines at $v_x^\star = \pm \sqrt{3}$ represent the radius of the sphere from Eq. \ref{['eq:radius-sphere-velspace-14mom']}. The energy axis in the Right panel assumes a unitary particle mass, $m = 1$.
• Figure 5: Top-Left: Junk subspace, physical realizability boundary and probed points in moment space. Other boxes: contours of the 14-moment maximum-entropy VDF in dimensionless velocity space, for two Gaussians, (G-1) and (G-2), and for cases (14f), (14g) and (14h). $R_{iijj}^\star$ and $Q_{ijk}^\star$ are taken at equilibrium, while the pressure anisotropy is increased from Left to Right.