Appendix: Thermal Lattice Boltzmann Model

The thermal lattice Boltzmann model used in this paper was constructed such that the lowest 26 moments of the equilibrium distribution match those of the Maxwell-Boltzmann distribution in order to obtain closure of the mass, momentum and energy equations (thermal only) [22,23]. The coefficients of the equilibrium distribution were worked out for a 4D FCHC lattice and then projected to 3D. The resulting 3D lattice vectors, $\vec{j}_i$ are all the permutations of $(\pm1,\pm1,0),(\pm1,0,0),(\pm1,\pm1,\pm1), (\pm 2,0,0)$ and (0,0,0) where $1 \leq i \leq 12$, $13 \leq i \leq 18$, $19 \leq i \leq 26$, $27 \leq i \leq 32$, and $33 \leq i \leq 34$ respectively (note due to a degeneracy in projecting from 4D to 3D there are two rest mass particles. The equilibrium distribution was solved for to fourth order in velocity and has the following form

$$\begin{eqnarray*}n^{eq}_i=t_i\rho(a_i+b_i\vec{j}_i\cdot \vec{v} + c_iv^2 +d_i (... ...{j}_i\cdot \vec{v})^3 +g_i(\vec{j}_i\cdot \vec{v})^2v^2 +h_iv^4) \end{eqnarray*}$$

where the coefficients are for $1 \leq i \leq 18$, $a_i=(2e-\frac{3}{2}e^2)/24$, $b_i=(2-\frac{3}{2}e)/12$, $c_i=(\frac{3}{2}e-2)/24$, d_i=(1-e)/4, e_i=-1/8, f_i=1/12, g_i=-1/16, h_i=1/48. For $19 \leq i \leq 32$, $a_i=(-e+\frac{3}{2}e^2)/48$, $b_i=(-1+\frac{3}{2}e)/24$, $c_i=(1-\frac{3}{2}e)/48$, d_i=(2e-1)/32, e_i=0, f_i=1/96, g_i=1/64, h_i=-1/96also $n^{eq}_{33}=t_{33}\rho(a_{33}+c_{33}v^2+h_{33}v^4)$ with $a_{33}=(-e+\frac{3}{2}e^2)/48$, $c_{33}=(1-\frac{3}{2}e)/48$, h₃₃=-1/96. and $n^{eq}_{34}=t_{34}\rho(a_{34}+c_{34}v^2+h_{34}v^4)$ with $a_{34}=(1+\frac{3}{4}e^2-\frac{3}{2}e)$, $c_{34}=-\frac{3}{4}(1-e)$, h₃₄=1/8. t_i=1 for $1 \leq i \leq 12$, t_i=2 for $12 \leq i \leq 26$, t_i=2 for $19 \leq i \leq 26$, and t_i=1 for $27 \leq i \leq 32$, t₃₃=2, and t₃₄=1.

In projection from 4D to 3D the assumption is made that there is no net flux and no gradients of any quantity in the fourth dimension. However, there is still an internal degree of freedom associated with a fourth component so that the internal energy, e, is actually given by

$$\begin{eqnarray*}\rho e=\frac{1}{2}\sum n^{eq}_i (\vec{j}_i-\vec{v})^2 + \frac{1}{2}\sum n^{eq}_i (j'_i)^2 \end{eqnarray*}$$

where the j' account for contributions to the internal energy associated with a fourth dimension. For this model, j'_i=0 for $1 \leq i \leq 12$, j'_i=1 for $18 \leq i \leq 26$, j'_i=0 for $27 \leq i \leq 32$, j'_i=2 for i = 33 , and j'_i=0 for i = 34.