Integer Lattice Gas with a sampling collision operator for the fluctuating Navier-Stokes Equation

TL;DR

This work advances integer lattice gas methods as competitive alternatives to lattice Boltzmann approaches for fluctuating hydrodynamics by constructing a 1D $D1Q3$ framework with a sampling collision operator. It derives a local equilibrium ensemble in moment space, shows that the Boltzmann limit is not unique due to correlations, and demonstrates that, in many cases, the BGK entropic lattice Boltzmann operator provides a good approximation while revealing regimes where correlations modify transport. The authors validate the method through non-equilibrium tests including a decaying isothermal sound wave and an isothermal SOD shock tube, analyze non-equilibrium fluctuations, and quantify computational efficiency gains with dynamic lookup tables and pre-generated random numbers. The results suggest the sampling lattice gas can match or exceed entropic LB performance in several regimes and retain physically meaningful fluctuations, offering a path toward higher-dimensional implementations and forcing extensions with potential practical impact for fluctuating hydrodynamics simulations.

Abstract

This paper constitutes a step in the direction of developing integer lattice gas methods as an attractive alternative to lattice Boltzmann methods. Here we show that to Boltzmann limit the one dimensional Blommel integer lattice gas is very close to entropic lattice Boltzmann. More interestingly the integer lattice gas retains additional correlations that prevent the existence of a well defined Boltzmann limit. In the analysis of the decaying sine wave we will see that in some situations the bulk viscosity can crucially depend on such correlations beyond the Boltzmann limit. A sampling collision operator, introduced here, can speed up the execution time to make the algorithm obtain comparable computational efficiency to entropic lattice Boltzmann methods.

Figures (18)

• Figure 1: The Local equilbrium ensemble of $\pi$ values for $N=9$ and $J=0$. A single lattice site was made to undergo 100,000 collisions for both Blommel and Wagner's algorithm and our sampling algorithm blommel2018integer. Then, a further 100,000,000 collisions were run, collecting the $\pi$ value after each. The histograms of these data sets are compared to the recursive and explicit formulae given by Eqs. (\ref{['eqn: Pi Dist Recursive']}) and (\ref{['eqn: Local Explicit Formula']}) respectively
• Figure 2: The averaged collision operator given an imposed average $\pi$ value for four lattice gas particle distributions compared to the BGK collision operator given an initial average ensemble of $\{\bar{N}=30,\bar{J}=0,\bar{\pi}=0\}$ with $\omega=\omega_{LB}=1$
• Figure 3: Normalized pre- and post-collision $\pi$ values of a decaying sound wave. The lattice Boltzmann system was initialized using Eq. (\ref{['eqn: sine density']}) with $\bar{N}^{eq}=1,000$ and $L=250$, and the lattice gas used these same values as a mean for its Poisson distributed initialization. After 50 iterations run at $\omega=\omega_{LB}=1$
• Figure 4: The extracted amplitude evolution of the decaying sound wave for various relaxation times for the lattice gas and lattice Boltzmann simulations over 850 iterations for a lattice size of 50. Systems were initialized otherwise identically to those in Fig. \ref{['fig: SinTheta']}, and run for $\omega=\omega_{LB}=1.0$, $\omega=\omega_{LB}=0.7$, and $\omega=\omega_{LB}=0.1$.
• Figure 5: The recovered viscosities of the lattice Boltzmann and lattice Gas corresponding to various relaxation times along with the theoretical scaling from Eq. (\ref{['eqn: Kinematic Viscosity']}). Each system was initialized and run identically to those in Fig. \ref{['fig: SinDecay']}. Along with the systems from Fig. \ref{['fig: SinDecay']}, systems were also run at $\omega=\omega_{LB}=0.3$ and $\omega=\omega_{LB}=0.5$. The $\lambda$ values were found via non-linear curve fitting on xmgrace with the analytical solution from Eq. (\ref{['eqn: SineDecaySolution']}) and calculated using Eq. (\ref{['eqn: nuAndlambda']}).