Lattice gas with molecular dynamics collision operator

We introduce a lattice gas implementation that is based on coarse-graining a molecular dynamics (MD) simulation. Such a lattice gas is similar to standard lattice gases, but its collision operator is informed by an underlying MD simulation. This can be considered an optimal lattice gas implementation because it allows for the representation of any system that can be simulated with MD. We show here that equilibrium behavior of the popular lattice Boltzmann algorithm is consistent with this optimal lattice gas. This comparison allows us to make a more accurate identification of the expressions for temperature and pressure in lattice Boltzmann simulations, which turn out to be related not only to the physical temperature and pressure but also to the lattice discretization. We show that for any spatial discretization, we need to choose a particular temporal discretization to recover the lattice Boltzmann equilibrium.

Figure 1

Sketch of the MDLG algorithm: a lattice (blue line) is superimposed on the domain of the MD simulation. Particles in the reciprocal lattice cells (indicated by the red boundaries) are associated with the corresponding lattice point. Particles then get associated with the $n_i$ for the $v_i$, which corresponds to their lattice displacement in the time interval $\mathrm{\Delta }t$.

Figure 2

Simple thought experiment for lattice gas representation of a particle moving with a constant velocity. Clearly the lattice gas definition of momentum will fluctuate as a function of time even though the underlying MD momentum is conserved. Averaging over all lattice placements, however, will recover the correct momentum.

Figure 3

The numbering convention for the velocities $v_i$ in two dimensions. The central point is 0 and corresponds to velocity $v_0=(0,0)$, and the other velocities are given by the connecting vector between the central point and the lattice point in question.

Figure 4

(a) Measured velocity correlation function from MD simulation data compared to the exponential fit. (b) Measured mean-square displacement from MD simulation data compared to the predicted value according to Eq. (47).

Figure 5

Measured equilibrium distributions $f_i^{\text{eq}}$ as a function of the mean-squared displacement measure a from Eq. (44). They are compared to the analytic solution given by Eq. (42). We find excellent agreement between the predicted and measured equilibrium distributions. The horizontal lines indicate the value of D2Q9 lattice Boltzmann weights, and the green vertical line indicates the value of $a^2=1/6$ for which these weights agree with the MDLG results.

Figure 6

Pairs of spatial and temporal discretizations $(\mathrm{\Delta }x,\mathrm{\Delta }t)$ that lead to equilibrium distributions of MDLG that are consistent with the lattice Boltzmann equilibrium distribution $f_i^0(\rho ,0)=w_i$. Notice the two scaling regimes of $\mathrm{\Delta }t\propto \mathrm{\Delta }x$ for the ballistic regime for small times and the diffusive regime ${\left(\mathrm{\Delta }t\right)}^2\propto \mathrm{\Delta }x$ for large times.

Figure 7

Dependence of the equilibrium distribution function on an imposed velocity $U$ for $\mathrm{\Delta }x=100\sigma$ and $\mathrm{\Delta }t=34.16\tau$, corresponding to the green line in Fig. 5. The prediction of Eq. (42) (solid lines) agrees perfectly with the measured averages of the $n_i$ (symbols). These results are compared to the lattice Boltzmann equilibrium distribution for D2Q9 (dashed lines). We find good agreement between the MDLG equilibrium distribution and the LB equilibrium distribution for velocities below about 0.2.

Figure 8

The normalized second moment of Eq. (67) as a function of the second moment $a^2$ from Eq. (44) of the mean-squared displacement probability. The value of $\mathrm{\Psi }$ is shown for different 11 different values of $u_x$ between 0 and 0.5. The values for $u_x=0$ correspond to the bottom points of the graph. For a Galilean-Invariant discrete equilibrium distribution moment the value of $\mathrm{\Psi }$ should be independent of $u$. This is compared to a standard lattice Boltzmann second moment with $\theta =1/3$.

Figure 9

Third velocity moment given by Eq. (68) for an unrestricted velocity set and a velocity set restricted to only a single shell of velocities for 10 equally spaced velocities $u_x$ between 0 and 0.001.

Figure 10

Two manifestations of the temperature $\theta$ from the second and third moments, given by Eqs. (70) and (71). We show the moments for the full velocity set and moments for a velocity set restricted to the first shell (thinner lines). For the restricted velocity set we only see agreement between the two moments for $\theta =1/3$ corresponding to $a^2=1/6$.

Figure 11

The difference of the discrete second moment $m_2$ of Eq. (78) minus $a^2$ converges quickly to a constant $1/6$.