Fluctuating lattice Boltzmann method for the diffusion equation

We derive a fluctuating lattice Boltzmann method for the diffusion equation. The derivation removes several shortcomings of previous derivations for fluctuating lattice Boltzmann methods for hydrodynamic systems. The comparative simplicity of this diffusive system highlights the basic features of this first exact derivation of a fluctuating lattice Boltzmann method.

Figure 1

Difference of the predicted equilibrium distribution of Eq. (19) and the measured equilibrium distribution as a function of the average density in the range of 0 to 120 for different relaxation times. The results are for $\theta =1/3$ averaged over ${10}^6$ iterations and show excellent agreement. Results shown are for the local noise implementation; the global noise implementation shows similarly small errors.

Figure 2

Deviation of Fourier modes from Eq. (79) for a lattice of $L=10$ averaged for $T=336\times {10}^6$ shows no discernible structure. Note that each of the $f(\mathbf{k},t)f(-\mathbf{k},t)$ to be averaged has values varying between about $\pm 50000$, so the remaining scale of $\pm 7$ after averaging is very small.

Figure 3

Correlators $\langle f_i{f}_j\rangle$ divided by the predicted result of Eq. (7) as a function of the mean densities. Ideally this expression would have a constant value of 1. Unsurprisingly for low densities some deviations are observed. It is noted that the deviation for the global noise amplitude (a) sets in below ${\rho }^{\mathrm{eq}}\approx 7$, whereas it only sets in at about ${\rho }^{\mathrm{eq}}\approx 3$ for the local noise implementation (b), and the agreement is quantitatively better.

Figure 4

Comparison of observed distribution functions for local and global noise amplitudes to the Poisson distribution. We show the distribution for $f_1$ for (a) $f_1^{\mathrm{eq}}=5$ (for ${\rho }^{\mathrm{eq}}=30$) and (b) $f_1^{\mathrm{eq}}=1/6$ (for ${\rho }^{\mathrm{eq}}=1$). The small dots represent a histogram with smaller binning.

Figure 5

Third moment correlators divided by the theoretical prediction $\zeta$ of Eq. (80) as a function of the mean densities for both global (a) and local (b) noise. The mixed terms ($\langle f_0{f}_0{f}_1\rangle$, $\langle f_0{f}_1{f}_1\rangle$) have more pronounced deviations from our expected value than the terms examined in Table 1.

Figure 6

Effect of different mobilities (a) and different temperature (b) in two different regions on the equilibrium behavior of the system. In the case of different temperatures with values ${\theta }_1=1/3$ and ${\theta }_2=1/6$ we find a density difference, whereas the case of different mobilities with diffusion constants $D_1=1/30$ and $D_2=1/3$ leaves the densities constant. The insets show the numerical solution divided by the analytical solution.

Figure 7

Time correlators of Eq. (87) for an $81\times 81$ lattice and ${\rho }^{\mathrm{eq}}=120$, $D=1/6$ for different $k$ values. Values have been averaged over $2\times {10}^6$ iterations.

Figure 8

Plot of dynamics of diffusion front with ${\rho }_1=120$ and ${\rho }_2=20$. There is good agreement between simulation and theory.

Figure 9

Plot of variance for global (a) and local (b) noise implementaions in $\rho$ in the form $\left(\rho -{\rho }^{\mathrm{eq}}\right)=\langle \rho \rangle$, with initial density domains of ${\rho }_1=20$ and ${\rho }_2=120$. The densities are calculated on a 100x10,000 lattice. The densities are averaged over the y-direction (each x-dimensional lattice space is averaged over 10,000 $y$ values). It is observed that applying noise to ${\rho }^{\mathrm{eq}}$ gives a near constant variance which does not vary in time.