Fluctuating multicomponent lattice Boltzmann model

Current implementations of fluctuating lattice Boltzmann equations (FLBEs) describe single component fluids. In this paper, a model based on the continuum kinetic Boltzmann equation for describing multicomponent fluids is extended to incorporate the effects of thermal fluctuations. The thus obtained fluctuating Boltzmann equation is first linearized to apply the theory of linear fluctuations, and expressions for the noise covariances are determined by invoking the fluctuation-dissipation theorem directly at the kinetic level. Crucial for our analysis is the projection of the Boltzmann equation onto the orthonormal Hermite basis. By integrating in space and time the fluctuating Boltzmann equation with a discrete number of velocities, the FLBE is obtained for both ideal and nonideal multicomponent fluids. Numerical simulations are specialized to the case where mean-field interactions are introduced on the lattice, indicating a proper thermalization of the system.

Figure 1

A schematic view of the fluctuating Boltzmann distribution $f\left(t\right)$, its ensemble average $\langle f\left(t\right)\rangle$, and its deviation from the equilibrium distribution $f^{{}_{{}^{\text{eq}}}}$. Note that, per definition, the equilibrium distribution function neither has any explicit time dependence nor exhibits any fluctuations. The ensemble averaged distribution function, $\langle f\left(t\right)\rangle$, on the other hand, is per construction free of fluctuations but may depend on time if the system is brought out of equilibrium by some perturbation. Since $\langle f\left(t\right)\rangle$ converges towards $f^{{}_{{}^{\text{eq}}}}$ for long times, the difference $\delta f\left(t\right)=f\left(t\right)-f^{{}_{{}^{\text{eq}}}}$ becomes identical to $f\left(t\right)-\langle f\left(t\right)\rangle$ asymptotically for $t\to \infty$.

Figure 2

Equilibrium ratio (ER) for the densities and the barycentric velocity as a function of wave-vector magnitude $k$. Simulation results are normalized according to the theoretical predictions of Eqs. (56) and (57), except for the cross-density correlation $S_{\rho ,{\rho }^{\prime }}\left(k\right)$, which is normalized by $k_{B}T$. The mutual interaction strength in Eq. (87) is set to $\mathcal{G}=0$ in all the numerical simulations (ideal gases). To appreciate the effects of the noise on the momentum modes [see the first two equations in (74)], we repeated the numerical simulations by setting such noise to zero, i.e., without stochastic diffusion fluxes (no-sdf).

Figure 3

Equilibrium ratio (ER) for the densities and the barycentric velocity as a function of wave-vector magnitude $k$. Simulation results are normalized according to the theoretical predictions of Eqs. (56) and (57), except for the cross-density correlation $S_{\rho ,{\rho }^{\prime }}\left(k\right)$, which is normalized by $k_{B}T$. The mutual interaction strength in Eq. (87) is set to $\mathcal{G}=0.4$ in all the numerical simulations. For the simulation parameters chosen (see text for details), the critical point for phase separation is found at ${\mathcal{G}}_c=1.0$ lbu. To appreciate the effects of the noise on the momentum modes [see the first two equations in (74)], we repeated the numerical simulations by setting such noise to zero, i.e., without stochastic diffusion fluxes (no-sdf).

Figure 4

Equilibrium ratio (ER) for the densities and the barycentric velocity as a function of wave-vector magnitude $k$. Simulation results are normalized according to the theoretical predictions of Eqs. (56) and (57), except for the cross-density correlation $S_{\rho ,{\rho }^{\prime }}\left(k\right)$, which is normalized by $k_{B}T$. The mutual interaction strength in Eq. (87) is set to $\mathcal{G}=0.85$ in all the numerical simulations. For the simulation parameters chosen (see text for details), the critical point for phase separation is found at ${\mathcal{G}}_c=1.0$ lbu. To appreciate the effects of the noise on the momentum modes [see the first two equations in (74)], we repeated the numerical simulations by setting such noise to zero, i.e., without stochastic diffusion fluxes (no-sdf).

Figure 5

The cross-density correlation $S_{\rho ,{\rho }^{\prime }}\left(k\right)$, i.e., the diagonal part of $\langle \delta \widehat{\rho }\left(k\right)\delta {\widehat{\rho }}^{\prime }(-k^{\prime })\rangle$ in the homogeneous case, normalized by $k_{B}T$ is reported as a function of the wave-vector magnitude $k$. The mutual interaction strength in Eq. (87) is set to $\mathcal{G}=0.4$ (squares), $\mathcal{G}=0.85$ (circles), $\mathcal{G}=0.95$ (triangles). Other simulation parameters are given in the text. The critical point for phase separation is found at ${\mathcal{G}}_c=1.0$ lbu. All the numerical simulations are performed with the noise on the momentum modes, i.e., with stochastic diffusion fluxes (sdf), according to Eqs. (74). The theoretical prediction of the right-hand side of Eq. (57) is also reported (red solid line). Correspondingly, we also show the prediction for the ideal-gases $(\mathcal{G}=0)$ case (blue dotted line).

Figure 6

Capillary fluctuations of a planar one-dimensional interface obtained with mean-field lattice interactions (see Sec. 6b). The equal-time spectrum of interfacial height fluctuations obtained from LBE simulations (circles) is compared with the theoretical structure factor (solid line) reported in Eq. (91). The wave vector in the interface region is indicated with $k$. Simulation parameters are reported in the text.