A fluctuating lattice Boltzmann formulation based on orthogonal central moments

TL;DR

This work develops a fluctuating lattice Boltzmann method built on an orthogonal central-moments basis to enforce exact fluctuation–dissipation balance at the lattice level. By projecting stochastic forcing into the CM space and pairing it with mode-dependent relaxation, the method yields diagonal equilibrium covariance and independent discrete Ornstein–Uhlenbeck processes for non-conserved modes, preserving equipartition and Galilean invariance. Explicit schemes for D2Q9 and D3Q27 (and a reduced lattice example) demonstrate exact equipartition, isotropy of fluctuations, correct scaling with $k_B T$ and $ ho$, and robustness to over-relaxation, outperforming fluctuating BGK in stability-limited regimes. The framework is lattice-agnostic and provides a structurally consistent description of fluctuating hydrodynamics on discrete lattices, with potential extensions to multiphase and thermal variants in future work.

Abstract

Thermal fluctuations play a central role in fluid dynamics at mesoscopic scales and must be incorporated into numerical schemes in a manner consistent with statistical mechanics. In this work, we develop a fluctuating lattice Boltzmann formulation based on an orthogonal central-moments-based representation. Stochastic forcing is introduced directly in the space of central moments (CMs) and consistently paired with mode-dependent relaxation, yielding a discrete kinetic model that satisfies the fluctuation-dissipation theorem exactly at the lattice level. Owing to the orthogonality of the basis, the equilibrium covariance matrix of the central moments is diagonal, and each non-conserved mode can be interpreted as an independent discrete Ornstein-Uhlenbeck process with variance fixed by equilibrium thermodynamics. The resulting formulation guarantees exact equipartition of kinetic energy at equilibrium, preserves Galilean invariance, and retains the enhanced numerical stability characteristic of CMs-based collision operators. Explicit fluctuating schemes are constructed for the D2Q9 and D3Q27 lattices. The extension to reduced-velocity discretisation is discussed too. A comprehensive set of numerical tests verifies correct thermalisation, isotropy of equilibrium statistics, and the expected scaling of velocity fluctuations with thermal energy, density, and relaxation time. In contrast to fluctuating BGK formulations, the present method remains stable and well posed in the over-relaxation regime, including in the immediate vicinity of the stability limit. These results demonstrate that CMs-based lattice Boltzmann methods provide a natural and robust framework for fluctuating hydrodynamics, in which dissipation, noise, and kinetic mode structure are consistently aligned at the discrete level.

Figures (7)

• Figure 1: Test 2: grid-convergence of the Taylor--Green vortex. Log--log plot of the $L_2$-norm of the velocity error versus grid resolution, showing a fitted slope of $-2$, consistent with second-order accuracy of the scheme in space and time. Findings obtained by the adoption of an orthogonal basis (blue orthogonal crosses) overlap those generated by a non-orthogonal one (grey diagonal crosses).
• Figure 2: Test 3. Time evolution of the domain-averaged velocity variances $\langle u_x^2\rangle$ (dark magenta line with triangles) and $\langle u_y^2\rangle$ (dark cyan line with inverted triangles) for a homogeneous system with thermal fluctuations enabled ($k_B T = 1/3000$, $\rho_0=1$). The black dashed line indicates the equipartition prediction. Both velocity components fluctuate around the expected value, confirming correct calibration of the stochastic forcing.
• Figure 3: Test 4: scaling of the equilibrium velocity variance with thermal energy. Time-averaged values of $\langle u_x^2\rangle$ (dark orange line with squares) and $\langle u_y^2\rangle$ (dark yellow line with circles) as a function of $k_B T$ for a homogeneous system at equilibrium. The black dashed line indicates the equipartition prediction $\langle u_\alpha^2\rangle = k_B T/\rho_0$ ($\rho_0=1$). Both velocity components follow the expected linear scaling.
• Figure 4: Test 5: scaling of the equilibrium velocity variance with density. Time-averaged values of $\langle u_x^2\rangle$ (goldenrod line with filled triangles) and $\langle u_y^2\rangle$ (dark spring green line with inverted filled triangles) as a function of the reference density $\rho_0$ for a homogeneous system at equilibrium, with fixed thermal energy $k_B T$. The black dashed line indicates the equipartition prediction $\langle u_\alpha^2\rangle = k_B T/\rho_0$. Both velocity components collapse onto the theoretical $k_B T/\rho_0$ scaling, demonstrating consistency with the equipartition theorem.
• Figure 5: Test 6: relative deviation $\psi$ from the equipartition target as a function of the relaxation time $\tau$. (a) Full $\tau$--range comparison between the BGK scheme (green line with diamonds) and the present formulation (red line with pentagons). While both methods recover comparable accuracy at moderate and large $\tau$, BGK exhibits a sharp increase in error as $\tau \to 0.5$. (b) Zoomed view near the stability limit $\tau=0.5$, showing the abrupt numerical breakdown of BGK, characterized by large excursions and the onset of NaN values, in contrast to the smooth and robust behaviour of the present method.