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.
Thermal fluctuations are an intrinsic feature of fluid motion at mesoscopic and microscopic scales, where stochastic effects coexist with deterministic hydrodynamic transport 1 . In such regimes, fluctuations influence equilibrium properties, transport coefficients, and spatio-temporal correlations, and must be accounted for to obtain physically consistent descriptions. Numerical methods aiming to resolve these phenomena are therefore required to respect both hydrodynamic conservation laws and the fluctuation-dissipation theorem (FDT) 2 .
The lattice Boltzmann method (LBM), derived from a discrete kinetic description, provides a natural framework for incorporating thermal fluctuations at the mesoscopic level 3 . In LBM, fluid motion is represented by collections of particle distribution functions (also know as populations) that stream and collide on a fixed Cartesian lattice 4 . By augmenting the kinetic dynamics with stochastic forcing, fluctuating lattice Boltzmann methods (FLBMs) are able to reproduce equilibrium velocity and stress fluctuations and recover fluctuating hydrodynamics in the macroscopic limit 5 . Over the past two decades, a variety of fluctuating LB formulations have been proposed, most of them based on single-relaxation-time or raw-moment multi-relaxation-time collision operators 6 . While successful in many applications, these approaches often suffer from limited robustness and degraded accuracy in regimes characterised by low viscosity, strong nonequilibrium effects, or stringent stability requirements.
Indeed, the choice of collision operator plays a cen-tral role in this context. Single-relaxation-time (also know as BGK, that is the acronym for Bhatnagar-Gross-Krook) models 7 offer conceptual simplicity but provide limited control over non-hydrodynamic modes, whose uncontrolled dynamics may contaminate fluctuation spectra and compromise numerical stability. Multi-relaxationtime (MRT) formulations 8 introduce additional flexibility through mode-dependent relaxation, yet when expressed in terms of raw moments they may exhibit spurious couplings between hydrodynamic and higher-order modes, particularly at finite velocities. These limitations are amplified in fluctuating settings, where improper mode coupling or inconsistent noise allocation readily leads to violations of equipartition, distorted fluctuation spectra, and unphysical correlations 9 . A significant advance in this direction has been achieved very recently by Lauricella et al. 10 , who introduced a regularised fluctuating lattice Boltzmann method based on a full Hermite expansion on the D3Q27 lattice. By reconstructing both equilibrium and nonequilibrium populations on an orthogonal Hermite basis and carefully controlling the relaxation of hydrodynamic and non-hydrodynamic (ghost) modes, this approach restores thermodynamic consistency and significantly improves numerical stability across a broad range of relaxation parameters. These results clearly demonstrate that accurate fluctuating hydrodynamics at the lattice level crucially relies on orthogonal mode decompositions and a clean separation between physical and non-physical degrees of freedom.
This insight has been further sharpened in a sub-sequent work by the same authors 11 , who proposed a ghost-mode filtered fluctuating lattice Boltzmann formulation. In this approach, ghost modes are prevented from carrying deterministic information altogether and are reduced to purely stochastic carriers of thermal noise, while hydrodynamic modes retain their physical dynamics. Remarkably, this minimalistic treatment yields fluctuation statistics comparable to fully regularised highorder schemes, underscoring that diagonal equilibrium covariance and strict mode decoupling are not merely advantageous, but essential for physically consistent fluctuating lattice Boltzmann formulations. Central-moments-based lattice Boltzmann methods (CM-LBMs) provide a natural and robust framework in which these requirements can be satisfied intrinsically 12,13 . By formulating the collision process in a reference frame moving with the local fluid velocity, such schemes improve Galilean invariance, suppress spurious velocity-dependent couplings, and significantly enhance numerical stability 14,15 . These advantages have been demonstrated in a wide range of demanding applications, including turbulent, multiphase, and strongly nonequilibrium flows 16,17 . Despite their favourable properties, the systematic construction of fluctuating lattice Boltzmann schemes within a central-moments framework has received comparatively limited attention.
Actually, incorporating thermal fluctuations into a CMs-based formulation poses specific challenges. Stochastic forcing must be introduced directly in moment space in a manner consistent with mode-dependent relaxation, while ensuring exact equilibrium statistics and strict compliance with the FDT. At the same time, the defining properties of central-moments schemes,
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