Lattice Boltzmann Models

• Lattice Boltzmann Models are computational algorithms that discretize mesoscopic kinetic equations to simulate fluid and continuum dynamics with high efficiency.
• Different collision operators such as BGK, MRT, and TRT are used to balance stability and accuracy in modeling multiphase, porous, and reactive systems.
• The local update structure of LBM supports parallel and scalable implementations with advanced boundary treatments for complex geometries.

The lattice Boltzmann model (LBM) is a class of computational algorithms for simulating the dynamics of fluids and other continuum systems. Instead of directly integrating macroscopic equations such as the Navier–Stokes or advection–diffusion equations, LBM operates by discretizing a mesoscopic kinetic equation for particle distribution functions on a lattice, allowing emergent recovery of the relevant hydrodynamics via moment closure. LBM has seen wide adoption in computational physics and engineering due to its flexibility for handling complex geometries, local update structure, and parallelizability. Applications encompass single-phase flows, multiphase and multicomponent systems, porous media, reactive flows, thermal transport, solid mechanics, and active matter, as well as emerging fields such as quantum and plasma hydrodynamics.

1. Historical Overview and Core Principles

The development of LBM traces to lattice gas automata (LGA) in the 1980s, which used Boolean particle populations and collision rules to mimic fluid behavior. Limitations of LGA, particularly statistical noise and lack of Galilean invariance, drove the evolution to LBM in the late 1980s and early 1990s. The fundamental advance was replacing Boolean occupation numbers with real-valued distribution functions, $f_i(\vec{x}, t)$, associated with discrete lattice velocities $\vec{c}_i$. The LBM iteratively propagates (streams) and relaxes (collides) these distributions:

$$f_i(\vec{x}+\vec{c}_i\Delta t, t+\Delta t) = f_i(\vec{x}, t) + \Omega_i(f(\vec{x}, t))$$

where $\Omega_i$ is a collision operator, typically the BGK (single-relaxation-time) or MRT (multi-relaxation-time) form. The local equilibrium $f_i^{eq}$ is designed such that prescribed hydrodynamic moments yield macroscopic density, momentum, and energy fields. Chapman–Enskog multiscale expansion demonstrates that, under appropriate assumptions (low Knudsen and Mach numbers), LBM recovers the incompressible/compressible Navier-Stokes equations, advection–diffusion, and other transport systems (Carenza et al., 2019).

LBM's strictly local, explicit update on regular grids lends itself to efficient parallel implementation. Complex boundary conditions, including curved, moving, and reactive interfaces, can be accommodated with physical or algorithmic treatments such as bounce-back, interpolated boundaries, and immersed boundary/interface approaches (Qin et al., 2020).

2. Algorithmic Variants and Collision Models

LBM variants arise primarily in the choice of collision operator, force/flux incorporation, grid topology, and equilibrium distributions:

• BGK/SRT (Single-Relaxation-Time): The canonical form relaxes distributions toward equilibrium at a single rate $\tau$, with kinematic viscosity $\nu = c_s^2 (\tau - 0.5)\delta t$. BGK is simple but can be unstable at low viscosity and in high Reynolds number flow.
• MRT (Multiple-Relaxation-Time): Distinct non-hydrodynamic moments relax at different rates, governed by a matrix $S$ in moment space (Rao et al., 2019). Proper selection of MRT parameters eliminates non-physical dependencies (e.g., permeability on $\tau$ in microtomographic simulations) and enhances stability, especially in heterogeneous or anisotropic media (Set B/D/E parameterizations).
• TRT (Two-Relaxation-Time): Separates even/odd moments for higher accuracy in certain settings.
• Central-moment and cumulant LBM: Address deficiencies in non-Galilean invariance and numerical artifacts by more careful moment handling.
• Force incorporation: Advanced Hermite expansion methods, accurate to arbitrary order, ensure correct momentum and especially energy transport, crucial in compressible/thermal flows (Li et al., 2023). Omission of higher-order force moments induces spurious heat flux at finite Mach number.
• Boundary treatments: Standard bounce-back for no-slip walls, interpolated or momentum-exchange boundaries for curved or moving walls, immersed boundary (IB-LBM) and immersed interface (II-LBM) for flexible/rigid solid bodies, each with distinct trade-offs in accuracy and locality (Qin et al., 2020, Millet et al., 3 Nov 2025).

3. Multiphase and Multicomponent Extensions

LBM frameworks for multiphase and multicomponent fluids enable pore-scale modeling of phase segregation, emulsions, and interfacial phenomena (Liu et al., 2014). Several major classes exist:

• Color-gradient (Rothman–Keller) models: Multiple distribution functions for each component, interface handled via color field and recoloring operations, direct control of interfacial tension.
• Pseudopotential (Shan-Chen) models: Non-ideal interactions via local potentials induce phase separation and capture interfacial tension. Widely used for single- and multi-component flows, but density ratios and spurious currents are limitations without advanced force schemes (Liu et al., 2014).
• Free-energy and phase-field (Cahn–Hilliard) models: Thermodynamically consistent, with interface thickness and tension parameters explicit in the free energy. Advanced stabilized diffuse-interface schemes with high-order isotropy and zero-flux boundary conditions extend density ratios to $O(10^3)$ and suppress spurious velocities.
• Mean-field and equation-of-state-based models: Address non-ideal fluid behavior, including critical properties and phase diagrams.
• Conservative Allen-Cahn interface-capturing: Newer approaches combine phase-field evolution with high-order, thread-safe LBM for robust HPC scaling at high property contrast (Lauricella et al., 1 Jan 2025).
• Solid particulates: Particle–fluid coupling for dense suspensions/margination is achieved via momentum-exchange or immersed-boundary variants.

4. Applications and Benchmarking

Porous Media and Permeability

LBM is standard for simulating flows in digital pore spaces from micro-CT images (Rao et al., 2019). Parameter-independent permeability prediction in the Stokes regime requires proper collision model selection (optimal MRT parameter sets), controlled resolution, and knowledge of bounceback boundary effects. For true continuum results, the Knudsen number ($Kn$) must satisfy $Kn \ll 1$; higher $Kn$ introduces kinetic/rarefaction artifacts, and LBM becomes invalid for modeling intrinsic permeability in ultra-tight porous media (Li et al., 2016). Upscaling via local fine-grid simulations enables coarse-grid models that preserve average fluxes, substantially reducing computational cost while retaining accuracy (Li et al., 2013).

Multiphase Flows and Complex Interfacial Systems

LBM excels in complex multiphase settings, capturing drainage/imbibition, fingering, trapping, wettability, and surfactant dynamics at the pore scale (Liu et al., 2014). For severe conditions (e.g., air–water density/viscosity ratios), stabilized phase-field models [Stabilized Diffuse-Interface] yield accurate interface representation; Allen-Cahn-based thread-safe LBM delivers high-fidelity, scalable simulations for droplets and bubbles with sharp interfaces at $10^3$ property ratios (Lauricella et al., 1 Jan 2025).

Soft Matter, Active Fluids, and Solid Mechanics

LBM accommodates polar, nematic, and chiral order via tensorial/field extensions, simulating a spectrum from active turbulence to self-propelled droplets and complex rheology (Carenza et al., 2019). In solid domains, LBM schemes can now handle elastic deformations via decomposition into dilatational and rotational wave equations, allowing Dirichlet and Neumann boundary conditions and benchmarking against FEM (Schlüter et al., 2022).

Reactive, Thermal, and Quantum Systems

Hermite-expansion-based body-force schemes allow accurate simulation of compressible and thermal flows with body forces, provided third (or higher) moment contributions are consistently included (Li et al., 2023). LBM has also been deployed for quantum hydrodynamics and plasma simulations (e.g., wakefield acceleration), where intrinsic regularization is beneficial for handling shocks and singularities (Parise et al., 2022).

5. Optimization, Scalability, and High-Performance Computing

LBM’s local update structure is matched to modern computational architectures. Key implementation strategies include:

• Data layout: Structure-of-arrays (SoA) for optimal bandwidth; blocked and list-based approaches for memory locality in sparse geometry.
• Kernel design: Alternating access (AA pattern), one-step (push/pull), and in-place schemes tailored to minimize memory movement and maximize SIMD vectorization (Wittmann et al., 2017).
• Roofline analysis: Performance is primarily memory-bandwidth-bound, and optimal kernels saturate hardware limits for both homogeneous and heterogeneous domains.
• Thread-safety and lock-free parallelization: Critical for utilizing large GPU clusters and shared memory environments (Lauricella et al., 1 Jan 2025).
• Automatic differentiation and machine learning integration: PyTorch-based frameworks now enable differentiation through simulation, model training (e.g., neural collision operators), and flow optimization (Bedrunka et al., 2021).

6. Limitations and Emerging Directions

LBM’s paradigm presents both strengths and challenges:

• Hydrodynamic regime restriction: Standard LBM is rigorously valid in the $Kn \ll 1$, low Mach regime. At higher Knudsen/Mach regimes, rarefaction or compressibility effects may require more elaborate kinetic models.
• Extension beyond fluids: Solid mechanics, quantum hydrodynamics, and hyperbolic systems require tailored equilibrium and collision schemes (Anandan et al., 8 Jan 2024).
• Boundary and interface handling: Accurate representation of curved or moving interfaces, singular force imposition, and wettability remains a research focus; methods include sharp-interface (II-LBM), immersed boundary, and penalty techniques (Millet et al., 3 Nov 2025, Qin et al., 2020).
• Physical unit simulation: “Dimensional LBM” directly simulates in SI units, removing the need for conversion and facilitating engineering integration (Martins et al., 2023).
• Minimal/lossless schemes: Formulations such as MacLAB eliminate the collision step, embedding viscosity via lattice parameters, and retaining correct hydrodynamics under suitable grid/Mach conditions (Zhou, 2019).

7. Summary Table: Established LBM Classes and Representative Applications

Model Class	Principal Features	Typical Applications
BGK/SRT LBM	Single relaxation; simplicity	Channel, turbulence, simple geometry
MRT/TRT LBM	Multiple relaxation; parameter control	Heterogeneous porous, high Re flows
Color-gradient/Pseudopotential	Diffuse-interface; multiphase	Pore-scale multiphase, emulsions
Phase-field/Free-energy/Stabilized	Phase-field, Cahn-Hilliard, high HDR	HDR droplet/bubble, wettability
Curvilinear LBM	General coordinate grids	Arbitrary geometry, adaptive mesh
Allen-Cahn multiphase	Conservative interface-capture	HPC multiphase, high contrast
Immersed boundary/interface	Fluid-structure interaction	Sedimenting particles, elastic membrane
Dimensional LBM	SI units, no conversion	Engineering, realistic properties
MacLAB	No collision step, macroscopic update	Large-scale, low-Mach/low-Re flows
Neural LBM/ML-integrated	Auto-diff, ML-trained operators	Turbulence, optimization, control

LBM continues to expand in both engineering and fundamental science, both as a standalone solver and a platform for hybrid, multiscale, and ML-integrated techniques. Ongoing research addresses physical rigor, scalability, interface modeling, and unification across physical regimes.