Latticegas

Lattice gas refers to a class of simplified fluid models in statistical mechanics in which a fluid is represented by particles moving on the sites of a lattice. Each lattice site stores a set of occupation numbers for discrete velocity directions. Time progresses in discrete steps, and the dynamics proceed in two phases: streaming, where particles move to neighboring sites along their respective directions; and collision, where particles at a site interact and are reassigned to new velocity directions according to rules that conserve mass and momentum. Occupation numbers are usually 0 or 1 for each direction, though some variants permit multiple occupants.

In two dimensions the classic realizations are the HPP model on a square lattice with four directions and the FHP model on a hexagonal (triangular) lattice with six directions. Collision rules are crafted to conserve local mass and momentum and to reproduce Navier–Stokes-like behavior in the continuum limit. Early LGA faced challenges with isotropy and Galilean invariance, but refined lattices and collision prescriptions have mitigated many of these issues. Lattice gas automata can be extended to more complex situations, including multi-species or multiphase variants, and to three dimensions with appropriate lattice geometries.

Applications of lattice gas models include qualitative and quantitative studies of fluid flow in simple regimes, porous media, and teaching environments where microscopic rules illustrate emergent hydrodynamic behavior. They also inspired the lattice Boltzmann method, which uses averaged distribution functions instead of discrete occupancy and has become a widely used tool in computational fluid dynamics. Limitations of LGA include significant statistical noise at small scales, limited Reynolds-number ranges, and lattice artifacts that require careful handling of boundaries and isotropy.