Lattice gas automaton

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Lattice gas automata (LGCA) are a simple way to simulate fluids using a grid and tiny particles. They were developed by Hardy–Pomeau–de Pazzis and Frisch–Hasslacher–Pomeau and helped shape lattice Boltzmann methods. LGCA can lead to the Navier–Stokes equations, but interest declined in the 1990s as lattice Boltzmann grew. A modern variant, BIO-LGCA, is still used to model collective cell movement in biology.

How it works
- Space is a lattice. Each site can hold particles moving in a fixed set of directions; each direction is a boolean (particle present or not).
- Time moves in steps with two stages: propagation and collision.
- Propagation: each particle moves to the neighboring site in its direction.
- Collision: if several particles reach a site, they interact according to rules that conserve mass and momentum. Sometimes there are multiple valid outcomes, so a random choice is used.
- The exclusion principle prevents two particles from traveling along the same link in the same direction. Opposite directions on the same link can pass through without colliding.
- Density at a site is the number of particles there; momentum comes from summing their velocities. Because these are discrete and noisy, results are often averaged over regions or across many runs.

Two classic 2D models
- HPP (square grid): particles move in four directions. Collisions redirect particles to move at right angles. This model has strong anisotropy and square-shaped vortices, due to limited symmetry.
- FHP (hexagonal-like grid): uses six directions (plus optional seventh “at rest” in later variants). FHP-I, II, and III differ in which collisions are allowed. More collision options (in II and III) let the model handle less viscous flows and higher Reynolds numbers.

3D and symmetry
- Extending LGCA to 3D is harder because you need a lattice with good symmetry. Early 3D approaches used more complex grids to keep the physics reasonable.

Pros and cons
- Pros: exact boolean states avoid floating-point rounding errors, and the framework is highly suitable for parallel computing.
- Cons: lacks Galilean invariance (results can depend on the reference frame), has statistical noise, and extending to 3D is challenging.

Applications
- LGCA laid the groundwork for lattice Boltzmann methods.
- BIO-LGCA is used today to model biological processes like collective cell migration, where momentum conservation isn’t required and cells can die or reproduce. Collisions in biology are often treated probabilistically to mimic local interactions.