Lattice Gas Automata

Lattice gas automata or lattice gas cellular automata are a type of cellular automaton used to simulate fluid flows. They were the precursor to the lattice Boltzmann methods. From lattice gas automata, it is possible to derive the macroscopic Navier-Stokes equations. Interest in lattice gas automaton methods levelled off in the early 1990s, as the interest in the lattice Boltzmann started to rise.

As a cellular automaton, these models comprise a lattice, where the sites on the lattice can take a certain number of different states. In lattice gas, the various states are particles with certain velocities. Evolution of the simulation is done in discrete time steps. After each time step, the state at a given site can be determined by the state of the site itself and neighboring sites, before the time step.

The state at each site is purely boolean. At a given site, there either is or is not a particle moving in each direction.

At each time step, two processes are carried out, propagation and collision.

In the propagation step, each particle will move to a neighboring site determined by the velocity that particle had. Barring any collisions, a particle with an upwards velocity will after the time step maintain that velocity, but be moved to the neighboring site above the original site. The so-called exclusion principle prevents two or more particles from travelling on the same link in the same direction.

In the collision step, collision rules are used to determine what happens if multiple particles reach the same site. These collision rules are required to maintain mass conservation, and conserve the total momentum; the block cellular automaton model can be used to achieve these conservation laws. Note that the exclusion principle does not prevent two particles from travelling on the same link in opposite directions, when this happens, the two particles pass each other without colliding.

In papers published in 1973 and 1976, Hardy, Pomeau and de Pazzis introduced the first Lattice Boltzmann model, which is called the HPP model after the authors. HPP model is a two-dimensional model of fluid particle interactions. In this model, the lattice is square, and the particles travel independently at a unit speed to the discrete time. The particles can move to any of the four sites whose cells share a common edge. Particles cannot move diagonally.

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