Simulating Single Particle Motion on Hexagonal Cellular Automata

Simulating single particle motion on hexagonal cellular automata serves as a classical approach for studying fluid dynamics and complex system behaviors. This MATLAB-based program implements particle motion simulation on hexagonal grids using FHP (Frisch-Hasslacher-Pomeau) rules, where the core algorithm involves initializing a hexagonal lattice structure and defining particle transition mechanisms through state matrices.

Cellular automata represent discrete computational models composed of cells arranged in regular lattices, where each cell updates its state based on neighboring cell states and predefined rules. The hexagonal grid's symmetry makes it particularly suitable for simulating particle collisions and flow patterns, with MATLAB implementation typically using coordinate mapping functions to handle the six-directional connectivity.

The FHP rules govern particle movement and interactions on hexagonal grids through three fundamental operations: propagation, collision, and boundary handling. Particles move along one of six possible directions encoded as binary states (e.g., direction vectors [1 0], [0.5 √3/2], etc.). Collision rules are implemented using conditional statements that modify direction bits when multiple particles meet, mimicking physical interactions through bitwise operations and state transition tables.

Program implementation involves key steps: grid initialization using meshgrid or custom coordinate systems, particle positioning through boolean matrices, rule application via vectorized operations for efficiency, and visualization using MATLAB's scatter or patch functions. Iterative updates using while/for loops demonstrate particle diffusion, collision dynamics, and pattern formation, with performance optimized through sparse matrix representations for large-scale simulations.

Beyond fluid dynamics, this simulation framework extends to multi-particle systems like traffic flow modeling and biological migration studies. MATLAB's matrix operations (e.g., circshift for neighbor access) and graphical capabilities (quiver plots for direction visualization) provide robust tools for such simulations, with potential enhancements through GPU acceleration for real-time dynamics observation.