Diffusion-Limited Aggregation Simulation Using Cellular Automata

Diffusion-Limited Aggregation (DLA) is a mathematical model describing numerous growth phenomena in nature, such as snowflake crystallization, electrodeposition, and bacterial colony expansion. Using Cellular Automata methods, we can simulate the aggregation behavior of adhesive particles in diffusion environments through discrete computational models.

Core Concept The fundamental mechanism of DLA involves simulating particles undergoing random walks (diffusion) that adhere to a sticky core upon contact and cease movement. Initially, a seed particle is placed at the center of the simulation space, while other particles are randomly released at the boundaries to diffuse. When a particle contacts any fixed aggregate, it stops moving and becomes part of the growing structure. Implementation typically requires tracking particle coordinates and neighbor detection algorithms.

Cellular Automata Implementation Framework Grid Discretization: Divide 2D space into discrete grid cells where each cell can be empty, contain an active particle, or hold a fixed aggregate. Code implementation often uses a 2D array with integer flags (0=empty, 1=active, 2=fixed). Particle Emission: Generate new particles at random positions along the grid boundary, marked as "active" state. This can be implemented using random number generators for coordinate selection. Random Walk: Active particles move randomly to one of four adjacent directions (up, down, left, right) at each time step. The algorithm typically employs random direction selection with uniform probability distribution. Adhesion Rule: If an active particle has any fixed aggregate in its Moore neighborhood (8 surrounding cells), it transitions to fixed state and stops moving. This requires checking neighboring cells using conditional statements. Termination Condition: Simulation stops when the aggregate reaches a predetermined size or all particles are depleted. Programmatically, this involves continuous size monitoring through counter variables.

Extensions and Optimizations Multi-Particle Concurrency: Simultaneously release multiple particles to improve simulation efficiency using parallel processing or batched operations. Boundary Control: Dynamically adjust particle emission regions to minimize computational waste through adaptive boundary algorithms. Anisotropy: Introduce directional preferences to simulate physical conditions like gravity effects by modifying random walk probabilities (e.g., downward bias).

This simulation not only visually demonstrates self-organizing fractal growth patterns but also applies to materials science and biofilm formation research. By adjusting adhesion probabilities or diffusion rules through parameterized functions, researchers can systematically explore how different parameters influence aggregation morphology.