Diffusion-Limited Aggregation (DLA)

The Diffusion-Limited Aggregation (DLA) model is a classical algorithm for simulating fractal structure growth, proposed by Witten and Sander in 1981. The core concept involves constructing complex fractal aggregates through gradually accumulating particles performing random walks. The fundamental procedure can be broken down into three key steps, which can be implemented in code using a grid-based approach with particle tracking: First, place a seed particle at the center of the simulation area as the initial aggregation nucleus, typically represented as a fixed coordinate in a 2D array. Next, release new particles randomly at the boundary region far from the seed, implementing their irregular Brownian motion (random walk) through step-by-step position updates using random direction vectors. When a moving particle contacts the existing aggregate (detected through neighborhood checks in the grid system), it stops moving and becomes part of the cluster. Repeating this process eventually forms dendritic structures with fractal characteristics. The crucial feature of the DLA model lies in its diffusion-limited growth process, where newly added particles' movement is entirely determined by random walks, with immediate attachment upon contact. This mechanism results in clusters exhibiting typical fractal properties like self-similarity, openness, and non-integer dimensions. The model finds applications not only in studying aggregation phenomena in physical chemistry but also in interdisciplinary research areas such as electrodeposition and bacterial colony growth. By adjusting parameters in the simulation code - such as particle release positions, step sizes for movement, or introducing anisotropic conditions through directional bias in random walks - different morphological DLA structures can be generated. This provides an important theoretical framework for understanding the formation of complex patterns in natural systems, with implementation typically involving Monte Carlo methods and neighbor detection algorithms.