Diffusion-limited aggregation (DLA) is the process whereby particles undergoing a
random walkA random walk, sometimes denoted RW, is a mathematical formalisation of a trajectory that consists of taking successive random steps. The results of random walk analysis have been applied to computer science, physics, ecology, economics, and a number of other fields as a fundamental model for...
due to
Brownian motionBrownian motion is the seemingly random movement of particles suspended in a fluid or the mathematical model used to describe such random movements, often called a particle theory....
cluster together to form aggregates of such particles. This theory, proposed by Witten and Sander in 1981 , is applicable to aggregation in any system where
diffusionMolecular diffusion, often called simply diffusion, is a net transport of molecules from a region of higher concentration to one of lower concentration by random molecular motion. The result of diffusion is a gradual mixing of material...
is the primary means of
transportIn physics, chemistry, biology and engineering, a transport phenomenon is any of various mechanisms by which particles or quantities move from one place to another. The laws which govern transport connect a flux with a "motive force". Three common examples of transport phenomena are diffusion,...
in the system. DLA can be observed in many systems such as electrodeposition, Hele-Shaw flow, mineral deposits, and dielectric breakdown.
The clusters formed in DLA processes are referred to as
Brownian treeA Brownian tree, whose name is derived from Robert Brown via Brownian motion, is a form of computer art that was briefly popular in the 1990s, when home computers started to have sufficient power to simulate Brownian motion...
s.
Diffusion-limited aggregation (DLA) is the process whereby particles undergoing a
random walkA random walk, sometimes denoted RW, is a mathematical formalisation of a trajectory that consists of taking successive random steps. The results of random walk analysis have been applied to computer science, physics, ecology, economics, and a number of other fields as a fundamental model for...
due to
Brownian motionBrownian motion is the seemingly random movement of particles suspended in a fluid or the mathematical model used to describe such random movements, often called a particle theory....
cluster together to form aggregates of such particles. This theory, proposed by Witten and Sander in 1981 , is applicable to aggregation in any system where
diffusionMolecular diffusion, often called simply diffusion, is a net transport of molecules from a region of higher concentration to one of lower concentration by random molecular motion. The result of diffusion is a gradual mixing of material...
is the primary means of
transportIn physics, chemistry, biology and engineering, a transport phenomenon is any of various mechanisms by which particles or quantities move from one place to another. The laws which govern transport connect a flux with a "motive force". Three common examples of transport phenomena are diffusion,...
in the system. DLA can be observed in many systems such as electrodeposition, Hele-Shaw flow, mineral deposits, and dielectric breakdown.
The clusters formed in DLA processes are referred to as
Brownian treeA Brownian tree, whose name is derived from Robert Brown via Brownian motion, is a form of computer art that was briefly popular in the 1990s, when home computers started to have sufficient power to simulate Brownian motion...
s. These clusters are an example of a
fractalA fractal is "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
. In 2-D these fractals exhibit a dimension of approximately 1.71 for free particles that are unrestricted by a lattice, however computer simulation of DLA on a lattice will change the
fractal dimensionIn fractal geometry, the fractal dimension, D, is a statistical quantity that gives an indication of how completely a fractal appears to fill space, as one zooms down to finer and finer scales. There are many specific definitions of fractal dimension. The most important theoretical fractal...
slightly for a DLA in the same embedding dimension. Some variations are also observed depending on the geometry of the growth, whether it be from a single point radially outward or from a plane or line for example.
Computer simulation of DLA is one of the primary means of studying this model. Several methods are available to accomplish this. Simulations can be done on a lattice of any desired geometry of embedding dimension, in fact this has been done in up to 8 dimensions , or the simulation can be done more along the lines of a standard molecular dynamics simulation where a particle is allowed to freely random walk until it gets within a certain critical range at which time it is pulled onto the cluster. Of critical importance is that the number of particles undergoing Brownian motion in the system is kept very low so that only the diffusive nature of the system is present.
Artwork based on diffusion-limited aggregation
The intricate and organic forms that can be generated with diffusion-limited aggregation algorithms have been explored by artists. Andy Lomas'
'Aggregation' series features ray-traced images and animations of aggregates containing more than 50 million particles. Lomas has presented the series at more than nine galleries and as a sketch at
SIGGRAPHSIGGRAPH is the name of the annual conference on computer graphics convened by the ACM SIGGRAPH organization. The first SIGGRAPH conference was in 1974. The conference is attended by tens of thousands of computer professionals...
2005, in which he describes his technique.
External links
Diffusion-Limited Aggregation: A Model for Pattern Formation
An elegant Java applet demonstration of DLA from Hong Kong University