The term "Diffusion Limited Aggregation" (DLA) [1] refers to a simple
growth algorithm in which individual particles are added
to a growing cluster through a diffusion-like process.
Starting from any suitable cluster seed fixed in
-dimensional space, a new particle is launched
at a random position far away from the cluster and undergoes Brownian motion.
When the randomly walking particle contacts the cluster,
it is stopped and incorporated to the cluster at its contacting position.
This process of launching a random walker and adding it to the cluster
on its first contact is repeated until the cluster
has reached a desired number of particles (mass), 
.
The DLA model generates strongly ramified clusters with a complex geometry.
Figure 1 displays a DLA cluster embedded in
two-dimensional space at different stages of growth.
The cluster consists of identical discs of diameter 
.
The cluster seed was a single disc.
Due to the non-deterministic nature of Brownian motion, each DLA cluster represents the individual outcome of a statistical process. Although the process itself is simple, its boundary conditions are not. The surface of the cluster constitutes the absorbing boundary for the diffusing particles, and changes while the cluster grows.
More specifically, DLA is a growth process
in which the growth rate of the cluster boundary is determined by the gradient
of a scalar field 
that fulfills the Laplace equation.
The boundary conditions are that 
on the cluster boundary, and 
at infinity.
The random walkers provide unbiased samples of the Laplace field ,
and the probability of contacting the cluster boundary at a given position
is proportional to 
.
In summary, DLA is a model for Laplacian growth in the presence of
statistical noise.
Figure 1:
A two-dimensional, off-lattice DLA cluster at six different stages of growth.
The cluster consists of identical discs of diameter , and is shown when
it contains 
, 
, 
, 
, 
, and
particles, respectively. The cluster seed was one single
disc.
According to the finite printer resolution, only a fraction
of all particles are shown for 
in order not to change the
visual appearance of the cluster.
Consequently, the DLA algorithm is a relevant model for many Laplacian growth phenomena in nature, including fluid-fluid displacement in Hele-Shaw cells [2] and in porous media [4][3], bacterial growth [5], electrodeposition [7][6], thin film deposits [8], dissolution of porous materials [10][9], and dielectric breakdown [11]. A more extensive discussion of DLA and its relation to natural growth processes may be found in Refs. [12] and [13].
DLA clusters are known to be fractals. To illustrate this we show in
Figure 2 the result of a measurement
of the average number of particles, 
, within a distance 
of
a given particle of a DLA cluster.
To obtain , 
particles 
, 
were randomly
chosen from the first particles of the cluster shown in Fig. 1.
When the cluster consisted of 
particles,
the number of particles within distance to particle , 
, was
counted for each of the selected particles.
Finally, was calculated as the average value of .
For 
, the relation between
and is consistent with the scaling law
Consequently, the particle density,
, is not
constant, but decreases when increasing the
length scale on which the density is measured. This scale dependence
of the mass density is apparent in Fig. 1, where larger clusters
clearly are more sparse than smaller ones.
The scaling exponent 
can be
interpreted as an effective fractal dimensionality of the cluster [14].
Figure 2:
Log-log plot of average mass within distance
to a given particle as a function of for two-dimensional,
off-lattice DLA.
Filled circles are experimental data, obtained
from the DLA cluster shown in Fig. 1
as explained in the text.
The line corresponds to the fit
,
indicating a fractal dimensionality of 
of the cluster.
From Fig. 2 a fractal dimensionality 
may be read off, in reasonable agreement with the accepted value
of 
for two-dimensional, off-lattice DLA
[15], obtained from the dependence of the cluster radii of
gyration on .
The two crossover length that define the regime in which
the scaling law holds are the particle diameter, , and the
radius of the complete cluster of particles,
, respectively.