Fractal Dimension & DLA Clusters

• In a system with a maximum of 94 lattice constants, the fractal dimension is estimated to be approximately 1.77, based on the calculation D = ln(3150) / ln(94).
• Simulations reveal that incorporating long jumps for distances greater than Rd does not significantly increase processing time when the annihilation circle's radius is enlarged.

Exercises

• When determining fractal dimension, the incomplete nature of the generated cluster means arms are still forming and particles are reaching interior regions.
• Stopping the simulation at a certain maximum radius neglects these particles.
• Therefore a denser actual cluster increases the fractal dimension.
• Determine the mass of the cluster by excluding particles beyond a distance r from the origin.
• Represent the mass as a function of r on a log-log plot to interpret the behavior.
• Varying the original Diffusion-Limited Aggregation (DLA) model may change the form and fractal dimension of the cluster.
• Introduce a parameter pstick into the dla.c program, representing the probability that a particle sticks to the cluster upon arrival at an adjacent site.
• If a particle does not initially stick to the cluster, it diffuses until hitting the cluster again, at which point there is a probability pstick of sticking.
• During diffusion, particles are prohibited from jumping to already occupied sites.

Percolation Basics

• Ideal crystals consist of atoms in a regular lattice arrangement.
• Solid-state physicists study non-ideal structures and materials with irregular structure.
• The percolation model describes porous materials through which particles can diffuse if continuous paths exist through the pores.
• The geometry of pores is determined by percolation theory, useful for teaching and contains physics related to phase transitions, critical phenomena, universality, and self-similar fractals.
• The percolation model can be simulated.

Physics Principles

• An alloy has two kinds of atoms, A and B, with the concentration of A atoms denoted by p.
• Both types of atoms are distributed randomly and form a regular crystal lattice.
• A atoms are magnetic and have short-range magnetic interaction, which is effective only if two A atoms are located next to each other.
• Without coupling with nonmagnetic B atoms, magnetic order forms if there is an infinite network of connected A atoms.
• Phase transitions can occur in infinite systems and a critical concentration of A atoms can result in a phase transition in finite systems.
• The size of interconnected A atoms, called a cluster, significantly impacts this process.
• Increasing the average size of clusters of A atoms increases until a cluster extends through the entire crystal at a threshold value pc, the percolation threshold or critical concentration.
• Above pc, the infinite cluster coexists with many small ones, but at p=1 all sites are occupied by A atoms so there is only one infinite cluster.
• Infinite crystals have a well-defined percolation threshold pc, and pc depends on the lattice type and coupling range, but not on the realization if crystal size is infinite.
• Finite crystals also have probability of a cluster connecting two opposite sides, depending on p.

Properties and Relationships

• Several properties of clusters include value of pc for different lattice structures, how the average amount of smaller clusters increases when p increases up to the threshold pc, density of cluster increasing above pc, cluster sizes, and possible infinite cluster structure at pc.
• Governing laws in the vicinity of the percolation threshold pc are particularly interesting.
• The average size R(s) of a cluster consisting of s particles is defined by R²(s) = (1 / s(s-1)) * Σ(ri - rj)², where i and j are particles of the cluster and ri is the position of the ith atom on the lattice.
• The average size ξ of the finite clusters is defined as ξ = √(R²(s))<∞.
• The angle brackets (...) < ∞ designates the average over all finite clusters.
• At the percolation threshold, ξ diverges, and close to pc, ξ ~ |p - pc|⁻ν.
• The exponent v is universal across all percolation models, and depends only on spatial dimension.
• In the planar case, v = 4/3 is known for different lattice types, ranges of couplings, and even the corresponding problem without a lattice, and for real two-dimensional alloys.
• For three dimensions, v≈ 0.88.
• The probability P(p) that an arbitrarily chosen A atom is part of the infinite cluster is zero for p < pc and increases continuously from P(pc) = 0 to P(1) = 1.
• Close to pc the exponent β describes a power law: P(p) ~ (p - pc)β (p ≥ pc).
• In two dimensions, β = 5/36 ≈ 0.14, meaning P(p) rises steeply near pc.
• The density of the infinite cluster vanishes at the percolation threshold P(pc) = 0, but still exists as a fractal.
• Considering a square section with the lattice sides of length L, the number of particles in this section which belong to the infinite cluster is M(L), where (M(L)) ∝ LD at pc.
• The angle brackets (...) designate the average over different sections; the mass of the cluster increases as a power of the length.

Fractal Dimension and Critical Exponents

• In two dimensions, the fractal dimension D = 91/48 ≈ 1.89, meaning the critical percolation cluster is more compact than the aggregate from the previous section, which was generated by diffusion.
• Universal critical exponents include v, β, and D, and many more properties are described by power laws with additional exponents near pc.
• Similarities are not independent but connected by scaling laws.
• Let M(p, L) be the average number of particles in a cluster that traverses a square with sides of length L.
• One can determine the number M as a function of p for different L values, and close to the critical point, the set of curves after suitable scaling can be represented by a single universal function f.
• The scales for M and L can be expressed by powers of either p - pc or the variable ξ, where L is measured in units of ξ, and M in units of a power of ξ, e.g., ξ^x.
• M(p, L) ~ ξ^x * f(L/ξ), where f is a function not known initially and x is a critical exponent.
• If L = kξ, x = D follows from (M(L)) ∝ LD, meaning L is measured in units of ξ and M in units of ξD, hence the name "scaling theory."
• From L = kξ, M(p, L) / Ld ~ ξD-d * f(k), where Ld is the number of lattice sites in the d-dimensional cube with edges of length L.
• M / Ld is the probability that a lattice site belongs to the percolation cluster, or pP(p), and for large values of L, M(p, L) / Ld ~ pP(p) ~ (p - pc)β.
• Equations (5.19), (5.23), and (5.24) lead to (p - pc)β ~ (p - pc)(D - d)v.
• One can conclude that β = (d – D) v and the three critical exponents are coupled by a scaling law. The knowledge of two exponents is sufficient for calculating all others.
• Defined only for infinite lattices, with the computer we can only fill finite lattices, so there is no phase transition, no sharply defined percolation thresholds, and no divergences.
• The theory of finite size scaling answers the question of how to deduce the universal critical laws of the infinite lattice from the properties of a finite system.
• Based on the scaling laws, theory of finite size scaling makes statements about dependence of the singularities on the lattice size L.
• Using the behavior of correlation length, M(p, L) |p - pc|-D ~ f((p - pc) L1/v), where f(x) =tilde f(x1/2).
• Concentration p and system size L are related to one another close to the critical point p=pc, L=∞.
• Fractal dimension D is already obtainable, and for L << ξ the finite system looks like the infinite one.
• Therefore, M(pc, L) ~ LD, showing one can numerically calculate the fractal dimension D from the increase of the number M(pc, L) of particles in the percolation cluster as a function of L. But how does one determine pc and the exponent v?
• A scaling law analogous to π(p, L) is established to find a cluster connecting two opposite faces, which therefore percolates, in a sample of size L.
• Obviously for small values, π (p, ∞) = { 0 for p < pc, and 1 for p > pc.
• For finite values of L, the step function is rounded off in the vicinity of Pc.
• Since π is constant, the scaling exponent is zero.
• Therfore (p,L) = g ((p-Pc) (L1/2) and a function g

Algorithm

• On a lattice, the percolation structure is easily generated numerically, where one
  • Chooses a square lattice with sites (i, j), i = 0, . . ., L − 1 and j = 0, ..., L − 1
  • Uses uniformly distributed random numbers r∈ [0,1].
• The algorithm is:
  • Loop through all (i, j)
  • Draw a random number r
  • Plot a point at the position (i, j) if r < p.

Numerical Generation

• The task of quantitatively answering suitable questions to then extract mathematical regularities shows that analyzing the percolation structure cannot be programmed easily.
• Answering whether the structure percolates can't be done easily.
• An algorithm to identify clusters of connected particles uses a naive method of drawing circles and connect all marked sites
• Developed in 1976 by Hoshen and Kopelman, A fast algorithm loops through the lattice and assigns cluster numbers
• Parts of the clusters have different numbers and the conflicts are resolved through bookkeeping
• One can generate clusters directly by a growth process created in 1976 by Leath
• The procedure:
  • Start by occupying the center of an empty lattice and then defining all adjacent sites as either occupied or free, with respective probabilities p and 1 − p.
  • This process is then iterated, converting for each occupied site the adjacent undefined sites to either free or occupied ones.
  • When no undefined sites adjacent to the cluster are left, the growth stops; if the resulting cluster connects two opposite sides of the lattice, it is a percolation cluster.
• Since each particle generates a statistical weight, the growth process creates clusters with the previous algorithm.
• Growth algorithm has a few steps:
• Label all sites of a square lattice with sides of length L as "undefined" and occupy the center.
• Pick an undefined site adjacent to the cluster.
• Draw a uniformly distributed random number r ∈ [0,1] and occupy the site if r < p. Otherwise, label the site as "vacant."
• Iterate 2 and 3 until no undefined sites adjacent to the cluster are left.

Growth Termination

• To make sure, the process terminates by labelling the sites around the lattice boundary as "vacant."
• The list needs to be updated during the growth process
• Simulating large lattices displays clusters and lists 3 cases with the numbers 0, 1, and 2: #define NOTDEFINED 0, #define OCCUPIED 1, #define VACANT 2.
• The lattice is defined by char array (L) (L) to contain the positions
• A list to will hold positions (i, j) by: struct {int x; int y;} list[PD];
• 5. The loop, item 4 is: While (! done ) {event(); for(k=0; list[countPD 1==NOTDEFINED){count++;list[count1==NOTDEFINED){count++;list[countincrement and all undefined boundaries can be generated through count.

Results

• Populating a square lattice produces mathematical regularities that only displays noise
• Growth algorithms can determine clusters on length scales and generates a fractal object.
• Growth process can be stopped prematurely
• Occurs frequently whenever P is less than PE
• Occurs almost exclusively whenever P is greater than PE