Regular Fractals Research Articles

Frustration and correlations in fractals

A technique for obtaining exact thermodynamic and correlation functions of regular fractals is illustrated by a development for the Ising system on the triangular Sierpińsky gasket. A generalised hyperscaling statement is obtained. The technique is then applied to the fully frustrated antiferromagnetic case, where the exact ground state entropy per spin is shown to be S(0)=0.4930…. This result shows that the introduction of (regularly disposed) holes into the triangular lattice does not relieve frustration, as is commonly supposed. Relations completely determining the correlation functions are obtained. That for the zero field pair correlation function of vertex spins at separation r is G−1(2r)=G−2(r)−G−+1, together with a relation which sets the correlation- length scale. Exact asymptotic expressions are derived for ferromagnetic correlations in both zero and non-zero field. The zero-field, zero-temperature correlation length of the fully frustrated system is shown to be ξ=0.79, representing extremely rapid decay of correlations.

Critical dynamics of the kinetic Ising model on regular fractals.

By the exact time-dependent renormalization-group method, the critical dynamics of the kinetic Ising model on a family of regular fractal models is studied for the diffusion-limited-aggregation (DLA) model cluster. The scaling law of the dynamics exponent, Z=${D}_{f}$+2/\ensuremath{\nu}, is found and it is shown that \ensuremath{\nu} (the static correlation exponent) is independent of the choice of regular DLA clusters. The dynamics exponent Z=${D}_{f}$+2/\ensuremath{\nu} is therefore proposed for a random DLA cluster.

Anomalous diffusion on regular and random models for diffusion-limited aggregation

Families of regular fractal models for Witten-Sander (DLA) clusters in d dimensions are proposed resembling the computer generated aggregates. The authors evaluate random walk dimensionalities exactly using resistance scaling ideas. The applicability of the conflicting conjectures of Alexander-Orbach (1982) (AO), dw=3df/2, and Aharony-Stauffer (1984) (AS), dw=df+1, to DLA is then examined. One finds that the dendritic nature of the fractals presented ensures that AS is satisfied exactly in all dimensions. An RSRG fugacity transformation method is applied to the simplest of the regular fractals in d=2. The result for dw is compared with the exact resistance scaling result to obtain an estimate of the error introduced by the RSRG. The RG method is then generalised to treat the random DLA problem. The resulting estimate for dw is in fortuitously good agreement with AS, the corresponding Monte Carlo data and the results above for the regular fractal models.

Calculation of the small-angle x-ray and neutron scattering from nonrandom (regular) fractals.

The intensity of the small-angle x-ray or neutron scattering has been calculated for two nonrandom (regular) fractals: the Menger sponge and a related fractal, called the fractal jack (a form similar to the metal six-pointed object used in the American children's game). The scatterers are assumed to be systems of independently scattering, randomly oriented identical nonrandom fractals constructed from a material with uniform density. The scattered intensity I(q) can be expressed as a function of qa, where q=4\ensuremath{\pi}${\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}1}$sin(theta/2); \ensuremath{\lambda} is the scattered wavelength; theta is the scattering angle, and a is the edge of the cube which is the starting approximant to the fractal. The calculations show that I(q) is a monotonically decreasing function on which maxima and minima are superimposed. For large qa the monotonic decay is proportional to ${q}^{\mathrm{\ensuremath{-}}D}$, where D is the fractal dimension. The first maximum for q>0 is a single peak located at q=${q}_{1}$. Groups of maxima are found at q${=3}^{k}$${q}_{1}$, where k is a positive integer greater than 1. The number of maxima within a group becomes greater as k increases. Numerical calculations of I(q) provide no evidence that the maxima and minima are damped and die out as q becomes larger. Thus I(q) for the two nonrandom fractals does not appear to approach the simple power-law scattering proportional to ${q}^{\mathrm{\ensuremath{-}}D}$ which is characteristic of the small-angle scattering from random fractals. The techniques developed to calculate I(q) for the Menger sponge and the fractal jack can also be employed to find the small-angle scattering from other nonrandom (regular) fractals.