Ancillary CPS Research Projects
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Diffusion-limited growth processes have attracted considerable recent attention because of the myriad of applications to the morphology of growing interfaces and because of fundamental issues associated with disorderly growth processes. The diffusion-limited aggregation (DLA) model, is a particularly attractive realization of such growth processes. DLA growth is completely characterized by assigning to each perimeter site i the number Pi, the probability that site i is the next to grow. Evidence suggests that the numbers Pi, form a multifractal set: that which cannot be characterized by a single exponent, but rather require an infinite hierarchy of exponents. Since the hottest tips of a DLA aggregate grow much faster than the deep fjords, the scaling of Pi, with respect to the aggregate size depends on the particular site. Analogies to percolation systems are made by assigning different sites in the DLA model to particular kinds of percolation bonds. Additionally, conditions for viscous-fingering experiments yield patterns that obey the same scale symmetry as DLA have been found, leading to new questions about the physics underlying the viscous-fingering phenomenon.
The physical properties of disordered media pose a wide range of fascinating questions and open problems, with both theoretical and experimental ramifications. Our recent work has focused on theoretical studies of percolation models of disordered materials. We have developed new tools, such as scaling and the renormalization group, and developed efficient (parallel) large-scale numerical simulation techniques. Recently, we have found that there is an underlying hierarchical structure which governs many aspects of disordered media. The implications of this hierarchical organization are far-reaching, as many well-established ideas from critical phenomena need to be reformulated. Instead of a single scale, or ``fractal dimension'' describing a disordered system, there can be a multiplicity of scales, or ``multifractality''. The insights gained from these advances have ramifications for problems such as fluid flow and chemical reactions in porous media, and transport processes in disordered materials.
One important application is the electrical conductance of disordered materials. This quantity is of relevance to a range of phenomena, such as the anomalous behavior of the elastic modulus and shear viscosity near the sol-gel transition. We have exploited the Einstein relation to connect the electrical conductance to properties of random walkers in the same disordered medium. By this equivalence, we can compute the electrical conductance of two-component composite media. Through the isomorphism between electrical conductance and viscoelastic properties of disordered networks, the mechanical behavior near the sol-get transition has been investigated.
At a more geometric level, ``exact'' closed-form solutions to descriptions of the structure of the incipient infinite cluster in percolation, and the infinite network that form just above the percolation threshold have been achieved. A hybrid model for this structure was proposed that incorporates both singly-connected and multiply-connected bonds. Fundamental studies of this ``links and blobs'' model, using both exact calculations and computer simulations, yielded a rigorous result for the geometry of the singly-connected bonds.
Recently, it was discovered by researchers in our group that non-coding DNA sequences exhibit scale-invariant long-range correlations quantitatively measured by a power law decay. The exponent characterizing the power law decay of the correlations is well-defined for infinite sequences. However, for DNA sequences the accuracy of the analysis is limited by the length of the available nucleotide chains (i.e., there are only a few samples of published nucleotide sequences with length >100,000 base pairs). The goal of this project is to investigate the possible broad-based implications of this discovery.
See our Main Research Projects Index